Talks of D2 Equations I

Stefano Bianchini, SISSA, Italy

DIFFERENTIABILITY OF THE FLOW FOR BV VECTOR FIELDS

Date: 2019-09-16 (Monday); Time: 14:15-14:55; Location: building B-8, room 0.10a.

Abstract

We show that the Regular Lagrangian Flow $$X(t,y)$$ generated by nearly incompressible BV vector fields admits a derivative $$\nabla X(t,y)$$ in the sense of measure. This matrix satisfies the ODE $\frac{d}{dt} \nabla X(t,y) = (D\mathbf{b}(t))_y \nabla X(t-,y)$ where $$(D\mathbf{b})_y$$ is the disintegration of the measure $$\int D\mathbf{b}(t) dt$$ w.r.t. the trajectories $$X(t,y)$$.

Yoshikazu Giga, University of Tokyo, Japan

ON THE LARGE TIME BEHAVIOR OF SOLUTIONS TO BIRTH AND SPREAD TYPE EQUATIONS

Joint work with Hiroyoshi Mitake, Takeshi Ohtsuka, and Hung V. Tran

Date: 2019-09-16 (Monday); Time: 15:00-15:40; Location: building B-8, room 0.10a.

Abstract

We consider a level-set eikonal-curvature flow equation with an external force. Such a problem is considered as a model to describe an evolution of height of crystal surface by two-dimensional nucleation or possibly some class of growths by screw dislocations. For applications, it is important to estimate growth rate. Without an external source term the solution only spreads horizontally and does not grow vertically so the source term plays a key role for the growth.

Although the large time behavior of parabolic equations are well studied, the equations we study are degenerate parabolic equations where no diffusion effect exists in the normal to each level-set of a solution. Thus, very little is known even for growth rate. Our goal is to describe our recent progress on such type of problems. Ealier results are presented in the paper by H. Mitake, H.V. Tran and the lecturer published in SIAM Math. Anal. in 2016. A review paper is published in Proc. Int. Cong. of Math. in 2018.

In this talk, we first show the existence of asymptotic speed called growth rate. We also discuss asymptotic profile as well as estimates on growth rate.

David Jerison, Massachusetts Institute of Technology, USA

THE TWO HYPERPLANE CONJECTURE

Date: 2019-09-18 (Wednesday); Time: 10:40-11:20; Location: building B-8, room 0.10a.

Abstract

I will introduce a conjecture that I call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. Emanuel Milman has shown that in its strongest, dimension-independent form, my conjecture implies the Hyperplane Conjecture of Kannan, Lovász and Simonovits in theoretical computer science, which says that the area of such an isoperimetric surface is comparable, by an absolute constant independent the convex body and its dimension, to the area of some hyperplane dividing the convex body in half. Their conjecture is closely related to several famous unsolved problems in high dimensional convex geometry. But unlike the hyperplane conjecture, the two-hyperplane conjecture has significance even in low dimensions.

I will relate the conjecture to qualitative and quantitative connectivity properties and regularity of area-minimizing surfaces, free boundaries and level sets of eigenfunctions, and report on work in progress with Guy David. The main theme of the talk is that the level sets of least energy solutions to scalar variational problems should be as simple as possible, but no simpler.

Sergiu Klainerman, Princeton University, USA

ON THE NONLINEAR STABILITY OF BLACK HOLES

Date: 2019-09-17 (Tuesday); Time: 14:15-14:55; Location: building B-8, room 0.10a.

Abstract

Black holes are precise mathematical solutions of the Einstein field equations mainly represented by the famous two parameter Kerr family including, as a particular case, the Schwarzschild solution. To correspond to physical reality, i.e. to be more than mathematical artifacts, these solutions have to be stable under small perturbations. While there is today no doubt concerning the physical reality of black holes, based both on observational data and numerical simulations, an actual proof of stability remains a fundamental challenge of Mathematical and Geometric Analysis.

In my talk I will formulate the precise mathematical problem of the nonlinear stability of the Kerr family and describe the main results known so far. In the second part of the talk I will describe my recent result with J. Szeftel "Global Nonlinear Stability of Schwarzschild Spacetime under Polarized perturbations" - arXiv:1711.07597. The result establishes the full nonlinear stability of Schwarzschild spacetime under axially symmetric, polarized perturbations, i.e. stability of solutions of the Einstein vacuum equations for asymptotically flat 1 + 3 dimensional Lorentzian metrics which admit a hyper-surface orthogonal space-like Killing vector-field with closed orbits. While building on the remarkable advances made in last 15 years on establishing quantitative linear stability, the paper introduces a series of new ideas among which we emphasize the $$\textit{general covariant modulation}$$ (GCM) procedure which allows us to construct, dynamically, the center of mass frame of the final state. The mass of the final state itself is tracked using the well known Hawking mass relative to a well adapted foliation itself connected to the center of mass frame. Our work here is the first to prove the nonlinear stability of Schwarzschild in a restricted class of nontrivial perturbations. To a large extent, the restriction to this class of perturbations is only needed to ensure that the final state of evolution is another Schwarzschild space.

Aleksandr Logunov, Princeton University, USA

ZERO SETS OF LAPLACE EIGENFUCNTIONS

Date: 2019-09-20 (Friday); Time: 15:00-15:40; Location: building B-8, room 0.10a.

Abstract

We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Laplace eigenfunctions on two dimensional sphere are restrictions of homogeneous harmonic polynomials of three variables onto the sphere. Zero sets of such functions are unions of smooth curves with equiangular intersections. Topology of the zero set could be quite complicated, but the total length of the zero set of any spherical harmonic of degree $$n$$ is comparable to $$n$$.

References

1. A. Logunov, E. Malinnikova, Review of Yau’s conjecture on zero sets of Laplace eigenfunctions, to appear in Current Developments in Mathematics.
2. A. Logunov, E. Malinnikova, Quantitative propagation of smallness for solutions of elliptic equations, Proceedings of the International Congress of Mathematicians–Rio de Janeiro, 2357–2378, 2018.
3. A. Logunov, E. Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three, 50 Years with Hardy Spaces, A Tribute to Victor Havin, 333–344, 2018.
4. A. Logunov, E. Malinnikova, Lecture notes on quantitative unique continuation for solutions of second order elliptic equations, to appear in IAS/Park City Mathematics series, AMS.
5. A. Logunov, 2018, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Annals of Mathematics, 187(1), 221–239.
6. A. Logunov, 2018, Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture, Annals of Mathematics, 187(1), 241–262.

Felix Otto, Max Planck Institute for Mathematics in the Sciences, Germany

EFFECTIVE BEHAVIOR OF RANDOM MEDIA

Date: 2019-09-19 (Thursday); Time: 15:00-15:40; Location: building B-8, room 0.10a.

Abstract

In engineering applications, heterogeneous media are often described in statistical terms. This partial knowledge is sufficient to determine the effective, i.e. large-scale behavior. This effective behavior may be inferred from the Representative Volume Element (RVE) method. I report on last years' progress on the quantitative understanding of what is called stochastic homogenization of elliptic partial differential equations: optimal error estimates of the RVE method, leading-order characterization of fluctuations, effective multipole expansions. Methods connect to elliptic regularity theory and to concentration of measure arguments.

Endre Süli, University of Oxford, UK

IMPLICITLY CONSTITUTED FLUID FLOW MODELS: ANALYSIS AND APPROXIMATION

Date: 2019-09-19 (Thursday); Time: 14:15-14:55; Location: building B-8, room 0.10a.

Abstract

Classical models describing the motion of Newtonian fluids, such as water, rely on the assumption that the Cauchy stress is a linear function of the symmetric part of the velocity gradient of the fluid. This assumption leads to the Navier-Stokes equations. It is known however that the framework of classical continuum mechanics, built upon a simple linear constitutive equation for the Cauchy stress, is too narrow to describe inelastic behavior of solid-like materials or viscoelastic properties of materials. Our starting point in this work is therefore a generalization of the classical framework of continuum mechanics, called the implicit constitutive theory, which was proposed recently in a series of papers by Rajagopal. The underlying principle of the implicit constitutive theory in the context of viscous flows is the following: instead of demanding that the Cauchy stress is an explicit (and, in particular, linear) function of the symmetric part of the velocity gradient, one may allow a nonlinear, implicit and not necessarily continuous relationship between these quantities. The resulting general theory therefore admits non-Newtonian fluid flow models with implicit and possibly discontinuous power-law-like rheology.

We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi-valued graph. Using a variety of weak compactness techniques, we show that when the graph of the stress-strain relationship is maximal monotone a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the discretization parameter, measuring the granularity of the finite element triangulation, tends to zero. When the graph is nonmonotone, a subsequence of the sequence of finite element solutions is shown to converge to a gradient Young-measure solution of the problem. A key new technical tool in the analysis is a finite element counterpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions. The talk is based on a series of papers with Miroslav Bulíček and Josef Málek (Prague), Miles Caddick (Oxford), Lars Diening (Bielefeld), Christian Kreuzer (Dortmund), and ongoing research with Alexei Gazca-Orozco (Oxford) and Tabea Tscherpel (Aachen).

References

1. M. Bulíček, J. Málek, E. Süli, Existence of global weak solutions to implicitly constituted kinetic models of incompressible dilute homogeneous polymers, Commun. Part. Diff. Eq. 38(5) (2013), 882-924.
2. M. Caddick, E. Süli, Numerical approximation of Young-measure solutions to parabolic systems of forward-backward type, Appl. Anal. Discr. Math., Accepted for publication on 17/02/2019.
3. L. Diening, C. Kreuzer, E. Süli, Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, SIAM J. Numer. Anal. 51(2) (2013), 984-1015.
4. C. Kreuzer, E. Süli, Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, ESAIM: Math. Model. Numer. Anal. 50(5) (2016), 1333-1369.
5. E. Süli, T. Tscherpel, Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids, IMA J. of Numer. Anal., https://doi.org/10.1093/imanum/dry097.

András Vasy, Stanford University, USA

GLOBAL ANALYSIS VIA MICROLOCAL TOOLS: FREDHOLM THEORY IN NON-ELLIPTIC SETTINGS

Joint work with Peter Hintz

Date: 2019-09-17 (Tuesday); Time: 15:00-15:40; Location: building B-8, room 0.10a.

Abstract

In this lecture I will describe a microlocal framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, introduced in [8] and extended in [6] and various other works. Examples in which such a framework (or a similar framework) has recently been useful include wave propagation on black hole spacetimes, which is the key analytic ingredient for showing the stability of black holes [5, 4], analysis of the resolvent of the generator of the flow for dynamical systems [2], which is the key tool for the analysis of the Ruelle zeta function [1], Feynman propagators in quantum field theory [3, 9] and inverse problems, namely boundary rigidity and tensor tomography [7].

References

1. Semyon Dyatlov and Maciej Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér. (4) 49(3) (2016), 543–577.
2. Frédéric Faure and Johannes Sjöstrand, Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys., 308(2) (2011), 325–364.
3. Jesse Gell-Redman, Nick Haber and András Vasy, The Feynman propagator on perturbations of Minkowski space, Comm. Math. Phys. 342(1) (2016), 333–384.
4. Peter Hintz and András Vasy, Linear stability of slowly rotating Kerr black holes, Preprint, arXiv:1906.00860, 2019.
5. Peter Hintz and András Vasy, The global non-linear stability of the Kerr–de Sitter family of black holes, Acta mathematica, 220 (2018), 1–206.
6. Peter Hintz and András Vasy, Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes, Anal. PDE 8(8) (2015), 1807–1890.
7. Plamen Stefanov, Gunther Uhlmann and András Vasy, Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge Preprint, arXiv:1702.03638, 2017.
8. András Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math., 194(2):381–513, 2013.
9. András Vasy and Michał Wrochna, Quantum Fields from Global Propagators on Asymptotically Minkowski and Extended de Sitter Spacetimes, Ann. Henri Poincaré 19(5) (2018), 1529–1586.

Luis Vega, University of the Basque Country & BCAM, Spain

THE VORTEX FILAMENT EQUATION, THE TALBOT EFFECT, AND NON-CIRCULAR JETS

Joint work with Valeria Banica and Francisco De La Hoz

Date: 2019-09-18 (Wednesday); Time: 11:25-12:05; Location: building B-8, room 0.10a.

Abstract

We will propose the vortex filament equation as a possible toy model for turbulence, in particular because of its striking similarity to the dynamics of non-circular jets. This similarity implies the existence of some type of Talbot effect due to the interaction of non-linear waves that propagate along the filament. Another consequence of this interaction is the existence of a new class of multi-fractal sets that can be seen as a generalization of the graph of Riemann's non-differentiable function. Theoretical and numerical arguments about the transfer of energy will be also given.

Enrique Zuazua, DeustoTech, Bilbao, Basque Country & Universidad Autónoma de Madrid, Spain

POPULATION DYNAMICS AND CONTROL

Date: 2019-09-20 (Friday); Time: 14:15-14:55; Location: building B-8, room 0.10a.

Abstract

Population dynamics is an old subject. Classical models in this field are written in terms of reaction-diffusion equations.

There is a wide literature concerning the dynamical properties of those systems. But much less is known from a control perspective. And control constitutes the ultimate proof of our understanding of a process.

This lecture will be devoted to present two recent results in this area. We first consider a bistable reaction-diffusion arising in the modelling of bilingual populations and then address models involving age structuring and spatial diffusion (of Lotka-McKendrick type).

As we shall see, both aspects require of an in depth understanding of the dynamics of the systems under consideration.

We shall present sharp results on our ability to steer the dynamics of those systems to a prescribed final configuration. Some open problems and future directions of research will also be presented.

This lecture is based on recent joint work in collaboration with D. Maity, C. Pouchol, E. Trélat, M. Tucsnak and J. Zhu.

References

1. E. Trélat, J. Zhu, E. Zuazua, Allee optimal control of a system in ecology, Mathematical Models and Methods in Applied Sciences 28(9) (2018), 1665-1697.
2. C. Pouchol, E. Trélat, E. Zuazua, Phase portrait control for 1D monostable and bistable reaction-diffusion equations, Nonlinearity, to appear (hal-01800382).
3. D. Maity, M. Tucsnak, E. Zuazua, Sharp Time Null Controllability of a Population Dynamics Model with Age Structuring and Diffusion, J. Math pures et appl., to appear (hal-01764865).

Giovanni Bellettini, University of Siena & ICTP, Italy

ON THE RELAXED AREA OF THE GRAPH OF NONSMOOTH MAPS FROM THE PLANE TO THE PLANE

Joint work with Alaa Elshorbagy, Maurizio Paolini, and Riccardo Scala

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 0.10a.

Abstract

In various problems concerning area-minimizing surfaces, such as the non parametric Plateau problem, it is natural to have at hand a concept of area for graphs of nonsmooth scalar functions. In this scalar context, the correct notion to consider turns out to be the $$L^1(\Omega)$$-relaxation of the classical area functional $$f \in \mathcal C^1(\Omega) \to \int_\Omega \sqrt{1 + \vert \nabla f\vert^2}~dx$$; such a notion has been characterized and admits an integral representation in the space of functions of bounded variation in $$\Omega$$ [5, 2]. We are interested in a similar problem for vector-valued maps, more precisely for maps $$u = (u_1,u_2) \to \mathbb{R}^2$$ from a plane domain $$\Omega$$ to the plane. In this situation the classical area functional is polyconvex and, provided the map $$u$$ is sufficiently smooth, say $$u \in \mathcal C^1(\Omega; \mathbb{R}^2)$$, reads as $A(u) := \int_\Omega \sqrt{1 + \vert \nabla u_1\vert^2 + \vert \nabla u_2\vert^2 + \Big( \frac{\partial u_1}{\partial x} \frac{\partial u_2}{\partial y} - \frac{\partial u_1}{\partial y} \frac{\partial u_2}{\partial x} \Big)^2 } ~dx dy.$ Its $$L^1(\Omega;\mathbb{R}^2)$$-relaxation $$\overline A$$ is the object of our interest (see also [4], [1]), in particular evaluated at discontinuous maps, the graphs of whice are therefore nonsmooth two-dimensional surfaces of codimension two. Assuming for simplicity $$\Omega$$ to be a disk centered at the origin in the source plane, we shall present some recent results [3] concerning piecewise constant maps $$u_T : \Omega \to \{\alpha, \beta, \gamma\}$$ taking three values and having a triple junction as a jump set; here $$\alpha, \beta, \gamma$$ are three noncollinear vectors in the target plane. The appearence of solutions of certain Plateau-type problems, inducing a nonlocality phenomenon on $$\overline A(u_T)$$, will be pointed out.

References

1. E. Acerbi, G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), 329-371.
2. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Mathematical Monographs, Oxford University Press, 2000.
3. G. Bellettini, A. Elshorbagy, M. Paolini, R. Scala, On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions, submitted.
4. E. De Giorgi, On the relaxation of functionals defined on cartesian manifolds, in Developments in Partial Differential Equations and Applications in Mathematical Physics (Ferrara 1992), Plenum Press, New York 1992.
5. E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984.

Sun-Sig Byun, Seoul National University, South Korea

GLOBAL GRADIENT ESTIMATES FOR NONLINEAR ELLIPTIC PROBLEMS WITH NONSTANDARD GROWTH

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 0.10a.

Abstract

We are concerned with a quasilinear elliptic equation with non-standard growth condition over a non-smooth domain. The nonlinearity involves a non-uniformly ellipticity property and the boundary of the domain is sufficiently flat. We prove a global regularity estimate for the gradient of solutions in the frame of a generalized Sobolev space under substantially more general assumptions.

Juan Calvo, University of Granada, Spain

A BRIEF PERSPECTIVE ON TEMPERED DIFFUSION EQUATIONS

Joint work with Antonio Marigonda and Giandomenico Orlandi

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 0.10a.

Abstract

Tempered diffusion equations (also termed ''flux-saturated'' or ''flux-limited'' diffusion equations) are a class of non-linear versions of the standard diffusion equation displaying a mixture of parabolic and hyperbolic features. After reviewing the entropy solution framework and the qualitative properties of such equations, we will discuss the way in which those models fit in the variational framework developed by Jordan, Kinderlehrer and Otto to study evolution problems in terms of optimal transport theory. Then we will present some anisotropic versions of the foregoing theory and conclude with a tentative application to Developmental Biology: cytoneme networks.

Giacomo Canevari, University of Verona, Italy

THE SET OF TOPOLOGICAL SINGULARITIES OF VECTOR-VALUED MAPS

Joint work with Giandomenico Orlandi

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 0.10a.

Abstract

We introduce [1] an operator $$\mathbf{S}$$ on vector-valued maps $$u\colon\Omega\subseteq\mathbb{R}^d\to\mathbb{R}^m$$, which has the ability to capture the relevant topological information carried by $$u$$. In particular, this operator is defined on maps that take values in a closed submanifold $$\mathcal{N}$$ of the Euclidean space $$\mathbb{R}^m$$, and coincides with the distributional Jacobian in case $$\mathcal{N}$$ is a sphere. More precisely, the range of $$\mathbf{S}$$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. We use $$\mathbf{S}$$ to characterise strong limits of smooth, $$\mathcal{N}$$-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière [2]. We present applications to the study of manifold-valued maps of bounded variation, and to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with $$\mathcal{N}$$-well potentials.

References

1. G. Canevari, G. Orland, opological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces, Calc. Var. Partial Dif. 58(2) (2019), 58-72.
2. M.R. Pakzad, T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal. 13(1) (2003), 223-257.

Matteo Cozzi, University of Bath, UK

REGULARITY AND RIGIDITY RESULTS FOR NONLOCAL MINIMAL GRAPHS

Joint work with Xavier Cabré, Alberto Farina, and Luca Lombardini

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 0.10a.

Abstract

Nonlocal minimal surfaces are hypersurfaces of Euclidean space that minimize the fractional perimeter, a geometric functional introduced in 2010 by Caffarelli, Roquejoffre & Savin in connection with phase transition problems displaying long-range interactions.

In this talk, I will focus on the particular case of nonlocal minimal graphs and present some recent results obtained on their regularity and classification.

Graziano Crasta, Sapienza University of Rome, Italy

BERNOULLI FREE BOUNDARY PROBLEM FOR THE INFINITY LAPLACIAN

Joint work with Ilaria Fragalà

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 0.10a.

Abstract

We study the following interior Bernoulli free boundary problem for the infinity Laplacian:$\begin{cases}\Delta_\infty u = 0 & \text{ in } \Omega^+(u) := \{x\in\Omega:\ u(x) > 0\},\\u = 1 & \text{ on } \partial \Omega,\\ |\nabla u| = \lambda & \text{ on } F(u) := \partial\Omega^+(u) \cap \Omega\, , \end{cases}$where $$\Omega$$ is an open bounded connected domain in $$\mathbb{R}^n$$ ($$n\geq 2$$), and $$\Delta_\infty$$ is the infinity Laplace operator.

Our results cover existence, uniqueness, and characterization of viscosity solutions (for $$\lambda$$ above a threshold representing the ''infinity Bernoulli constant''), their regularity, and their relationship with the solutions to the interior Bernoulli problem for the $$p$$-Laplacian.

References

1. G. Crasta, I. Fragalà, Bernoulli free boundary problem for the infinity Laplacian, arXiv:1804.08573.
2. G. Crasta, I. Fragalà, On the supremal version of the Alt-Caffarelli minimization problem, arXiv:1811.12810.

Antonio De Rosa, New York University, USA

SOLUTIONS TO TWO CONJECTURES IN BRANCHED TRANSPORT: STABILITY AND REGULARITY OF OPTIMAL PATHS

Joint work with Maria Colombo and Andrea Marchese

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 0.10a.

Abstract

Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $$\alpha \in (0,1)$$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $$\alpha$$ is bigger than the critical threshold $$1-\frac 1n$$, where $$n$$ is the dimension of the ambient space $$\mathbb R^n$$. In [2, 3] we prove it holds for every exponent $$\alpha \in (0,1)$$ and we provide a counterexample for $$\alpha=0$$. Thus we completely solve a conjecture of Bernot, Caselles and Morel, see [1, Problem 15.1]. Moreover, the robustness of our argument allows us to prove stability for more general lower semicontinuous cost functionals, called $$H$$-masses, introduced in [6] and also studied in [5]. Furthermore, in [4] we prove the stability for the mailing problem, which was completely open in the literature, solving a second conjecture in [1, Remark 6.13]. We use the latter result to show the regularity of the optimal networks, partially answering [1, Problem 15.5].

References

1. M. Bernot, V. Caselles, and J.M. Morel, Optimal transportation networks. Models and theory, Lecture Notes in Mathematics, 1955, Springer, Berlin, 2009.
2. M. Colombo, A. De Rosa, and A. Marchese, Improved stability of optimal traffic paths, Calc. Var. Partial Differential Equations 57 (2018), 28 pp.
3. M. Colombo, A. De Rosa, and A. Marchese, On the well-posedness of branched transportation, Available on arXiv: https://arxiv.org/abs/1904.03683, (2019).
4. M. Colombo, A. De Rosa, and A. Marchese, Stability for the mailing problem, J. Math. Pures Appl. (2018).
5. M. Colombo, A. De Rosa, A. Marchese, and S. Stuvard On the lower semicontinuous envelope of functionals defined on polyhedral chains, Nonlinear Analysis 163 (2017), 201-215.
6. T. De Pauw and R. Hardt, Size minimization and approximating problems, Calc. Var. Partial Differential Equations 17 (2003), 405-442.

Franz Gmeineder, Universität Bonn, Germany

ON TRACES AND SOME FINE PROPERTIES OF FUNCTIONS OF BOUNDED $$A$$-VARIATION

Joint work with Lars Diening

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 0.10a.

Abstract

In this talk I address an interplay between trace theorems and selected fine properties of functions of bounded $$A$$-variation. In the case of $$BV$$-functions, the singular part of the gradients splits into the jump- and the Cantor part. We establish that, within the framework of functions of bounded $$A$$-variation, such a splitting requires refinement, and explain the connections to the corresponding trace theory.

Jonas Hirsch, Universität Leipzig, Germany

DIMENSIONAL ESTIMATES AND RECTIFIABILITY FOR MEASURES SATISFYING LINEAR PDE CONSTRAINTS

Joint work with Adolfo Arroyo-Rabasa, Guido De Philippis, and Filip Rindler

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 0.10a.

Abstract

We are going to present an rectifiability result for measures satisfying a linear PDE constraint. The presented rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. For instance it includes the the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD).

More precisely Let $$\mathcal{A}$$ be a $$k^{\text{th}}$$-order linear constant-coefficient PDE operator acting on $$\mathbb R^m$$-valued functions on $$\mathbb{R}^d$$ via $\mathcal{A} \varphi := \sum_{|\alpha|\le k}A_{\alpha} \partial^\alpha\varphi \text{ for all }\varphi\in C^\infty(\mathbb{R}^d;\mathbb{R}^m),$ where $$A_{\alpha}\in \mathbb{R}^{n \times m}$$ are (constant) matrices, $$\alpha=(\alpha_1,\dots,\alpha_d)\in (\mathbb N \cup \{0\})^d$$ is a multi-index and $$\partial^\alpha:=\partial_1^{\alpha_1}\ldots\partial_d^{\alpha_d}$$. We also assume that at least one $$A_\alpha$$ with $$\lvert \alpha\rvert = k$$ is non-zero. An $$\mathbb{R}^m$$-valued Radon measure $$\mu \in \mathcal M(U;\mathbb{R}^m)$$ defined on an open set $$U\subset \mathbb{R}^d$$ is said to be $$\mathcal{A}$$-free if $\mathcal{A} \mu=0 \qquad\text{in the sense of distributions on $$U$$.}\tag{1}$

Using the Lebesgue-Radon-Nikodým theorem we may define the polar of $$\mu$$ by $\frac{{\rm d}\mu}{{\rm d}|\mu|}(x):=\lim_{r\to 0} \frac{\mu(B_r(x))}{|\mu|(B_r(x))}\,.$

In the pioneering work [1], G. De Philippis and F. Rindler established a strong constraint on the direction on the polar on the singular part of an $$\mathcal{A}$$-free measure.

In this talk we are going to present a refinement of this pioniering result, the direction of the polar is further constrained on "lower dimensional parts" of the measure, [2].

As a consequence in the particular case of divergence-free tensors we are able to obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.

References

1. G. De Philippis, F. Rindler, On the structure of $$\mathcal{A}$$-free measures and applications, Ann. of Math. (2) 184(3) (2016), 1017–1039.
2. A. Arroyo-Rabasa, G. De Philippis and F. Rindler, Dimensional estimates and rectifiability for measures satisfying linear PDE constraints, Geometric and Functional Analysis 184 (2019), 1420-8970.

Emanuel Indrei, Purdue University, USA

THE GEOMETRY OF THE FREE BOUNDARY NEAR THE FIXED BOUNDARY GENERATED BY A FULLY NONLINEAR UNIFORMLY ELLIPTIC OPERATOR

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 0.10a.

Abstract

The dynamics of how the free boundary intersects the fixed boundary has been the object of study in the classical dam problem which is a mathematical model describing the filtration of water through a porous medium split into a wet and dry part. By localizing around a point at the intersection of free and fixed boundary, one is led to the following problem $\begin{cases} F(D^2 u)=\chi_\Omega & \text{in }B_{1}^{+}\\ u=0 & \text{on }B'_{1} \end{cases}$ where $$\Omega = \big(\{u \ne 0\} \cup \{\nabla u \neq 0\} \big) \cap \{x_n>0\}\subset \mathbb{R}_+^n$$, $$B'_{1}=\{x_n=0\}\cap \overline{B_1^+}$$, and $$F$$ is a convex $$C^1$$ fully nonlinear uniformly elliptic operator. This talk focuses on the regularity problem for the free boundary $$\Gamma=\partial \Omega \cap B_{1}^{+}$$.

References

1. E. Indrei, Boundary regularity and non-transversal intersection for the fully nonlinear obstacle problem, Comm. Pure Appl. Math. 72 (2019), 1459-1473.

Vesa Julin, University of Jyväskylä, Finland

CLASSICAL SOLUTION TO THE FRACTIONAL MEAN CURVATURE FLOW

Joint work with Domenico La Manna

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 0.10a.

Abstract

I will discuss about recent developments in the study of fractional mean curvature flow and introduce our recent work which is the first short time existence result of the smooth solution to the fractional mean curvature flow.

Michał Kowalczyk, University of Chile, Chile

MAXIMAL SOLUTION OF THE LIOUVILLE EQUATION IN DOUBLY CONNECTED DOMAINS

Joint work with Angela Pistoia and Giusi Vaira

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 0.10a.

Abstract

In this talk I will discuss a new existence result for the widely studied Liouville problem $$\Delta u+\lambda^2 e^u=0$$ in a bounded, two dimensional, doubly connected domain with Dirichlet boundary conditions. I will show that for a sequence of $$\lambda_n \to 0$$ this equation has solutions that blow-up in in the whole domain. Profiles of the blowing-up solutions are related to a free boundary problem which gives a solution to an optimal partition problem for the given domain. I will also describe the role of the free boundary problem in other classical equations such as the mean field model or the prescribed Gaussian curvature equation.

Alpár R. Mészáros, University of California, Los Angeles, USA

SOBOLEV ESTIMATES FOR FIRST ORDER MEAN FIELD GAMES AND PLANNING PROBLEMS

Joint work with Jameson Graber, Francisco Silva, and Daniela Tonon

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 0.10a.

Abstract

In this talk we discuss Sobolev estimates for weak solutions of first order variational Mean Field Game systems (in the sense of Lasry-Lions) with coupling terms that are local functions of the density variable. Under some coercivity conditions on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity conditions on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates in the space variable are fully global in time, while the ones involving the time variable are local in time. In the same time we show how to obtain the same estimates for the mean field planning problem (introduced by P.-L. Lions). Our methods have their roots in Brenier's work on the regularity of the pressure field arising in weak solutions of the incompressible Euler equations (see [2]), which was improved later by Ambrosio-Figalli in [1]. The talk is based on the works [3, 4].

References

1. L. Ambrosio, A. Figalli, On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations, Calc. Var. Partial Differential Equations 31(4) (2008), 497–509.
2. Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Commun. Pure Appl. Math. 52(4) (1999), 411–452.
3. P.J. Graber, A.R Mészáros, Sobolev regularity for first order Mean Field Games, Ann. Inst. H. Poincaré Anal. Non Linéaire 35(6) (2018), 1557–1576.
4. P.J. Graber, A.R Mészáros, F.J. Silva, D. Tonon, The planning problem in Mean Field Games as regularized mass transport, Calc. Var. Partial Differential Equations (2019), to appear.

Robin Neumayer, Northwestern University, USA

ANISOTROPIC LIQUID DROP MODELS

Joint work with Rustum Choksi and Ihsan Topaloglu

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 0.10a.

Abstract

Anisotropic surface energies are a natural generalization of the perimeter functional that arise, for instance, in scaling limits for certain probabilistic models on lattices. Smoothness and ellipticity assumptions are sometimes imposed on the energy to improve analytic aspects of associated isoperimetric problems, but these assumptions are not always desirable for some applications nor checkable when the problem comes from a scaling limit. We consider an anisotropic variant of a model for atomic nuclei and show that minimizers behave in a fundamentally different way depending on whether or not the energy is smooth and elliptic. This is joint work with Choksi and Topaloglu.

Giandomenico Orlandi, University of Verona, Italy

ENERGY MINIMIZING MAPS INTO FINSLER MANIFOLDS AND OPTIMAL ONE-DIMENSIONAL NETWORKS

Joint work with Sisto Baldo and Annalisa Massaccesi

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 0.10a.

Abstract

We consider certain natural variational problems for maps valued into manifolds equipped with a Finsler norm, study their relaxation and show the emergence of optimal one dimensional networks where energy concentrates, providing a link with the classical Steiner Tree problem or, more generally, with Gilbert-Steiner irrigation-type problems.

Filip Rindler, University of Warwick, UK

THEME & VARIATIONS ON $${\rm div}\, \mu = \sigma$$

Joint work with A. Arroyo-Rabasa, G. De Philippis, J. Hirsch, and A. Marchese

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 0.10a.

Abstract

The PDE $${\rm div}\, \mu = \sigma$$ for (vector) measures $$\mu$$ and $$\sigma$$ appears - sometimes in a slightly hidden way - in many different problems of geometric measure theory and the calculus of variations, for instance in the structure theory of normal currents, Lipschitz functions and varifolds. In this talk I will survey a number of recent results about this equation and other related PDEs. As applications, I will discuss the structure of singularities of solutions, dimensional estimates, and several versions of Rademacher's theorem (both in Euclidean and non-Euclidean settings).

Matteo Rizzi, University of Chile, Chile

THE CAHN-HILLIARD EQUATION: EXISTENCE RESULTS AND QUALITATIVE PROPERTIES

Joint work with Michal Kowalczyk

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 0.10a.

Abstract

In the talk I will present the construction of a family $$\{u_\varepsilon\}$$ of solutions to the Cahn-Hilliard equation $-\varepsilon\Delta u_\varepsilon=\varepsilon^{-1}(u_\varepsilon-u_\varepsilon^3)-\ell_\varepsilon, \qquad\ell_\varepsilon\in\mathbb{R},$ whose zero level set is prescribed and approaches, as $$\varepsilon\to 0$$, a given complete, embedded, $$k$$-ended constant mean curvature surface. Moreover, I will present some classification results, dealing with properties such as boundedness, monotonicity and radial symmetry.

References

1. M. Kowalczyk, M. Rizzi, Multiple Delaunay ends solutions of the Cahn-Hilliard equation, accepted by Communications in Partial differential equations.
2. M. Rizzi, Radial and cylindrical symmetry of solutions to the Cahn-Hilliard equation, submitted to Calculus of variations and PDEs.

Thomas Schmidt, Universität Hamburg, Germany

OBSTACLE PROBLEMS FOR BV MINIMIZERS, AND WEAK SUPERSOLUTIONS TO THE $$1$$-LAPLACE EQUATION

Joint work with Lisa Beck and Christoph Scheven

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 0.10a.

Abstract

The talk will be concerned with Dirichlet and obstacle problems for the total variation, the area integral, and possibly more general variational integrals on the space of functions of bounded variation. In particular, it is planned to discuss duality-based connections to (super)solutions of PDEs of $$1$$-Laplace and minimal surface type.

References

1. L. Beck, T. Schmidt, Convex duality and uniqueness for $$\mathrm{BV}$$-minimizers, J. Funct. Anal. 268 (2015), 3061-3107.
2. C. Scheven, T. Schmidt, $$\mathrm{BV}$$ supersolutions to equations of $$1$$-Laplace and minimal surface type, J. Differ. Equations 261 (2016), 1904-1932.
3. C. Scheven, T. Schmidt, On the dual formulation of obstacle problems for the total variation and the area functional, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35 (2018), 1175-1207.

Federico Stra, Ecole Polytechnique Fédérale de Lausanne, Switzerland

RECENT ADVANCES IN MULTI-MARGINAL OT APPLIED TO DFT

Joint work with M. Colombo and S. Di Marino

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 0.10a.

Abstract

Multi-marginal optimal transport has been adopted to approximate the electron-electron interaction energy in the context of Density Functional Theory (DFT).

In this talk I will review the OT formulation of the DFT problem and mention what conditions ensure the finiteness and the continuity of multi-marginal optimal transport with repulsive cost (expressed in terms of a suitable concentration property of the measure).

Finally I will present some recent results regarding the $$\Gamma$$-convergence of the functionals for the study of the semiclassical limit of ground states.

References

1. M. Colombo, S. Di Marino, F. Stra, Continuity of multi-marginal optimal transport with repulsive cost, SIAM J. Math. Anal. 51(4) (2019), 2903–2926.
2. M. Colombo, S. Di Marino, F. Stra, Semiclassical limit of ground state in DFT, in preparation.

Bozhidar Velichkov, Université Grenoble-Alpes, France

ON THE LOGARITHMIC EPIPERIMETRIC INEQUALITY

Joint work with Maria Colombo and Luca Spolaor

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 0.10a.

Abstract

This talk is dedicted to some recent advances on the regularity of the free boundaries arising in variational minimization problems. In particular, we will present a new variational approach for the analysis of singularities: the logarithmic epiperimetric inequalities, which was already applied to several different free boundary problems: the obstacle problem [2], the thin-obstacle problem [3, 7], the one-phase Alt-Caffarelli problem [4], to area-minimizing currents [5], and to parabolic free boundary problems [8].

The focus of this talk is on the classical obstacle problem:$\qquad\min\Big\{\int_{B_1} \!\!\big(|\nabla u|^2+u\big)\,dx\,:\, u\in H^1(B_1),\ u\ge 0\ \text{in}\ B_1,\ u\ \text{is prescribed on}\ \partial B_1\Big\}.\qquad{\rm(OB)}$It is well-known that, given a solution $$u$$ and setting $$\Omega_u:=\{u>0\}$$, the free boundary $$\partial\Omega_u\cap B_1$$ can be decomposed into a regular part, $$Reg\,(\partial\Omega_u)$$, and a singular part, $$Sing\,(\partial\Omega_u)$$, where

$$\bullet$$ $$Reg\,(\partial\Omega_u)$$ is a smooth manifold (this result was proved by Caffarelli in [1]);

$$\bullet$$ $$Sing\,(\partial\Omega_u)$$ are the points, at which the Lebesgue density of the set $$\{u=0\}$$ vanishes.

In this talk, we will prove a logarithmic epiperimetric inequality for the Weiss' boundary adjusted energy, from which we will deduce that the singular set $$Sing\,(\partial\Omega_u)$$ is $$C^{1,\log}$$-regular, that is, it is contained into the countable union of $$C^{1,\log}$$-regular manifolds. This results was first proved in [2]. Recently, Figalli and Serra [6] showed that this result is also optimal in the sense that the logarithmic modulus of continuity cannot be improved for general singularities.

References

1. L. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155–184.
2. M. Colombo, L. Spolaor, B. Velichkov, A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal. 28 (2018), 1029–1061.
3. M. Colombo, L. Spolaor, B. Velichkov, Direct epiperimetric inequalities for the thin obstacle problem and applications, Comm. Pure. Appl. Math. (to appear).
4. M. Engelstein, L. Spolaor, B. Velichkov, Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional, Preprint arXiv:1801.09276.
5. M. Engelstein, L. Spolaor, B. Velichkov (Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currents, Geom. Topol. 23 (2019), 513–540.
6. A. Figalli, J. Serra, On the fine structure of the free boundary for the classical obstacle problem, Invent. Math. 215 (2019), 311–366.
7. S. Jeon, A. Petrosyan, Almost minimizers for the thin obstacle problem, Preprint arXiv 1905.11956.
8. W. Shi, An epiperimetric inequality approach to the parabolic Signorini problem, Preprint arXiv 1810.11791.

Anna Zatorska-Goldstein, University of Warsaw, Poland

RENORMALIZED SOLUTIONS FOR PARABOLIC PROBLEMS WITH ANISOTROPIC STRUCTURE AND NONSTANDARD GROWTH

Joint work with Iwona Chlebicka and Piotr Gwiazda

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 0.10a.

Abstract

We investigate nonlinear parabolic problems where the ellipticity and the growth conditions for the leading part of the operator is driven by an inhomogeneous and anisotropic function of the Orlicz type. We establish the existence and the uniqueness of renormalized solutions when merely integrable data are allowed. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Our results cover in particular the case of variable exponent growth, with the exponent depending both on time and space variables.

References

1. P. Gwiazda, I. Skrzypczak, A. Zatorska-Goldstein, Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space, J. Differential Equations 264(1) (2018), 341–377.
2. I. Chlebicka, P. Gwiazda, A. Zatorska-Goldstein, Well-posedness of parabolic equations in the non- reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions, J. Differential Equations 265(11) (2018), 5716–5766.
3. I. Chlebicka, P. Gwiazda, A. Zatorska-Goldstein, Renormalized solutions to parabolic equations in time and space anisotropic Musielak-Orlicz spaces in absence of Lavrentiev’s phenomenon, J. Differential Equations 267(2) (2019), 1129–1166.
4. I. Chlebicka, P. Gwiazda, A. Zatorska-Goldstein, Parabolic equation in time and space anisotropic Musielak-Orlicz spaces in absence of Lavrentiev’s phenomenon, Ann. Inst. H. Poincaré Anal. Non Linéaire (2019), https://doi.org/10.1016/j.anihpc.2019.01.003.

Thomas Alazard, École Normale Supérieure Paris-Saclay, France

OBSERVABILITY, CONTROL AND STABILIZATION OF WATER WAVES

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 0.18.

Abstract

The main questions studied in this talk are the generation and the absorption of water waves in a wave tank. We want to understand which water waves can be generated by blowing above a localized portion of the free surface of a liquid. We also want to understand how to damp water waves when they reach the boundary of a numerical wave tank.

For $$2D$$ water waves in a rectangular tank, the fluid domain $$\Omega(t)$$ is of the form $\Omega(t)=\left\{\, (x,y)\,:\, x\in [0,L], h\le y\le \eta(t,x)\,\right\},$ where $$x$$ (resp. $$y$$) is the horizontal (resp. vertical) space variable, $$L$$ is the width of the tank, $$h$$ its depth and $$\eta$$ is the free surface elevation. The equations which dictate the motion are the incompressible Euler equations with free surface. This is a system of two nonlinear equations: the incompressible Euler equation for the velocity potential $$\phi\colon \Omega\rightarrow \mathbb{R}$$ (so that the velocity is $$v=\nabla_{x,y}\phi$$) and a kinematic equation for $$\eta$$ which states that the free surface moves with the fluid. Zakharov discovered that $$\eta$$ is conjugated to the trace $$\psi(t,x)=\phi(t,x,\eta(t,x))$$ of the velocity potential on the free surface: the equations have the hamiltonian form $\frac{\partial \eta}{\partial t}=\frac{\delta\mathcal{H}}{\delta \psi},\quad \frac{\partial \psi}{\partial t} =-\frac{\delta\mathcal{H}}{\delta \eta}-P_{ext},\tag{1}$ where $$P_{ext}$$ is an external pressure and $$\mathcal{H}$$ is the energy $\mathcal{H}=\frac{g}{2}\int_0^L\eta^2(t,x)\, dx+\int_0^L \big(\sqrt{1+(\partial_x \eta(t,x))^2}-1\big)\, dx+ \frac{1}{2}\iint_{\Omega(t)}\left\vert \nabla_{x,y}\phi(t,x,y)\right\vert^2\, dx dy.$

$$\bullet$$ The first problem is the following : given a time $$T>0$$, a final state $$(\eta_{final},\psi_{final})$$ in some space of regular functions, a non empty interval $$\omega=(a,b)\subset [0,L]$$, is-it possible to find a function $$P_{ext}(t,x)$$ supported in $$[0,T]\times \omega$$ such that the solution to (1) with initial data $$(\eta_{in},\psi_{in})=(0,0)$$ satisfies $$(\eta,\psi)\arrowvert_{t=T}=(\eta_{final},\psi_{final})$$? I will present a local controllability result obtained with Pietro Baldi and Daniel Han-Kwan ([1]).

$$\bullet$$ We then consider the stabilization problem. The goal here is to find a pressure law, relating $$P_{ext}$$ to the unknown $$(\eta,\psi)$$ and supported inside a small subset of $$[0,L]$$, such that $$\mathcal{H}$$ is decreasing and converges to zero. I will explain how to use the multiplier method of C. Morawetz and J.L. Lions to study this problem ([2, 3, 4]).

References

1. T. Alazard, P. Baldi, D. Han-Kwan, Control for water waves, J. Eur. Math. Soc. 20 (2018), 657–745.
2. T. Alazard, Boundary observability of gravity water waves, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 751–779.
3. T. Alazard, Stabilization of gravity water waves, J. Math. Pures Appl. 114 (2018), 51–84.
4. T. Alazard, Stabilization of the water-wave equations with surface tension, Annals of PDE 2 (2017), 41 pp.

Fabio Ancona, University of Padua, Italy

ONE-SIDE BOUNDARY CONTROLLABILITY OF THE $$P$$-SYSTEM

Joint work with Olivier Glass and Khai T. Nguyen

Date: 2019-09-17 (Tuesday); Time: 16:55-17:15; Location: building B-8, room 0.18.

Abstract

We consider the equations for one-dimensional isentropic compressible gases on an interval, in Eulerian or in Lagrangian coordinates (known as the the $$p$$-system). On one side of the interval it is imposed a fixed boundary condition (for instance the null velocity), while on the other side of the interval the boundary condition is treated as a control that one can choose to influence the system. We prove a result of controllability toward constants states in the context of (discontinuous) weak entropy solutions. Namely, we prove that it is possible, starting from an initial state small in $$BV$$, to reach any constant state compatible with the boundary conditions. This type of result was previously obtained in [2] in the context of boundary controls acting on both boundaries. These results are in sharp contrast with what happens for some other $$2 \times 2$$ strictly hyperbolic systems with genuinely nonlinear characteristics fields for which Bressan and Coclite [1] showed that, in general, exact controllability to constant states is not possible, even when controlling on both sides of the interval.

References

1. A. Bressan, G.M. Coclite, On the boundary control of systems of conservation laws, SIAM J. Control Optim. 41 (2002), 607-622.
2. O. Glass, On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc. (JEMS) 9(3) (2007), 427-486.

Petr Čoupek, Charles University, Czech Republic

AN ERGODIC CONTROL PROBLEM WITH ROSENBLATT NOISE

Joint work with Tyrone E. Duncan, Bohdan Maslowski, and Bożenna Pasik-Duncan

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 0.18.

Abstract

The talk will be devoted to an infinite time horizon linear-quadratic control problem for a stochastic differential equation with additive Rosenblatt noise.

Rosenblatt processes arise naturally as limits of suitably normalized sums of long-range dependent random variables in a non-central limit theorem. These continuous processes are self-similar, have stationary increments and exhibit long memory; however, unlike the family of regular fractional Brownian motions, they are not Gaussian. This last property makes their analysis somewhat intriguing and it is also the reason why they received considerable attention in recent years.

Initially, some recent results on stochastic calculus for Rosenblatt and related fractional processes will be presented in the talk. Subsequently, the ergodic control problem will be formulated and solved; and the optimal control as well as the optimal cost will be given explicitly.

Charles Dapogny, Université Grenoble-Alpes, France

SHAPE AND TOPOLOGY OPTIMIZATION VIA A LEVEL-SET BASED MESH EVOLUTION METHOD

Joint work with Grégoire Allaire, Florian Feppon, and Pascal Frey

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 0.18.

Abstract

The purpose of this presentation is to introduce a robust front-tracking method for dealing with arbitrary motions of shapes, even dramatic ones (e.g. featuring topological changes); although this method is illustrated in the particular context of shape optimization, it naturally applies to a wide range of inverse problems and reconstruction algorithms.

The presented method combines two different means of representing shapes: on the one hand, they are meshed explicitly, which allows for efficient mechanical calculations by means of any standard Finite Element solver; on the other hand, they are represented by means of the level set method, a format under which it is easy to track their evolution. The cornerstone of our method is a pair of efficient algorithms for switching from either of these representations to the other.

Several numerical examples are discussed in two and three space dimensions, in the 'classical' physical setting of linear elastic structures, but also in more involved situations involving e.g. fluid-structure interactions.

(Left) Optimal design of a mast withstanding an incoming flow; (right) optimal design of a bridge.

References

1. G. Allaire, F. Jouve and A.M. Toader, Structural optimization using shape sensitivity analysis and a level-set method, J. Comput. Phys. 194 (2004), 363-393.
2. G. Allaire, C. Dapogny and P. Frey, Shape optimization with a level set based mesh evolution method, Comput. Meths. Appl. Mech. Engrg. 282 (2014), 22-53.
3. F. Feppon, G. Allaire, F. Bordeu, J. Cortial and C. Dapogny, Shape optimization of a coupled thermal fluid-structure problem in a level set mesh evolution framework, SeMA journal (2019), HAL preprint: hal-01686770.

Luz de Teresa, National Autonomous University of Mexico, Mexico

A STACKELBERG STRATEGY TO CONTROL A HEAT EQUATION

Joint work with Enrique Fernández-Cara and José Antonio Villa

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 0.18.

Abstract

In this conference we present the control problem that arises with the application of multiple strategies to the control of parabolic equations. We assume that we can act on the system through a hierarchy of controls. The leader control has an optimization objective and the follower a null control objective. We discuss the differences that arise when the leader control has a null controllability objective while the follower an optimization one. Literature, as far as we know, treats the second problem, see [1, 2, 3, 4].

References

1. F.D. Araruna, E. Fernández-Cara and M.C. Santos, Stackelberg-Nash Exact Controllability For Linear And Semilinear Parabolic Equations, ESAIM:COCV 21 (2015), 835-856.
2. F.D. Araruna, E. Fernández-Cara, S. Guerrero and M.C. Santos, New results on the Stackelberg–Nash exact control of linear parabolic equations, System and Control Letters 104 (2017), 78-85.
3. V. Hernández-Santamaría, L. de Teresa, Some Remarks on the Hierarchic Control for Coupled Parabolic PDEs, Capítulo en “Recent Advances in PDEs: Analysis, Numerics and Control". Editado por: Doubova, A., González-Burgos, M., Guillén-González, F., Marín Beltrán, M. Springer 2018.
4. C. Montoya, L. de Teresa, Robust Stackelberg controllability for the Navier-Stokes equations, Nonlinear Differ. Equ. Appl. 25(46) (2018), https://doi.org/10.1007/s00030-018-0537-3.

Giorgio Fabbri, CNRS, France

HJB EQUATIONS IN INFINITE DIMENSION AND OPTIMAL CONTROL OF STOCHASTIC EVOLUTION EQUATIONS VIA GENERALIZED FUKUSHIMA DECOMPOSITION

Joint work with Francesco Russo

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 0.18.

Abstract

In this talk we present the result of the paper [1].

We consider a stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space. Thanks to the identification of the mild solution of the state equation as $$\nu$$-weak Dirichlet process, the value processes is proved to be a real weak Dirichlet process. The uniqueness of the corresponding decomposition is used to prove a verification theorem.

Through that technique some of the required assumptions are milder than those employed in previous contributions about non-regular solutions of Hamilton-Jacobi-Bellman equations. We present some explicit example.

References

1. G. Fabbri, F. Russo, HJB equations in infinite dimension and optimal control of stochastic evolution equations via generalized Fukushima decomposition, SIAM Journal on Control and Optimization 55(6) (2016), 4072-4091.

Marco Fuhrman, University of Milan, Italy

OPTIMAL CONTROL OF STOCHASTIC EVOLUTION EQUATIONS VIA RANDOMIZATION AND BACKWARD SDES

Joint work with Emanuela Gussetti

Date: 2019-09-16 (Monday); Time: 12:05-12:25; Location: building B-8, room 0.18.

Abstract

Backward Stochastic Differential Equations (BSDEs) have been successfully applied to represent the value of optimal control problems for controlled stochastic differential equations. Since in the classical framework several restrictions on the scope of applicability of this method remained, in recent times several approaches have been devised to obtain the desired probabilistic representation in more general situations.

We will review the so called randomization method, originally introduced by B. Bouchard in the framework of optimal switching problems, which consists in introducing an auxiliary, "randomized" problem with the same value as the original one, where the control process is replaced by an exogenous random point process, and optimization is performed over a family of equivalent probability measures. The value of the randomized problem is then represented by means of a special class of BSDEs with a constraint on one of the unknown processes.

This methodology will be applied in the framework of controlled evolution equations (with immediate applications to controlled SPDEs), a case for which very few results are known so far.

Irena Lasiecka, University of Memphis, USA

GLOBAL SOLUTIONS AND STABILITY FOR A 3-D FLUID-STRUCTURE INTERACTION

Date: 2019-09-17 (Tuesday); Time: 16:20-16:50; Location: building B-8, room 0.18.

Abstract

We consider an interface problem consisting of a 3-D fluid equation interacting with a 3-D dynamic elasticity. The interface is moving according to the speed of the fluid. The PDE system is modeled by system of partial differential equations describing motion of an elastic body inside an incompressible fluid. The fluid is governed by Navier-Stokes equation while the structure is represented by the system of dynamic elasticity with weak dissipation. The interface between the two environments undergoes oscillations which lead to moving frame configuration, the latter giving rise to a quasilinear system. Short time local existence of solutions has been shown in [1]. The aim of the talk is to discuss global [in time] solutions under small disturbance hypothesis. Stability [in time] of such solutions is also considered along with some control problems related to minimization of the vorticity. The problem is motivated by applications arising in bio-mechanics, aeroelasticity and industrial processes. In the presence of weak damping affecting the solid the control-to-observation map is proved global-so that the size of the data can be chosen uniformly in time. This allows consideration of an infinite time horizon control problem. The latter depends on the global existence results obtained in [2].

References

1. D. Coutand, S. Shkoller, Motion of an elastic solid inside an incompressible fluid, Archives of Rational Mechanics and Analysis 176 (2008), 1173-1207.
2. M. Ignatova, I. Kukavica, I. Lasiecka, A. Tuffaha, Small data global existence for a fluid structure model, Nonlinearity; 30 (2017), 848-898.

Antoine Laurain, University of São Paulo, Brazil

SHAPE DESIGN FOR SUPERCONDUCTORS GOVERNED BY H(CURL)-ELLIPTIC VARIATIONAL INEQUALITIES

Joint work with Malte Winckler and Irwin Yousept

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 0.18.

Abstract

We study a shape optimization problem governed by an elliptic $$\operatorname{curl}$$-$$\operatorname {curl}$$ variational inequality (VI) of the second kind. We present a Moreau-Yosida type regularization for the dual formulation of the VI that guarantees the Gâteaux-differentiability of the regularized dual variable. Then, for a fixed regularization parameter, the existence of an optimal shape for the corresponding problem is proved by means of a compactness theorem. Then we analyze the sensitivity of the regularized objective functional by rigorously computing the corresponding shape derivative using the averaged adjoint method, a lagrangian-type formulation. We also give a stability estimate for the shape derivative with respect to the regularization parameter, and show the strong convergence of the optimal shapes and the corresponding states for the regularized problem towards a solution to the problem without regularization. Finally, we present the numerical algorithm based on the distributed shape derivative coupled with the level set method, and we apply it to problems stemming from the type-II (HTS) superconductivity.

References

1. I. Yousept, Hyperbolic Maxwell Variational Inequalities for Bean’s Critical-State Model in Type-II Superconductivity, SIAM J. Numer. Anal. 55 (2017), 2444-2464.
2. A. Laurain, K. Sturm, Distributed shape derivative via averaged adjoint method and applications, ESAIM Math. Model. Numer. Anal. 50 (2016), 1241-1267.

Pierre Lissy, CEREMADE, Université Paris-Dauphine, France

LOCAL CONTROLLABILITY TO THE TRAJECTORIES OF THE FOKKER-PLANCK EQUATION WITH A LOCALIZED CONTROL

Joint work with Michel Duprez

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 0.18.

Abstract

We will present a new result on the control of the Fokker-Planck equation, posed on a smooth bounded domain of $$\mathbb{R}^d$$ $$(d \ge 1)$$. More precisely, the control is the drift force, localized on a small open subset. We prove that this system is locally null controllable to regular nonzero trajectories, with potentially strictly less that $$d$$ controls. The results are obtained thanks to a linearization method based on a standard inverse mapping procedure and the fictitious control method. The main novelties of the present article are twofold. Firstly, we propose an alternative strategy to the standard fictitious control method: the algebraic solvability is performed and used directly on the adjoint problem. Secondly, we prove a new Carleman inequality for the heat equation with one order space-varying coefficients: the right-hand side is the gradient of the solution localized on a subset (rather than the solution itself), and the left-hand side contains arbitrary high derivatives of the solution.

Dominika Machowska, University of Łódź, Poland

COMPETITION IN DEFENSIVE AND OFFENSIVE ADVERTISING STARTEGIES IN A SEGMENTED MARKET

Joint work with Andrzej Nowakowski

Date: 2019-09-17 (Tuesday); Time: 17:20-17:40; Location: building B-8, room 0.18.

Abstract

We propose the new goodwill model à la Nerlove-Arrow defined on a competitive segmented market. Based on the dual dynamic approach, we give the sufficient condition under which the open-loop equilibrium exists for the new game. We also introduce $$\varepsilon$$-open loop equilibrium as a basis for the numerical algorithm using a construction of the optimal solution in the finite steps. The numerical algorithm enables an analysis of how the level of the homogeneity of given competitive products and customer recommendations modify optimal goodwill and the total profit of each player.

Bohdan Maslowski, Charles University, Czech Republic

FILTERING AND OPTIMAL CONTROL AND FOR GAUSS-VOLTERRA PROCESSES IN HILBERT SPACES

Joint work with Tyrone E. Duncan, Bożenna Pasik-Duncan and Vít Kubelka

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 0.18.

Abstract

Kalman-Bucy type filter and some methods of parameter estimation are studied in the case when signals are Hilbert space-valued Gaussian processes.  The corresponding integral equations are derived for the optimal estimate and covariance of the error. Some basic properties of the filter are discussed. These general results are illustrated by examples  of  linear SPDEs where the noise terms are Gauss-Volterra processes (in particular, fractional Brownian motions). Also, some optimal control results for such systems are recalled for the case of quadratic cost functionals. In both cases, the results are compared with the standard ones for Gauss-Markov systems.

Dario Mazzoleni, Catholic University of Brescia, Italy

ASYMPTOTIC SPHERICAL SHAPES IN SPECTRAL OPTIMIZATION PROBLEMS

Joint work with Benedetta Pellacci and Gianmaria Verzini

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-8, room 0.18.

Abstract

We study the positive principal eigenvalue of a weighted problem associated with the Neumann-Laplacian settled in a box $$\Omega\subset \mathbb{R}^N$$, which arises from the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the sign-changing weight, one is lead to consider a shape optimization problem, which is known to admit spherical optimal shapes only in trivial cases. We investigate if spherical shapes can be recovered in the limit when the negative part of the weight diverges. First of all, we show that the shape optimization problem appearing in the limit is the so called $$\textit{spectral drop}$$ problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Thanks to $$\alpha$$-symmetrization techniques on cones, it can be proved that optimal shapes for the spectral drop problem are spherical for suitable choices of the box, the most interesting case being when $$\Omega$$ is a convex polytope, and in this case a quantitative analysis of the convergence can be performed. Finally, for a smooth $$\Omega$$, we show that small volume spectral drops are asymptotically spherical, with center at points with high mean curvature.

References

1. D. Mazzoleni, B. Pellacci, G. Verzini, Asymptotic spherical shapes in some spectral optimization problems, Preprint arXiv:1811.01623.
2. D. Mazzoleni, B. Pellacci, G. Verzini, Quantitative analysis of a singularly perturbed shape optimization problem in a polygon, Preprint arXiv:1902.05844.

Alberto Mercado, Federico Santa María Technical University, Chile & Institut de Mathématiques de Toulouse, France

BOUNDARY CONTROLLABILITY OF A COUPLED FOURTH-SECOND ORDER PARABOLIC SYSTEM

Joint work with Nicolás Carreño and Eduardo Cerpa

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 0.18.

Abstract

We study a control system coupling fourth and second-order parabolic equations, when we only control the second-order partial differential equation through a boundary condition $$h$$: $\left\lbrace \begin{array}{ll} u_t(t,x) + u_{xxxx}(t,x) = v(t,x),&t\in(0,T),\,x\in(0,\pi), \\ v_t(t,x) - d v_{xx}(t,x) = 0, &t\in(0,T),\,x\in(0,\pi), \\ u(t,0)=u_{xx}(t,0) = 0,&t\in(0,T),\\ u(t,\pi)=u_{xx}(t,\pi) = 0,&t\in(0,T), \\ v(t,0) = h(t),\, v(t,\pi)=0,&t\in(0,T). \end{array}\right.\tag{1}$

Following the methods introduced in [1], we obtain positive and negative results for approximate- and null-controllability, depending on the diophantine approximation properties of the diffusion coefficient $$d$$. In particular, we prove that, if $$\sqrt d$$ has finite irrationality measure (also called Liouville-Roth constant), then system (1) is null controllable in any time $$T>0$$.

References

1. F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, and L. de Teresa, Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences, J. Funct. Anal. 267 (2014), 2077-2151.
2. N. Carreño, E. Cerpa, and A. Mercado, Boundary controllability of a cascade system coupling fourth- and second-order parabolic equations, Preprint.

Iván Moyano, University of Cambridge, UK

SPECTRAL INEQUALITIES FOR THE SCHRÖDINGER OPERATOR $$-\Delta_x + V(x)$$ in $$\mathbb{R}^d$$

Joint work with Gilles Lebeau

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 0.18.

Abstract

In this talk, we will first review some classical results on the so-called ’spectral inequalities’, which yield a sharp quantification of the unique continuation of the spectral family associated with the Laplace-Beltrami operator in a compact manifold. In a second part, we will discuss how to obtain the spectral inequalities associated to the Schrodinger operator $$-\Delta_x + V(x)$$, in $$\mathbb{R}^d$$, in any dimension $$d\geq 1$$, where $$V=V(x)$$ is a real analytic potential. In particular, we can handle some long- range potentials.

Akambadath Keerthiyil Nandakumaran, Indian Institute of Science, India

ASYMPTOTIC ANALYSIS OF OPTIMAL CONTROL PROBLEMS IN OSCILLATORY DOMAINS

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 0.18.

Abstract

Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. Hence, it has applications in various branches of science and engineering. This study is basically developed from material science in the creation of composite materials though the present application is much far and wide. It has applications in composite media, porous domains, laminar structures, domains with rapidly oscillating boundaries, to name a few. The PDE problems posed on such complicated domains lead to the analysis of homogenization. It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium. Mathematically, it is a kind of asymptotic analysis. There are various methods developed in the last 50 years to understand the mathematical homogenization theory.

In this talk, we discuss the asymptotic analysis of various optimal control problems defined in domains who boundary is rapidly (highly) oscillating. Such complex domains appears in many real life applications like heat radiators, flows in channels with rough boundaries, propagation of electro-magnetic waves in regions having rough interface, absorption diffusion in biological structures, acoustic vibrations in medium with narrow channels etc. In the first part, we briefly present the work which we are carrying out in my group (see and later we present some specific results. We introduce the so called unfolding operators which we have developed for the problems under study through which we characterize the optimal controls. Finally, we do a homogenization process and obtain the limit control problem.

References

1. S. Aiyappan, A.K. Nandakumaran, R. Prakash, Locally periodic unfolding operator for highly oscillating rough domains, Annali di Matematica Pura ed Applicata (1923 -), https://doi.org/10.1007/s10231-019-00848-7.
2. S. Aiyappan, A.K. Nandakumaran, R. Prakash, Semi-linear optimal control problem on a smooth oscillating domain, Communications in Contemporary Mathematics, DOI: 10.1142/S0219199719500299 (26 pages).
3. R. Mahadevan, A.K. Nandakumaran, R. Prakash, Homogenization of an elliptic equation in a domain with oscillating boundary with non-homogeneous non-linear boundary conditions, Appl. Math. Optim., https://doi.org/10.1007/s00245-018-9499-4 (34 pages).
4. S. Aiyappan, A.K. Nandakumaran, R. Prakash, Generalization of Unfolding Operator for Highly Oscillating Smooth Boundary Domains and Homogenization, Calculus of Variations and PDE, (2018), 57-86.
5. S. Aiyappan, A.K. Nandakumaran, Optimal Control Problem in a Domain with Branched Structure and Homogenization, Mathematical Methods in Applied Sciences 40(8) (2017), 3173-3189.
6. A.K. Nandakumaran, R. Prakash and B.C. Sarda, Periodic Controls in an Oscillating Domain: Controls via Unfolding and Homogenization, SIAM Journal on Control and Optimization 53(5) (2015), 3245-3269.

Andrzej Nowakowski, University of Łódź, Poland

CONTROL OF BLOWUP AND APPROXIMATE OPTIMALITY CONDITIONS FOR THE WAVE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 0.18.

Abstract

The model problem we study is a semilinear wave equation to which we add a control $$u$$ in a source function and on the boundary: $x_{tt} -\Delta x =f (t ,z ,x ,u)\;\;\text{in}\;\mathbb{R}^{ +} \times D ,$ $x (t ,z) =u(t,z)\;\;\text{\ }\;\text{on}\;\mathbb{R}^{ +} \times \; \partial D ,$ $x (0 ,z) =v_{0} (z), \ x_{t}(0,z)=v_{1}(z).$ This type of problems arise in a great variety of situations. In the paper we propose the problem of controlling a system which may blow up in finite time. We want to minimize the blowup time. To this effect sufficient optimality conditions for controlled blowup time are derived in terms of new dynamic programming methodology. We define $$\varepsilon$$-optimal value function and we construct sufficient $$\varepsilon$$-optimal conditions for that function again in terms of new dynamic programming inequality.

Guillaume Olive, Jagiellonian University in Kraków, Poland

MINIMAL CONTROL TIME FOR ONE-DIMENSIONAL FIRST-ORDER HYPERBOLIC SYSTEMS

Joint work with Long Hu

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 0.18.

Abstract

The goal of this talk is to present some recent results in [2] concerning the exact controllability of one-dimensional first-order linear hyperbolic systems when all the controls are acting on the same side of the boundary. We show that the minimal time needed to control the system is given by an explicit and easy-to-compute formula with respect to all the coupling parameters of the system. The proof relies on the introduction of a canonical UL-decomposition and the compactness-uniqueness method.

References

1. M. Duprez, G. Olive, Compact perturbations of controlled systems, Math. Control Relat. Fields 8 (2018), 397-410.
2. L. Hu, G. Olive, Minimal time for the exact controllability of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls, preprint (2019), 47 pages.
3. A.F. Neves, H.S. Ribeiro, and O. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal. 67 (1986), 320-344.
4. D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), 639-739.
5. N. Weck, A remark on controllability for symmetric hyperbolic systems in one space dimension, SIAM J. Control Optim. 20 (1982), 1-8.

Yannick Privat, Université de Strasbourg, France

OPTIMAL RESOURCES CONFIGURATIONS IN POPULATION DYNAMICS

Joint work with Jimmy Lamboley, Antoine Laurain, and Grégoire Nadin

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 0.18.

Abstract

In this work, we are interested in the analysis of optimal resources configurations (typically foodstuff) necessary for a species to survive. For that purpose, we use a logistic equation to model the evolution of population density involving a term standing for the heterogeneous spreading (in space) of resources. The principal issue investigated in this talk writes: How to spread in an optimal way resources in a closed habitat? This problem can be recast as the one of minimizing the principal eigenvalue of an operator with respect to the domain occupied by resources, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. In particular, we investigate the optimality of balls.

Lionel Rosier, MINES ParisTech, PSL Research University, France

EXACT CONTROLLABILITY OF NONLINEAR HEAT EQUATIONS IN SPACES OF ANALYTIC FUNCTIONS

Joint work with Camille Laurent

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 0.18.

Abstract

It is by now well known that the use of Carleman estimates allows to establish the controllability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. That issue has obtained very recently almost sharp results in the linear case (see [4, 1, 2]). In this talk, we investigate the set of reachable states for a nonlinear heat equation in dimension one. The nonlinear part is assumed to be an analytic function of the spatial variable $$x$$, the unknown $$y$$, and its derivative $$y_x$$. By investigating carefully a nonlinear Cauchy problem in $$x$$ in some space of Gevrey functions, and the relationship between the jet of space derivatives and the jet of time derivatives, we derive an exact controllability result for small initial and final data that can be extended as analytic functions on some ball of the complex plane. It time allows, works in progress about the reachable states for KdV and for ZK will be outlined.

References

1. J. Dardé, S. Ervedoza, On the reachable set for the one-dimensional heat equation, SIAM J. Control Optim. 56(3) (2018), 1692-1715.
2. A. Hartman, K. Kellay, M. Tucsnak, From the reachable space of the heat equation to Hilbert spaces of holomorphic functions, to appear in JEMS.
3. C. Laurent, L. Rosier, Exact controllability of nonlinear heat equations in spaces of analytic functions, arXiv:1812.06637, submitted.
4. P. Martin, L. Rosier, P. Rouchon, On the reachable states for the boundary control of the heat equation, Appl. Math. Res. Express. AMRX 2 (2016), 181-216.
5. P. Martin, I. Rivas, L. Rosier, P. Rouchon, Exact controllability of a linear Korteweg-de Vries equation by the flatness approach, to appear in SIAM J. Control Optim.

Ionel Roventa, University of Craiova, Romania

OPTIMAL APPROXIMATION OF INTERNAL CONTROLS FOR A WAVE-TYPE PROBLEM WITH FRACTIONAL LAPLACIAN USING FINITE-DIFFERENCE METHOD

Joint work with Pierre Lissy

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 0.18.

Abstract

We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. Even if the initial condition is filtered, the control will excite all frequencies. This creates a lot of technical difficulties, because the spectral is not uniform with respect to the discretization step $$h$$. The proof is mainly based on a (non-classic) moment method. For more general uniform controllability results by using filtered spaces and resolvent estimates, the interested reader is referred to [2, 3, 7, 8].

Mathematically speaking, our model can be seen as an intermediate case between the cases of the wave equation and the beam equation. Our strategy consists of an appropriate filtering technique, introduced in [5] and notably used in [1, 4, 6] in the context of wave or beam equation, which consists in relaxing the control requirement by controlling only the low-frequency part of the solution. This approach will be considered here.

References

1. N. Cîndea, S. Micu, I. and Rovenţa, Boundary controllability for finite-differences semidiscretizations of a clamped beam equation, SIAM J. Control Optim. 55(2) (2017), 785–817.
2. S. Ervedoza, Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes, Numer. Math. 113 (2009), 377–415.
3. S. Ervedoza, Observability in arbitrary small time for discrete conservative linear systems, Some Problems in Nonlinear Hyperbolic Equations, ed. Tatsien Li, Yuejun Peng and Bopeng Rao, Series in Contemporary Mathematics CAM15, 283–309.
4. P. Lissy and I. Rovenţa, Optimal filtration for the approximation of boundary controls for the one- dimensional wave equation using finite-difference method, Math. Comp. 88(315) (2019), 273–291.
5. S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numer. Math. 91 (2002), 723–768.
6. S. Micu, I. Rovenţa, and L.E. Temereanca, Approximation of the controls for the linear beam equation, Math. Control Signals Systems 28(2) (2016), Art. 12, 53 pp.
7. L. Miller, Resolvent conditions for the control of unitary groups and their approximations, Journal of Spectral Theory 2 (2012), 1-55.
8. E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev. 47 (2005), 197-243 (electronic).

Amjad Tuffaha, American University of Sharjah, United Arab Emirates

ON BOUNDARY CONTROL OF THE POISSON EQUATION WITH THE THIRD BOUNDARY CONDITION

Joint work with Alip Mohamed

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 0.18.

Abstract

In this talk, we consider an optimal control problem involving the Poisson equation on the unit disk in $$\mathbb{C}$$ subject to the third boundary condition and where the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter $$\lambda$$ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when $$\lambda$$ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed.

References

1. A. Mohamed, A. Tuffaha, On Boundary Control of the Poisson Equation with the third boundary condition, Journal of Mathematical Analysis and Applications 459(1) (2018), 217–235.

Hans Zwart, University of Twente, Netherlands

DIFFERENTIAL ALGEBRAIC PORT-HAMILTONIAN EQUATIONS

Joint work with Volker Mehrmann

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 0.18.

Abstract

Port-Hamiltonian (pH) models can be used to describe physical systems which interact with their environment. Examples include ordinary and partial differential equations. By now the theory is quite complete with results ranging from control, approximation, and well-posedness of partial differential equations, [2]. Recently, the theory has been extended to differential equations with constraints, i.e., differential algebraic equations (DAE). We refer to [3] for the algebraic set-up of these systems and to [1] for (time-varying) DAE's with a finite-dimensional state space. Following on this, we study (time-invariant) port-Hamiltonian DAE on an infinite-dimensional state space. Our main focus is to show existence of (mild) solutions for this class of systems. So we consider the following abstract differential equation $E \dot{x}(t) = AQ x(t), \qquad x(0)=x_0$ where $$E$$ and $$Q$$ are bounded operators on $$X$$ satisfying $$E^*Q = Q^*E$$, and $$A$$ is the infinitesimal generator of a contraction semigroup on the Hilbert space $$X$$. Since $$A$$ is the the infinitesimal generator of a contraction semigroup, the above equation possesses a solution when $$E=Q=I$$. If $$E$$ and $$Q$$ are are non-invertible, then this needs not to hold. We present some sufficient conditions under which the above DAE possesses a unique solution. Furthermore, we show that our results are related to boundary triplets and passive systems.

References

1. C. Beattie, Christopher, V. Mehrmann, H. Xu, Hongguo, H. Zwart, Linear port-Hamiltonian descriptor systems,, Mathematics of Control, Signals, and Systems 30 (2018), 17-27.
2. B. Jacob, H. Zwart, Linear port-Hamiltonian Systems on Infinite-Dimensional Spaces, Birkhäuser/Springer, Basel, 2012.
3. A. van der Schaft, B. Maschke, Generalized port-Hamiltonian DAE systems, Systems & Control Letters 121 (2018), 31-37.

Joackim Bernier, Université Paul Sabatier, France

EXACT SPLITTINGS OF LINEAR QUADRATIC PDES

Joint work with Paul Alphonse, Fernando Casas, Nicolas Crouseilles, and Yingzhe Li

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-8, room 3.21.

Abstract

Usually, the higher the order of a splitting method is, the larger its number of steps is. However, we will see that for many linear ODEs, the resolution of an inverse problem provides an exact splitting, involving some modified vector fields, with the same number of steps as the usual low order methods (i.e. Lie or Strang splittings). Applying the Fourier integral operators theory (developed by Hörmander in [4]), we will see how these decompositions can be transposed at the level of the quadratic linear PDEs. I will show how this construction provides some new efficient splittings for many PDEs like the Schrödinger equations in rotating frames or the Vlasov equations with a rotation motion.

References

1. P. Alphonse, J. Bernier, Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects, in preparation.
2. J. Bernier, F. Casas, N. Crouseilles, Splitting methods for rotations: application to Vlasov equations, preprint available on HAL.
3. J. Bernier, Y. Li, N. Crouseilles, Splitting methods for Schrödinger equations in rotating frames, in preparation.
4. L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449.

Daniele Boffi, University of Pavia, Italy

A FICTITIOUS DOMAIN APPROACH TO FLUID-STRUCTURE INTERACTION PROBLEMS

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 3.21.

Abstract

We review a distributed Lagrange multiplier formulation of the Finite Element Immersed Boundary Method for the numerical approximation of the interaction between fluids and solids (see [1, 2]). The discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided. Optimal convergence estimates are proved for its finite element space discretization. The model, originally introduced for the coupling of incompressible fluids and solids, can be extended to include the simulation of compressible structures [3].

Recent research [4] investigates several time marching strategies for the proposed method.

References

1. D. Boffi, N. Cavallini, L. Gastaldi, The Finite Element Immersed Boundary Method with Distributed Lagrange multiplier, SIAM Journal on Numerical Analysis 53(6) (2015), 2584–2604.
2. D. Boffi, L. Gastaldi, A fictitious domain approach with distributed Lagrange multiplier for fluid-structure interactions, Numerische Mathematik 135(3) (2017), 711–732.
3. D. Boffi, L. Gastaldi, L. Heltai, A distributed Lagrange formulation of the Finite Element Immersed Boundary Method for fluids interacting with compressible solids, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications, D. Boffi, L. Pavarino, G. Rozza, S. Sacchi, C. Vergara eds., SEMA SIMAI Springer Series 16 (2018), 1–21.
4. D. Boffi, L Gastaldi, S. Wolf, Higher order time stepping schemes for fluid-structure interaction problems, in preparation.

Félix del Teso, BCAM, Spain

A UNIFIED PDE/NUMERICAL FRAMEWORK FOR NONLOCAL AND LOCAL EQUATIONS OF POROUS MEDIUM TYPE

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 3.21.

Abstract

We consider the following problem of porous medium type: $\left\{ \begin{split} \partial_t u(x,t)-(L^\sigma +\mathcal{L}^{\mu})\left[\varphi(u)\right](x,t)=f(x,t), \qquad & (x,t)\in \mathbb{R}^N\times (0,\infty),\\ u(x,t)=u_0(x),\hspace{5cm}& x\in \mathbb{R}^N, \end{split} \right.$ where $$\varphi: \mathbb{R}^N\to\mathbb{R}$$ is continuous and nondecreasing, and $\begin{split} L^\sigma[v](x)&=\text{Tr}\left(\sigma\sigma^T D^2v(x)\right), \hspace{2.8cm} \textbf{(local diffusion)}\\ \mathcal{L}^\mu[v](x)&=\text{P.V.} \int_{|z|>0}\left(v(x+z)-v(x)\right)d\mu(z), \textbf{ (nonlocal diffusion)} \end{split}$ with $$\sigma\in \mathbb{R}^{N\times p}$$ and $$\mu$$ symmetric measure s.t. $$\int \min\{|z|^2,1\}d\mu(z)<+\infty$$.

We will present a general overview of some of the results obtained in collaboration with J. Endal and E.R. Jakobsen:

$$\bullet$$ Uniqueness of distributional solutions.

$$\bullet$$ Continuous dependence on $$L^\sigma +\mathcal{L}^{\mu}$$, $$\varphi$$ and $$u_0$$.

$$\bullet$$ Unified theory of monotone numerical schemes of finite difference type. Here we use the fact that operators in the class of $$\mathcal{L}^\mu$$ includes discretizations of $$L^\sigma+\mathcal{L}^\mu$$. This fact allows us to use a pure PDE approach.

$$\bullet$$ We also propose a branch of discretizations and schemes and analyze their accuracy.

$$\bullet$$ As a consequence of numerics, we obtain existence of distributional solutions together with interesting properties like $$L^1$$-contraction, $$C([0,T],L^1_{\text{loc}}(\mathbb{R}^N))$$ regularity, energy estimates, ...

Martin Eigel, WIAS, Germany

A STATISTICAL LEARNING APPROACH FOR PARAMETRIC PDES

Joint work with Reinhold Schneider and Philipp Trunschke

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 3.21.

Abstract

Parametric PDEs (as encountered in the popular field of Uncertainty Quantification) are computationally complex due to the high dimensionality of the models describing random data. Common numerical approaches are Monte Carlo methods for statistical quantities of interest and functional approximations, representing the entire solution manifold in some function space. Assuming sufficient regularity (or sparsity), the latter attain high theoretical convergence rates. In practice, this can be realised e.g. by employing some kind of (a posteriori) error control in the computations. However, the implementation usually is non-trivial and does not generalise easily.

We examine a non-intrusive "Variational Monte Carlo'' (VMC) method based on statistical learning theory. This provides a combination of deterministic and statistical convergence results. The Galerkin solution can be computed with high probability using a tensor recovery algorithm on a training set of generated solution realisations. The representation in efficient hierarchical tensor formats tames the "curse of dimensionality''. Similarly, a residual a posteriori error estimator can be reconstructed easily, steering all discretisation parameters.

References

1. M. Eigel, R. Schneider, P. Trunschke, S. Wolf, Variational Monte Carlo - Bridging Concepts of Machine Learning and High Dimensional PDEs, Advances in Comp. Math., (to be published 2019).

Juan Galvis, National University of Colombia, Colombia

ROBUST SOLVERS FOR HIGH-CONTRAST MULTISCALE PROBLEMS

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 3.21.

Abstract

We present some recent developments in the numerical approximation of elliptic partial differential equations with high-contrast multiscale coefficients. In particular we review recently introduced robust upscaling technique known as the generalized multiscale finite element method (GMsFEM). We also present the design of robust two-levels domain decomposition methods that use the GMsFEM method as a second level. In order to show the benefits of using the proposed methodology several appliations are cosidered: two-phase flow in high-contrast multiscale porous media, the free boundary dam problem in heterogeneous media and an elasticity problem in topology optimization.

References

1. J. Galvis, C. Vásquez, L.F. Contreras, Numerical upscaling of the free boundary dam problem in multiscale high-contrast media, Submitted. 2019.
2. E. Abreu, C. Diaz, J. Galvis, A convergence analysis of Generalized Multiscale Finite Element Methods, 1990.
3. B. Lazarov, S. Serrano, M. Zambrano, J. Galvis, Fast multiscale contrast independent preconditioner for linear elastic topology optimization problems, in preparation, 2019.
4. Y. Efendiev, J. Galvis, T.Y. Hou, Generalized multiscale finite element methods (GMsFEM), Journal of Computational Physics 251 (2013), 116-135.

Thirupathi Gudi, Indian Institute of Science, India

APPROXIMATION OF DIRICHLET BOUNDARY CONTROL PROBLEM USING ENERGY SPACES

Joint work with Sudipto Chowdhury and Akambadath K. Nandakumaran

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 3.21.

Abstract

In this talk, we present an alternative energy space based approach for formulating the Dirichlet boundary control problem and then propose a finite element based numerical method for approximating its solution numerically. A priori error estimates of optimal order in the energy norm as well as in the $$L_2$$ norm will be discussed. Furthermore, we discuss on deriving a reliable and efficient a posteriori error estimator using an auxiliary problem for adaptive mesh refinement. The theoretical results will be illustrated by some numerical experiments.

References

1. S. Chowdhury, T. Gudi, A.K. Nandakumaran, Error bounds for a Dirichlet boundary control problem based on energy spaces, Journal Math. Comp. 86 (2017), 1103-1126.

Johnny Guzmán, Brown University, USA

SMOOTH FINITE ELEMENT EXACT SEQUENCE ON POWELL-SABIN SPLITS

Joint work with Anna Lischke and Michael Neilan

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 3.21.

Abstract

Starting with  $$C^1$$ finite elements on a Powell-Sabin split we show how to construct a de Rham exact sequence. The lowest order FEM in our family of FEM is piecewise quadratics. An interesting feature in the exact sequence is that Powell-Sabin splits introduce  singular vertices and naturally we have to constrain the space of 2-forms in two dimensions. We will indicate possible generalization to higher dimensions.

Helmut Harbrecht, University of Basel, Switzerland

ANALYSIS OF TENSOR APPROXIMATION SCHEMES FOR CONTINUOUS FUNCTIONS

Joint work with Michael Griebel

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 3.21.

Abstract

This talk is concerned with the analysis of tensor approximation schemes for continuous functions in high dimensions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate weights.

References

1. M. Griebel, H. Harbrecht, Analysis of tensor approximation schemes for continuous functions, arXiv:1509.09058, (2019).

Patrick Henning, Royal Institute of Technology (KTH), Sweden

THE NUMERICAL SOLUTION OF NONLINEAR SCHRÖDINGER EQUATIONS WITH APPLICATIONS TO SUPERFLUIDITY

Joint work with Daniel Peterseim and Johan Wärnegård

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 3.21.

Abstract

In this talk we consider the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass- and energy-conserving variant of the Crank-Nicolson method in time. The usage of finite elements becomes necessary if the equation contains terms that dramatically reduce the overall regularity of the exact solution. Examples of such terms are rough potentials or disorder potentials as appearing in many physical applications. We present some analytical results that show how the reduced regularity of the exact solution could affect the expected convergence rates and how it results in possible coupling conditions between the spatial mesh size and the time step size. We will also demonstrate the importance of numerical energy-conservation in applications with low-regularity by simulating the phase transition of a Mott insulator into a superfluid.

References

1. P. Henning, D. Peterseim, Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials, M3AS Math. Models Methods Appl. Sci. 27(11) (2017), 2147-2184.
2. P. Henning, J. Wärnegård, Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation, ArXiv e-print 1804.10547, 2018.
3. P. Henning, D. Peterseim, Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency, ArXiv e-print 1812.00835, 2018.

Helge Holden, Norwegian University of Science and Technology, Norway

ON TRAFFIC MODELING AND THE BRAESS PARADOX

Joint work with Nils Henrik Risebro and Rinaldo Colombo

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 3.21.

Abstract

We will discuss models for vehicular traffic flow on networks. The models include both the Lighthill-Whitham-Richards (LWR) model and Follow-the-Leader (FtL) models. Emphasis will be on the Braess paradox [1] in which adding a road to a traffic network can make travel times worse for all drivers, and we will show one way of studying the Braess paradox with an LWR model [2].

Furthermore, we will show how we can consider the FtL model as a discretization of the LWR model [3, 4]. Finally, we will also discuss a novel model for multi-lane traffic [5].

References

1. D. Braess, Über ein Paradoxon aus der Verkehrsplanung, Unternehmensforschung 12 (1968), 258-268.
2. R. Colombo, H. Holden, On the Braess paradox with nonlinear dynamics and control theory, J. Optimization Theory and Appl. 168 (2016), 216-230.
3. H. Holden, N.H. Risebro, Continuum limit of Follow-the-Leader models - a short proof, Discrete Contin. Dyn. Syst. 38 (2018), 715-722.
4. H. Holden, N.H. Risebro, Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, Networks & Heterogeneous Media 13 (2018), 409-421.
5. H. Holden, N.H. Risebro, Models for dense multilane vehicular traffic, arXiv:1812.01361.

Yanghong Huang, University of Manchester, UK

SPECTRAL RELATIONS OF THE GENERATOR TO ALPHA-STABLE PROCESSES AND RELATED SPECTRAL METHODS

Joint work with Xiao Wang

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 3.21.

Abstract

The majority of special functions and orthogonal polynomials are well-known to be associated with second order differential equations arising from mathematical physics, and are usually have to be extended to Merjie G function or Fox H function. In this talk, classical Jacobi polynomials are shown to establish spectral relations of the generator of alpha-stable processes, generalising the fractional Laplacian in one dimension to the non-symmetric case. The resulting spectral relations will be used to characterised the singularity near the boundary and the regularity of the solution to the Dirichlet problem, together with the development of a higher order spectral methods.

Bangti Jin, University College London, UK

TIME-STEPPING SCHEMES FOR FRACTIONAL DIFFUSION

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 3.21.

Abstract

Overall the last decade, a large number of time stepping schemes have been developed for time-fractional diffusion problems. These schemes can be generally divided into: finite difference type, convolution quadrature type and discontinuous Galerkin methods. Many of these methods are developed by assuming that the solution is sufficiently smooth, which however is generally not true. In this talk, I will describe recent works in analyzing and developing robust numerical schemes that do not assume solution regularity directly, but only data regularity.

Theodoros Katsaounis, KAUST, Saudi Arabia & University of Crete, Greece

A POSTERIORI ERROR CONTROL AND ADAPTIVITY FOR SCHRÖDINGER EQUATIONS

Joint work with Irene Kyza

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 3.21.

Abstract

In this talk I will present some recent results on aposteriori error estimation for linear and nonlinear Schrodinger equations. We use finite element discretizations and the Crank Nicolson time stepping scheme. For the derivation of the estimates we use the reconstruction technique and linear and nonlinear stability arguments as in the continuous problem. Based on these aposteriori estimators we further design and analyse a time-space adaptive algorithm. Various numerical experiments verify and complement our theoretical results.

Piotr Krzyżanowski, University of Warsaw, Poland

ROBUST PRECONDITIONING FOR DISCONTINUOUS GALERKIN DICRETIZATIONS OF DIFUSSION PROBLEM WITH HIGH CONTRAST COEFFICIENTS

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 3.21.

Abstract

We consider a diffusion problem in a heterogeneous medium, with prescribed transmission properties. We discuss preconditioners for iterative solutions of algebraic systems arising from problem discretizations of discontinuous Galerkin type.

In particular, a diffusion problem through a thin membrane is discussed. A nonoverlapping domain decomposition based preconditioner is introduced, and its convergence properties are discussed and verified in numerical experiments. In particular, the convergence rate is shown independent of the contrast in the diffusion coefficient, the number of inclusions and of the transmission parameter as well.

Shingyu Leung, Hong Kong University of Science and Technology, Hong Kong

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS ON MANIFOLDS AND POINT CLOUDS

Joint work with Hongkai Zhao, Meng Wang, and Ningchen Ying

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 3.21.

Abstract

We present recent numerical methods for solving partial differential equations on manifolds and point clouds. In the first part of the talk, we introduce a new and simple discretization, named the Modified Virtual Grid Difference (MVGD), for numerical approximation of the Laplace-Beltrami operator on manifolds sampled by point clouds. We first introduce a local virtual grid with a scale adapted to the sampling density centered at each point. Then we propose a modified finite difference scheme on the virtual grid to discretize the LB operator. The new discretization provides more diagonal dominance to the resulting linear system and improves its conditioning. In the second part, we present a local regularized least squares radial basis function (RLS-RBF) method for solving partial differential equations on irregular domains or on manifolds. The idea extends the standard RBF method by replacing the interpolation in the reconstruction with the least squares fitting approximation.

References

1. M. Wang, S. Leung and H. Zhao, Modified Virtual Grid Difference for Discretizing the Laplace-Beltrami Operator on Point Clouds, SIAM J. Sci. Comput. 40 (2018).
2. N. Ying and S. Leung, A Local Regularized Least Squares Radial Basis Function (RLS-RBF) Method for Differential Operators in Meshless Domain, in preparation.

Michael Multerer, USI Lugano, Switzerland

MULTILEVEL QUADRATURE FOR ELLIPTIC PARAMETRIC PARTIAL DIFFERENTIAL EQUATIONS

Joint work with Michael Griebel and Helmut Harbrecht

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 3.21.

Abstract

Multilevel quadrature methods for parametric operator equations such as the multilevel (quasi-) Monte Carlo method are closely related to the sparse tensor product approximation between the spatial variable and the parameter. In this presentation, we employ this fact and reverse the multilevel quadrature method via the sparse grid construction by applying differences of quadrature rules to finite element discretizations of increasing resolution. Besides being algorithmically more efficient if the underlying quadrature rules are nested, this way of performing the sparse tensor product approximation enables the easy use of non-nested and even adaptively refined finite element meshes. Especially, we present an error and regularity analysis of the fully discrete solution, taking into account the effect of polygonal approximations to a curved physical domain and the numerical approximation of the bilinear form. Numerical results in three spatial dimensions are provided to illustrate the approach.

References

1. M. Griebel, H. Harbrecht and M. Multerer, Multilevel quadrature for elliptic parametric partial differential equations in case of polygonal approximations of curved domains, arXiv:1509.09058, 2018.

Gulcin M. Muslu, Istanbul Technical University, Turkey

THE GENERALIZED FRACTIONAL BENJAMIN-BONA-MAHONY EQUATION: ANALYTICAL AND NUMERICAL RESULTS

Joint work with Goksu Oruc and Handan Borluk

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 3.21.

Abstract

The generalized fractional Benjamin-Bona-Mahony (gfBBM) equation models the propagation of small amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. The equation involves two fractional terms unlike the well-known fBBM equation. In this talk, we present the local existence and uniqueness of the solutions for the Cauchy problem. The sufficient conditions for the existence of solitary wave solutions are discussed. The Petviashvili method is proposed for the generation of the solitary wave solutions and their evolution in time is investigated by Fourier spectral method, numerically. The efficiency of the numerical methods is tested and the relation between nonlinearity and fractional dispersion will be presented by various numerical experiments.

This work was supported by Research Fund of the Istanbul Technical University. Project Number:42257.

Michael Neilan, University of Pittsburgh, USA

EXACT SEQUENCES OF PIECEWISE POLYNOMIALS ON ALFELD SPLITS

Joint work with Guosheng Fu and Johnny Guzmán

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 3.21.

Abstract

We develop exact polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an $$n$$-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.

Alexander Ostermann, University of Innsbruck, Austria

LOW-REGULARITY INTEGRATORS FOR NONLINEAR SCHRÖDINGER EQUATIONS

Joint work with Frédéric Rousset, Marvin Knöller and Katharina Schratz

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 3.21.

Abstract

Nonlinear Schrödinger equations are usually solved by pseudo-spectral methods, where the time integration is performed by splitting schemes or exponential integrators. Notwithstanding the benefits of this approach, its successful application requires additional regularity of the solution. For instance, second-order Strang splitting requires four additional derivatives for the solution of the cubic nonlinear Schrödinger equation. Similar statements can be made about other dispersive equations like the Korteweg–de Vries or the Boussinesq equation.

In this talk, we introduce as an alternative low-regularity Fourier integrators. They are obtained from Duhamel’s formula in the following way: first, a Lawson-type transformation eliminates the leading linear term and second, the dominant nonlinear terms are integrated exactly in Fourier space. For cubic nonlinear Schrödinger equations, first-order convergence of such methods only requires the boundedness of one additional derivative of the solution, and second-order convergence the boundedness of two derivatives. For details, see [1, 2].

Moreover, a filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation is presented. This scheme has better convergence rates at low regularity than any known scheme in the literature so far. To prove this superior error behavior, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of $$L^2$$. We are able to establish a global error estimate in $$L^2$$ for $$H^1$$ solutions, which is roughly of order $$\tau^{ \frac12 + \frac{5-d}{12} }$$ in dimension $$d \leq 3$$ with $$\tau$$ denoting the time step size. For details, see [3].

References

1. M. Knöller, A. Ostermann, K. Schratz, A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data, Preprint, arXiv:1807.01254, to appear in SIAM J. Numer. Anal. (2019).
2. A. Ostermann, K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math. 18 (2018), 731-755.
3. A. Ostermann, F. Rousset, K. Schratz, Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity, Preprint, arXiv:1902.06779 (2019).

David Pardo, University of the Basque Country & BCAM, Spain

EFFECTIVE COMPRESSIONAL WAVE VELOCITY ESTIMATION FOR POROUS ROCKS

Joint work with Ángel Javier Omella, Julen Alvarez-Aramberri, Magdalena Strugaru, Vincent Darrigrand, Carlos Santos, and Héctor González

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 3.21.

Abstract

In geophysics, it is of paramount importance to characterize the effective compressional wave velocity of the Earth's crust layers. In this work, we propose a set of numerical methods and techniques to estimate the effective compressional wave velocities of highly heterogeneous porous rocks along the entire frequency spectrum [1, 2]. To do so, we incorporate the internal structure of the rock at the pore scale and the properties of each of its constituents (density and primary wave velocity). For the low/medium frequency spectrum, we solve the acoustic equation in the frequency domain by the Finite Element Method (FEM), and we postprocess the solution along straight lines to estimate the homogenized compressional wave velocity. To obtain accurate results, we show the necessity to extend the domain by repeating the rock sufficient times with respect to the excitation frequency. Due to this requirement on the computational domain size, we consider non-fitting meshes [3], in which each finite element includes highly-discontinuous material properties. The use of non-fitting meshes allows us to reduce the number of degrees of freedom with respect to the use of traditional conforming fitting meshes. We take advantage of having to repeat the rock when precomputing blocks of the stiffness matrix to reduce the computational cost. At high frequencies, we solve the Eikonal equation by the Fast Marching Method (FMM) [4] to estimate the effective compressional wave velocity. The performance of the proposed methods are illustrated with different numerical experiments over synthetic and real porous rocks where the formations are provided by X-ray micro computed tomography.

References

1. Á.J. Omella, J. Alvarez-Aramberri, M.Strugaru, V. Darrigrand, D. Pardo, H. González, C. Santos, A Simulation Method for the Computation of the Effective P-Wave Velocity in Porous Rocks. Part 1: 1D Analysis, Submitted to Computational Geosciences (2019).
2. Á.J. Omella, M.Strugaru, J. Alvarez-Aramberri, V. Darrigrand, D. Pardo, H. González, C. Santos, A Simulation Method for the Computation of the Effective P-Wave Velocity in Porous Rocks. Part 2: 2D and 3D Analysis, Submitted to Computational Geosciences (2019).
3. T. Chaumont-Frelet, D. Pardo, Á. Rodríguez-Rozas, Finite element simulations of logging-while-drilling and extra-deep azimuthal resistivity measurements using non-fitting grids, Computational Geosciences 22(5) (2018), 1161-1174.
4. J.A. Sethian, A.M. Popovici, 3-D traveltime computation using the fast marching method, Geophysics 64 (1999), 516-523.

Daniel Peterseim, Universität Augsburg, Germany

SPARSE COMPRESSION OF EXPECTED SOLUTION OPERATORS

Joint work with Michael Feischl

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 3.21.

Abstract

We show that the expected solution operator of a prototypical linear elliptic partial differential operator with random diffusion coefficient is well approximated by a computable sparse matrix. This result holds true without structural assumptions on the random coefficient such as stationarity, ergodicity or any characteristic length of correlation. The constructive proof is based localized orthogonal multiresolution decompositions of the solution space for each realization of the random coefficient. The decompositions lead to a block-diagonal representation of the random operator with well-conditioned sparse blocks. Hence, an approximate inversion is achieved by a few steps of some standard iterative solver. The resulting approximate solution operator can be reinterpreted in terms of classical Haar wavelets without loss of sparsity. The expectation of the Haar representation can be computed without difficulty using appropriate sampling techniques. The overall construction leads to a computationally efficient method for the direct approximation of the expected solution operator which is relevant for stochastic homogenization and uncertainty quantification.

References

1. M. Feischl, D. Peterseim, Sparse Compression of Expected Solution Operators, ArXiv e-prints 1807.01741 (2018), 1-18.

Marcus Sarkis, Worcester Polytechnic Institute, USA

ROBUST MODEL REDUCTIONS FOR DARCY EQUATIONS WITH HIGH-CONTRAST COEFFICIENTS

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 3.21.

Abstract

Major progress has been made recently to make preconditioners robust with respect to variation of coefficients. A reason for this success is the adaptive selection of primal constraints based on localized generalized eigenvalue problems. In this talk we discuss how to transfer this technique to the field of discretizations. Given a target accuracy, we design a robust model reduction by delocalizing multiscale basis functions and establish a priori energy error estimates with such target accuracy with hidden constants independently of the coefficients.

Katharina Schratz, Heriot-Watt University, UK

NONLINEAR FOURIER INTEGRATORS FOR DISPERSIVE EQUATIONS

Joint work with Alexander Ostermann and Frédéric Rousset

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 3.21.

Abstract

A large toolbox of numerical schemes for nonlinear dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene since the underlying PDEs have very complicated solutions exhibiting high oscillations and loss of regularity. This leads to huge errors, massive computational costs and ultimately provokes the failure of classical schemes. Nevertheless, non-smooth phenomena play a fundamental role in modern physical modeling (e.g., blow-up phenomena, turbulences, high frequencies, low dispersion limits, etc.) which makes it an essential task to develop suitable numerical schemes. In this talk I present a new class of nonlinear Fourier integrators which offer strong geometric structure at low regularity and high oscillations. The key idea in the construction of the new schemes is to tackle and hardwire the underlying structure of resonances into the numerical discretization – linking the finite dimensional discretization to powerful existence results of nonlinear dispersive PDEs in low regularity spaces.

Christos Xenophontos, University of Cyprus, Cyprus

APPROXIMATION OF FOURTH ORDER TWO-POINT SINGULARLY PERTURBED PROBLEMS OF REACTION-DIFFUSION TYPE

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 3.21.

Abstract

We consider fourth order two-point singularly perturbed problems of reaction-diffusion type and the approximation of their solution by Galerkin's method. We consider both $$hp$$ Finite Elements (FEs) and Isogeometric Analysis (IGA). We first present regularity results which show that the solution may be decomposed into a smooth part, two boundary layers at the endpoints and a (negligible) remainder. Estimates for each part in the decomposition are obtained, which are explicit in the order of differentiation and the singular perturbation parameter [1]. Guided by these results, we construct an approximation using the so-called Spectral Boundary Layer mesh in FEs [2] and knot-vector in IGA [3], which converges independently of the singular perturbation parameter. When the error is measured in the energy norm associated with the problem, the convergence rate is exponential, as the degree of the approximating polynomials is increased. Numerical examples illustrating the theory will also be presented.

References

1. P. Constantinou, $$hp$$ Finite Element Methods for Fourth Order Singularly Perturbed Problems, Ph.D. Dissertation, Department of Mathematics and Statistics, University of Cyprus, 2019.
2. J.M. Melenk, C. Xenophontos, L. Oberbroeckling, Robust exponential convergence of hp-FEM for singularly perturbed systems of reaction-diffusion equations with multiple scales, IMA J. Num. Anal. 33 (2013), 609-628.
3. K. Liotati, C. Xenophontos, Isogeometric Analysis for singularly perturbed problems in 1-D: a numerical study, BAIL2018 Conference (Glascow, Scotland, 2018), 2019 (to appear).

Dirk Blömker, Universität Augsburg, Germany

STOCHASTIC INTERFACE MOTION IN THE CAHN-HILLIARD EQUATION

Joint work with Dimitra Antonopoulou and Georgia Karali

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 2.18.

Abstract

We study the two and three dimensional stochastic Cahn–Hilliard equation in the sharp interface limit. It is given by\begin{equation*}\partial_t u = -\Delta v+ \varepsilon^\sigma\partial_t {W},\\v = -\displaystyle\frac{F^{\prime}(u)}{\varepsilon}+\varepsilon \Delta u,\end{equation*}subject to Neumann boundary conditions on a bounded domain $$\mathcal{D}$$.

Here $$u: \mathcal{D}\times[0,T]\to [-1,1]$$ is the scalar concentration field of one of the components in a separation process,for example of binary alloys. The function $$F$$ is a double well potential, for example with $$F'(u)=u-u^3$$. The noise is given by a spatially smooth Wiener process $$W$$.The small parameter $$\varepsilon>0$$ measures the width of transition layers generated during phase separation.

Using formal asymptotic expansions, in $$[1]$$ we identify the limit. In the case $$\sigma=1$$ our results indicate that the stochastic Cahn–Hilliard equation converges to a two-phase Hele-Shaw (or Mullins-Sekerka) problem with stochastic forcing on the transition layers. For the interface $$\Gamma(t)=\{ x\in\mathcal{D}\ :\ u(t,x)=0\}$$ in the limit $$\varepsilon\to0$$ we obtain \begin{equation*}\left\{\begin{aligned}&\Delta v=0 \,\, \mbox{ in } \mathcal{D} \backslash\Gamma,\\&\partial_n v=0 \,\, \mbox{ on }\partial\mathcal{D},\\&v=\lambda H+ W\,\, \mbox{ on } \Gamma,\\&V= [\partial_n v] \,\, \mbox{ on } \Gamma,\end{aligned}\right.\end{equation*}where $$H$$ is the mean curvature and $$V$$ the normal velocity of $$\Gamma$$.The jump term $$[\partial_n v]$$ denotes the average of the normal derivative of $$v$$ from both sides of $$\Gamma$$.

In a joint work with S. Yokoyama (Tokyo) we show that the stochastic Hele-Shaw problem has a local smooth solution,given that the initial surface $$\Gamma(0)$$ is a smooth closed hypersurface that does not touch the boundary of $$\mathcal{D}$$.

In the case when the noise is sufficiently small (i.e. $$\sigma>1$$ sufficiently large), in $$[1]$$ we can prove rigorously that the limit is a deterministic Hele-Shaw problem without $$W$$. The main reason for $$\sigma$$ being very large for this result is due to the lack of sufficiently good spectral estimates for the linearized Cahn-Hilliard operator. These are currently only verified in $$H^{-1}$$.

References

1. D.C. Antonopoulou, D. Blömker, G.D. Karali, The sharp interface limit for the stochastic Cahn–Hilliard equation, Ann. Inst. H. Poincaré Probab. Statist. 54(1) (2018), 280-298.

Ajay Chandra, Imperial College London, UK

STOCHASTIC NON-ABELIAN YANG-MILLS HEAT FLOW

Joint work with Martin Hairer and Hao Shen

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 2.18.

Abstract

I will start by describing how a mathematician can think of the problem of constructing a Yang-Mills quantum field theory and how one approach to this problem involves working with singular SPDE. I will then describe results obtained in ongoing joint work with Martin Hairer and Hao Shen regarding local existence and gauge covariance for a singular SPDE that should correspond to a non-Abelian Yang-Mills quantum field theory.

Jean-François Chassagneux, Université Paris Diderot, France

WEAK ERROR EXPANSION FOR MEAN-FIELD SDE

Joint work with Lukasz Szpruch and Alvin Tse

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 2.18.

Abstract

In this work, we study the weak approximation error by particle system of Mean Field SDE. We prove an expansion of this error in terms of the number of particle. Our strategy of proof follows the approach of Talay-Tubaro for weak approximation of SDE by an Euler Scheme. We thus consider a PDE on the Wasserstein space (called the Master Equation in mean-field games literature) and, relying on smoothness properties of the solution, obtain our expansion. We also prove the required smoothness properties under sufficient conditions on the coefficient function.

François Delarue, Université Nice-Sophia-Antipolis, France

MEASURE-VALUED DIFFUSION PROCESSES AND MEAN FIELD GAMES

Joint work with Alekos Cecchin

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 2.18.

Abstract

The purpose of the talk is to address smoothing effect of diffusion processes with values in the space of probability measures, especially when the latter is constructed above a finite set. The motivation comes from the theory of mean field games and of mean field control, which is dedicated to the analysis of equilibria within large population of rational agents and which has been growing fast since the earlier works of Lasry and Lions [3, 4] (see also the recent monographs [1,2]). One key fact is that such equilibria may be described by a stochastic measure-valued process when the whole population is subjected to a common source of noise. In this framework, equilibria turn out to be unique if the common noise induces sufficiently strong regularizing properties onto the space of probability measures. While the latter mostly regards the influence of a common noise onto the equilibria, it also raises interesting questions on the case without common noise: We here show that, by letting the influence of the common noise tend to zero, we may select, in some cases (known as potential cases), some specific equilibria among all the possible ones.

References

1. R. Carmona and F. Delarue, robabilistic Theory of Mean Field Games: vol. I, Mean Field FBSDEs, Control, and Games, Stochastic Analysis and Applications. Springer Verlag, 2018.
2. R. Carmona and F. Delarue, robabilistic Theory of Mean Field Games: vol. II, Mean Field Games with Common Noise and Master Equations, Stochastic Analysis and Applications. Springer Verlag, 2018.
3. J.M. Lasry and P.L. Lions, Jeux à champ moyen I. Le cas stationnaire, Comptes Rendus de l’Académie des Sciences de Paris, ser. A 343(9) (2006).
4. J.M. Lasry and P.L. Lions, Jeux à champ moyen II. Horizon fini et contrôle optimal, Comptes Rendus de l’Académie des Sciences de Paris, ser. A 343(10) (2006).

Peter K. Friz, Technische Universität Berlin & WIAS, Germany

ON ROUGH SEMIMARTINGALES

Joint work with K. Le, A. Hocquet, and P. Zorin-Kranich

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 2.18.

Abstract

We introduce the new class of rough semimartingales (RSM) with motivation from filtering and SPDE theory. Under natural assumptions, RSM have a unique decomposition. Moreover, RSM are stable under composition with regular functions and stochastic / rough integration. RSM provide further a natural framework to study classes of Markov processes (which are not semimartingales) and we introduce the rough martingale problem.

Paul Gassiat, CEREMADE, Université Paris-Dauphine, France

ASYMPTOTIC FORMULAE IN ROUGH VOLATILITY MODELS

Joint work with Peter Friz and Paolo Pigato

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 2.18.

Abstract

Stochastic volatility models where the volatility behaves similarly to a fractional Brownian motion of Hurst index $$H < 1/2$$ ("rough volatility") have recently been the subject of considerable interest from the mathematical finance community, due to their ability to reproduce important features observed in market prices. In this talk I will present a result on asymptotics of short-dated call option prices in such models. The proof is based on combining the Laplace method on Wiener space with rough path type techniques.

Matthew Griffiths, King’s College London, UK

CYLINDRICAL LÉVY PROCESSES AND LÉVY SPACE-TIME WHITE NOISES

Joint work with Markus Riedle

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 2.18.

Abstract

It is well known (e.g. [1]) that the canonical cylindrical Brownian motion and the Gaussian space-time white noise correspond to each other. In this talk we consider the analogue relation between cylindrical Lévy processes and Lévy space-time white noises. Since there does not exist a “canonical” cylindrical Lévy process the situation is quite different from the Gaussian case. We then apply the established relations by embedding cylindrical Lévy processes in certain Besov spaces, which may be seen as a first result analysing the regular (or irregular) behaviour of the jumps of a cylindrical Lévy process.

References

1. G. Kallianpur, J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, Lecture Notes-Monographs 26 Institute of Mathematical Statistics, Hayward, California, 1995.

Erika Hausenblas, Montanuniversität Leoben, Austria

A SCHAUDER-TYCHANOFF TYPE STOCHASTIC FIXPOINT THEOREM AND A COUPLED STOCHASTIC SYSTEM FOR PATTERN FORMATION

Joint work with Mechthild Thalhammer, Tsiry Avisoa Randrianasolo, and Jonas Toelle

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 2.18.

Abstract

Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in a specific situation, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray–Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear.

In the derivation of a macroscopic model such as the deterministic Gray–Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g. fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations.

In the talk, we first present a stochastic Schauder-Tychanoff type Theorem, then we present as an application the existence of solution of the stochastic Gray-Scott system. Finally, we present some numerical results of the stochastic Gray–Scott equations driven by independent spatially time-homogeneous Wiener processes. The numerical simulations based on the application of a time-adaptive first-order operator splitting method and the fast Fourier transform illustrate the formation of patterns in the deterministic case and their variation under the influence of stochastic noise.

Jean-François Jabir, National Research University Higher School of Economics, Russia

LAGRANGIAN STOCHASTIC MODELS FOR TURBULENT FLOWS AND RELATED PROBLEMS

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 2.18.

Abstract

Lagrangian stochastic models for turbulence define a particular family of Langevin stochastic differential equations, endowing some specific nonlinearities of McKean type, that were originally introduced in the framework of computational fluid dynamics to describe and simulate the motions of a generic particle of a fluid flow. Although these stochastic models are currently applied in various engineering problems, Lagrangian stochastic models for turbulent flows display a certain number of original mathematical problems broadly linked to existence and uniqueness problems for singular McKean-Vlasov dynamics and the validation of related particle approximations; the modeling of boundary conditions for Langevin models; the introduction of distributions constraints ... The first part of the talk will be dedicated to a short presentation of practical interest and the characteristic theoretical problems related to these Lagrangian stochastic models while some resolutions to these problems, in simplified situations, will be discussed in the rest of the talk.

References

1. M. Bossy, J.-F. Jabir, D. Talay, n conditional McKean Lagrangian stochastic models, Probab. Theory Related Fields 151(1-2) (2011), 319-351.
2. M. Bossy, J.-F. Jabir, Lagrangian stochastic models with specular boundary condition, Journal of Functional Analysis 268(6) (2015), 1309-1381.
3. M. Bossy, J. Fontbona, J.-F. Jabir, P.-E. Jabin, Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain, Communications in Partial Differential Equations 38(7) (2013), 1141-1182.
4. M. Bossy, J.-F. Jabir, Particle approximation for Lagrangian Stochastic Models with specular boundary condition, Electron. Commun. Probab. 23 (2018), 1-14.

Kristin Kirchner, ETH Zürich, Switzerland

CONVERGENCE OF GALERKIN APPROXIMATIONS FOR FRACTIONAL STOCHASTIC PDES

Joint work with David Bolin, Mihály Kovács, and Sonja G. Cox

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-8, room 2.18.

Abstract

Many models in spatial statistics are based on Gaussian Matérn fields. Motivated by the relation between this class of Gaussian random fields and stochastic partial differential equations (PDEs), we consider the numerical solution of stochastic PDEs with additive spatial white noise on a bounded Euclidean domain $$\mathcal{D}\subset\mathbb{R}^d$$. The non-local differential operator is given by the fractional power $$\mathcal{A}^\beta$$, $$\beta \gt 0$$, of a second-order elliptic differential operator $$\mathcal{A}$$.

We propose an approximation which combines recent Galerkin techniques for deterministic fractional-order PDEs with an efficient way to simulate white noise. Under minimal regularity assumptions on the differential operator $$\mathcal{A}$$, in [1, 2, 3] we perform an error analysis for this approximation showing (i) strong mean-square convergence in $$L_2(\mathcal{D})$$, (ii) weak convergence, and (iii) convergence in Sobolev spaces: for the approximation of the random field in $$L_q(\Omega;H^s(\mathcal{D}))$$, where $$q\in(0,\infty)$$ and $$s\in[0,1]$$, as well as for the covariance function of the approximation in the mixed Sobolev space $$H^{s,s}(\mathcal{D}\times\mathcal{D})$$ at explicit and sharp rates.

For the motivating example of Gaussian Matérn fields, where $$\mathcal{A} =-\Delta + \kappa^2$$ and $$\kappa \equiv \operatorname{const.}$$, we perform several numerical experiments for various values of the fractional exponent $$\beta\gt 0$$ in dimensions $$d\in\{1,2\}$$, which attest the theoretical results.

References

1. D. Bolin, K. Kirchner, M. Kovács, Numerical solution of fractional elliptic stochastic PDEs with spatial white noise, IMA J. Numer. Anal., electronic (2018), n/a–n/a.
2. D. Bolin, K. Kirchner, M. Kovács, Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise, BIT Numer. Math. 58 (2018), 881–906.
3. S.G. Cox, K. Kirchner, Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields, Preprint, arXiv:1904.06569, 2019.

Tomasz Klimsiak, Nicolaus Copernicus University in Toruń, Poland

ASYMPTOTICS FOR LOGISTIC-TYPE EQUATIONS WITH DIRICHLET FRACTIONAL LAPLACIAN

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 2.18.

Abstract

We will present results on the asymptotics, as $$t\rightarrow \infty$$ and $$p\rightarrow \infty$$, of solutions of the following problem: $\frac{\partial u}{\partial t}-(\Delta^{\alpha/2})_{|D}u=a u-bu^p,\quad u(0,\cdot)=\varphi.\quad(*)$ Here $$D$$ is a bounded Lipschitz domain in $$\mathbb R^d$$, $$a>0, p>1, \alpha\in (0,2)$$, and $$b,\varphi$$ are bounded positive nontrivial Borel functions on $$D$$. We show that for suitable $$a$$ the limit function does not depend on the order of limits and is a unique solution of an obstacle problem.

Equations and systems of type $$(*)$$ with nonlocal operators appear in many models of population biology. Dirichlet fractional Laplacian in $$(*)$$ is designed to describe nonlocal dispersal strategy of animals (see [3]). Stationary equations of type $$(*)$$ have been studied recently in [1] with strictly positive $$b$$.

Asymptotics as $$p\rightarrow \infty$$ for solutions to stationary counterpart to $$(*)$$ with classical Laplacian was studied for the first time in [2]. In this paper it is observed that for large $$p$$ solutions of this stationary problem behave like solutions of certain steady-state predator-pray models. The methods used in [2] extensively exploit the local character of Dirichlet Laplacian and can not be applied to the case of fractional Laplacian. We present a new method (see [4], [5]) based on the notion of ultracontractivity and probabilistic potential theory. The method we introduce may be applied to a wide class of nonlocal operators.

References

1. L. Caffarelli, S. Dipierro, E, Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models 10 (2017), 141–170.
2. E. Dancer, Y. Du, On a free boundary problem arising from population biology, Indiana Univ. Math. J. 52 (2003), 51–67.
3. N.E. Humphries, N. Queiroz, J.R.M. Dyer et al., Environmental context explains Lévy and Brownian movement patterns of marine predators, Nature 465 (2010), 1066–1069.
4. T. Klimsiak, Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator, arXiv:1905.01667v1 (2019).
5. T. Klimsiak, Uniqueness for an obstacle problem arising from logistic-type equations with fractional Laplacian, arXiv:1905.01666v1 (2019).

Stig Larsson, Chalmers University of Technology & University of Gothenburg, Sweden

BACKWARD EULER-MARUYAMA METHOD FOR SDES WITH MULTIVALUED DRIFT COEFFICIENT

Joint work with Monika Eisenmann, Raphael Kruse, and Mihály Kovács

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 2.18.

Abstract

We consider the numerical approximation of a multivalued SDE $\begin{cases} \mathrm{d} X(t) + f(X(t)) \, \mathrm{d} t \ni g(t) \, \mathrm{d} W(t), \quad t \in (0,T],\\ X(0) = X_0, \end{cases}$ where the mapping $$f \colon \mathbf{R}^d \to 2^{\mathbf{R}^d}$$ is maximal monotone, of at most polynomial growth, coercive, and fulfills the condition $\langle f_v - f_z , z-w \rangle \leq \langle f_v - f_w , v-w \rangle,$ for every $$v,w,z \in \mathbf{R}^d$$, $$f_v \in f(v)$$, $$f_w \in f(w)$$, and $$f_z \in f(z)$$ as proposed in [1]. Under these low regularity assumptions on the drift coefficient, we can prove well definedness of the backward Euler method, as well as the strong convergence with a rate of $$\frac{1}{4}$$, if $$g$$ lies in a suitable Hölder space.

References

1. R.H. Nochetto, G. Savaré, and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), 525-589.

Xue-Mei Li, Imperial College London, UK

AVERAGING DYNAMICS DRIVEN BY FRACTIONAL NOISE

Joint work with Martin Hairer

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 2.18.

Abstract

Two scale stochastic equations model evolutions of random motions under the influence of fast motions. An overwhelming amount of efforts have been devoted to where the fast motion is assumed to have a Markov property or have independent increments, time series data begs to differ. I hope to explain the effective dynamics for a slow/fast systems where the slow system is driven by a fractional Brownian motion, and perhaps also to touch on further developments.

Fabio Nobile, Ecole Polytechnique Fédérale de Lausanne, Switzerland

DYNAMICAL LOW RANK APPROXIMATION OF RANDOM TIME DEPENDENT PDES

Joint work with Yoshihito Kazashi and Eva Vidlicková

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 2.18.

Abstract

Partial differential equations with stochastic coefficients and input data arise in many applications in which the data of the PDE need to be described in terms of random variables/fields due either to a lack of knowledge of the system or to its inherent variability. The numerical approximation of statistics of the solution poses several challenges when the number of random parameters is large and/or the parameter-to-solution map is complex, and effective surrogate or reduced models are of great need in this context.

In this talk we consider time dependent PDEs with few random parameters and seek for an approximate solution in separable form that can be written at each time instant as a linear combination of linearly independent spatial functions multiplied by linearly independent random variables (low rank approximation) in the spirit of a truncated Karhunen-Loève expansion. Since the optimal deterministic and stochastic modes can significantly change over time, we consider here a dynamical approach where those modes are computed on the fly as solutions of suitable evolution equations. From a geometrical point of view, this corresponds to constraining the original dynamics to the manifold of fixed rank functions, i.e. functions that can be written in separable form with a fixed number of terms. Equivalently, the original equations are projected onto the tangent space to the manifold of fixed rank functions along the approximate trajectory, similarly to the Dirac-Frenkel variational principle in quantum mechanics.

We discuss the construction of the method, present an existence result for a random semi-linear evolutionary equation, and discuss practical numerical aspects for several time dependent PDEs with random parameters, including the heat equation with a random diffusion coefficient; the incompressible Navier-Stokes equations with random Dirichlet boundary conditions; the wave equation with random wave speed.

References

1. E. Musharbash, F. Nobile, T. Zhou, Error analysis of the dynamically orthogonal approximation of time dependent random PDEs, SIAM J. Sci. Comp. 37(2) (2015), A776-A810.
2. E. Musharbash, F. Nobile, Dual Dynamically Orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions, J. Comput. Physics 354 (2018), 135-162.
3. E. Musharbash, F. Nobile, ymplectic dynamical low rank approximation of wave equations with random parameters, MATHICSE Technical Report 18.2017 École Polytechnique Fédérale de Lausanne.
4. Y. Kazashi, F. Nobile, Improved stability of optimal traffic paths, Existence of dynamical low rank approximation for random semi-linear evolutionary equations on the maximal interval, in preparation.

Olivier Menoukeu Pamen, University of Liverpool, UK & African Institute for Mathematical Sciences, Ghana

EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS OF SINGULAR SDES

Joint work with Salah Mohammed and Ludovic Tangpi

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 2.18.

Abstract

In this talk, we are interested in existence and uniqueness of strong solutions for stochastic differential equations with irregular drift coefficients. The driving noise is a $$d$$-dimensional Brownian motion. The method relies on Malliavin calculus and as a byproduct, we obtain Malliavin differentiability of the solutions. Existence of Sobolev differentiable flows in small time will also be discussed.

References

1. V.E. Benes, Existence of optimal stochastic control laws, SIAM J. Control Optim., 9 (1971), 446–475.
2. G. Da Prato, P. Malliavin, D. Nualart, Compact families of Wiener functionals, C. R. Acad. Sci. Paris, Sér. I 315 (1992), 1287–1291.
3. A.M. Davie, Uniqueness of solutions of stochastic differential equations, Int. Math. Res. 24, Article ID rnm 124, (2007), 26 pp.
4. H.-J. Engelbert and W. Schmidt, Strong Markov continuous local martingales and solutions of one- dimensional stochastic differential equations, I, II, III. Math. Nachr., 143, 144, 151 (1989, 1991), 167–184, 241–281, 149–197.
5. E. Fedrizzi and F. Flandoli, Hölder Flow and Differentiability for SDEs with Nonregular Drift, Stochastic Analysis and Applications, 31 (2013), 708–736.
6. I. Gyöngy, N. V. Krylov, Existence of strong solutions for Itô’s stochastic equations via approximations, Probab. Theory Relat. Fields 105 (1996), 143-158.
7. N.V. Krylov, M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Prob. Theory Rel. Fields 131(2) (2005), 154-196.
8. H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.
9. A. Lanconelli, F. Proske, On explicit strong solutions of Itô-SDE’s and the Donsker delta function of a diffusion, Infin. Dimen. Anal. Quant. Prob. related Topics, 7 (2004).
10. O. Menoukeu-Pamen, T. Meyer-Brandis, T. Nilssen, F. Proske, T. Zhang, A variational approach to the construction and malliavin differentiability of strong solutions of SDE’s, Math. Ann. 357 ( 2013), 761–799.
11. T. Meyer-Brandis, F. Proske, Construction of strong solutions of SDE’s via Malliavin calculus, Journal of Funct. Anal. 258 (2010), 3922–3953.
12. S.E.A. Mohammed, M.K.R. Scheutzow, Spatial estimates for stochastic flows in Euclidean space, Annals of Probability, 26(1) (1998), 56–77.
13. S.E.A. Mohammed, T. Nilssen, F. Proske, Sobolev differentiable stochastic flows for sde’s with singular coefficients: Applications to the stochastic transport equation, Annals of Probability, 43(3) (2015), 1535–1576.
14. T. Nilssen, One-dimensional SDE’s with discontinuous, unbounded drift and continuously differentiable solutions to the stochastic transport equation, Technical Report 6, University of Oslo, 2012.
15. D. Nualart, The Malliavin Calculus and Related Topics, Springer, 1995.
16. F. Proske, Stochastic differential equations-some new ideas, Stochastics 79 (2007), 563-600.
17. A.Y. Veretennikov, On the strong solutions of stochastic differential equations, Theory Probab. Appl. 24 (1979), 354–366.
18. A.K. Zvonkin, A transformation of the state space of a diffusion process that removes the drift, Math.USSR (Sbornik) 22 (1974), 129–149.

Grigorios Pavliotis, Imperial College London, UK

LONG TIME BEHAVIOUR, PHASE TRANSITIONS AND FLUCTUATIONS FOR THE McKEAN-VLASOV EQUATION

Joint work with José A. Carrillo, Matias Gonzalo Delgadino, Susana N. Gomes, Rishabh S. Gvalani, and André Schlichting

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 2.18.

Abstract

We study the long time behaviour, the number and structure of stationary solutions and fluctuations for the McKean-Vlasov equation, a nonlinear nonlocal Fokker-Planck type equation that describes the mean field limit of a system of weakly interacting diffusions. We consider two cases: the McKean-Vlasov equation in a multiscale confining potential with quadratic, Curie-Weiss, interaction (the so-called Dasai-Zwanzig model), and the McKean-Vlasov dynamics on the torus with periodic boundary conditions and with a localized interaction. Our main objectives are the study of convergence to a stationary state, the construction of the bifurcation diagram for the stationary problem and the study of fluctuations around the McKean-Vlasov limit, in particular past the phase transition. The application of our work to the study of models for opinion formation and of synchronization for Kuramoto-type models is also discussed. This talk is based on the recent works [1, 2].

References

1. J.A. Carrillo, R.S. Gvalani, G. A. Pavliotis, and A. Schlichting, Long-time behaviour and phase transitions for the Mckean–Vlasov equation on the torus, (2018).
2. S.N. Gomes and G.A. Pavliotis, Mean field limits for interacting diffusions in a two-scale potential, J. Nonlinear Science (2018).

Nicolas Perkowski, Max Planck Institute for Mathematics in the Sciences, Germany

A ROUGH SUPERBROWNIAN MOTION

Joint work with Tommaso Cornelis Rosati

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 2.18.

Abstract

We consider a 1- or 2-dimensional branching random walk in a small random potential and show that its large scale behavior is described by a new stochastic process, which we call rough superBrownian motion. This process is a mixture of the classical superBrownian motion and the continuous parabolic Anderson model (PAM), where the superBrownian part captures fluctuations caused by the branching mechanism and the PAM part describes the large scale behavior of the random potential. We use pathwise arguments to deal with the PAM-singularity, and martingale tools to deal with the singularity from the superBrownian part. We also study the survival properties of the rough superBrownian motion and show that it behaves very differently from its classical counterpart.

Philipp Schoenbauer, Imperial College London, UK

A SUPPORT THEOREM FOR SINGULAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Joint work with Martin Hairer

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 2.18.

Abstract

The purpose of this talk is to present a far-reaching generalization of the support theorem of Stroock and Varadhan. Recall that, given a stochastic differential equation (SDE), this theorem considers all ordinary differential equations (ODEs) which formally look the same as the SDE, but with the noise replaced by an arbitrary smooth function. The support of the SDE is then shown to be the closure of the set of all solutions to these ODEs. We prove a support theorem in the same spirit for stochastic partial differential equations (SPDEs). As part of our analysis we establish the “correct” way to deal with the issue of divergent renormalisation constants in such a description. (This issue makes it difficult to even guess the correct formulation of a support theorem.) Our approach applies to a range of interesting (singular) SPDEs, among them the stochastic quantization equations and the generalised KPZ equations. As an important corollary, we show the uniqueness of the invariant measure for the 3D stochastic quantization equation.

Anders Szepessy, Royal Institute of Technology (KTH), Sweden

LANGEVIN DYNAMICS DERIVED FROM QUANTUM MECHANICS AND ITS RELATION TO STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Joint work with Håkon Hoel

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 2.18.

Abstract

In the study of stochastic partial differential equation one may wonder what is the noise? Often the stochasticity modelled in partial differential equations has its origin in thermal fluctuations.

Starting from a quantum formulation of a molecular system coupled to a heat bath, I will show that ab initio Langevin dynamics, with a certain rank one friction matrix determined by the coupling, approximates the quantum system more accurately than any Hamiltonian system, for large mass ratio between the system and heat bath nuclei.

will also give an example  of course-graining a stochastic molecular dynamics equation to obtain a continuum stochastic partial differential equation for phase transitions.

References

1. A. Kammonen, P. Plecháč , M. Sandberg, A. Szepessy, Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics, nn. Henri Poincaré 19 (2018), 2727-2781.
2. P. Plecháč , M. Sandberg, A. Szepessy, The classical limit of quantum observables in conservation laws of fluid dynamics, arXiv:1702.04368.
3. E. von Schwerin, A. Szepessy, A stochastic phase-field model determined from molecular dynamics, ESAIM: Mathematical Modelling and Numerical Analysis 44 (2010), 627-646.

Lukasz Szpruch, University of Edinburgh, UK

MEAN-FIELD LANGEVIN DYNAMICS AND ENERGY LANDSCAPE OF NEURAL NETWORKS

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 2.18.

Abstract

In this paper we present a probabilistic analysis of the long-time behaviour of the nonlocal, diffusive equations with a gradient flow structure in 2-Wasserstein metric. Our work is motivated by a desire to provide a theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of deep neural networks. The key insight is that the certain class of the finite dimensional non-convex problems becomes convex when lifted to infinite dimensional space of measures. We leverage this observation and show that the flow of marginal laws induced by the Mean Field Langevin Dynamics equation converges to the stationary distribution which is exactly the minimiser of the energy functional. At the heart of our analysis is a pathwise perspective on Otto calculus used in gradient flow literature which is of independent interest. Our proof of convergence to stationary probability measure is novel and it relies on a generalisation of LaSalle's invariance principle.

Antoine Tambue, Western Norway University of Applied Sciences, Norway

MAGNUS-TYPE INTEGRATOR FOR THE FINITE ELEMENT DISCRETIZATION OF SEMILINEAR PARABOLIC NON-AUTONOMOUS SPDEs DRIVEN BY MULTIPLICATIVE NOISE

Joint work with Jean Daniel Mukam

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 2.18.

Abstract

This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE driven by multiplicative noise by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $$L^2$$ norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order $$\mathcal{O}\left(h^2\left(1+\max(0,\ln\left(t_m/h^2\right)\right)+\Delta t^{1/2}\right).$$ Numerical simulations to illustrate our theoretical finding are provided.

References

1. A. Tambue, J.D. Mukam, Magnus-type Integrator for the Finite Element Discretization of Semilinear Parabolic non-Autonomous SPDEs Driven by multiplicative noise, arXiv:1809.04438v1, 2018.

Hendrik Weber, University of Bath, UK

A-PRIORI BOUNDS FOR SINGULAR SPDES

Joint work with Ajay Chandra and Augustin Moinat

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 2.18.

Abstract

The theory of regularity structures is a powerful tool to develop a stable solution theory for a whole class of stochastic PDEs arising in statistical mechanics and quantum field theory. Initiated in Hairer's groundbreaking work in 2013, in only a few years an astonishingly general solution theory covering essentially all equations which satisfy a certain scaling condition (subcriticality or super-renormalizability), has been developed. However, up to now, most results only gave control over solutions for small times and on bounded spatial domains.

The aim if this talk is to present a method to prove a priori estimates in the framework regularity structures. These bounds complement the short time existence and uniqueness theory to obtain control of solutions globally in time and on unbounded domains. Our bounds are implemented in the example of the dynamic $$\Phi^4$$ equation, which is formally given by $(\partial_t - \Delta) u = -u^3 + \infty u +\xi.$ This equation is subcritical if the distribution $$\xi$$ is of class $$C^{-3+\frac{\delta}{2}}$$ for $$\delta>0$$, and we obtain bounds for all such $$\xi$$. An analogy to the regularity of white noise suggests to interpret this as a solution theory for $$\Phi^4$$ in dimension $$4-\delta$$.

References

1. A. Chandra, A. Moinat, H. Weber, A priori bounds for $$\Phi^4$$ in the subcritical regime, in preparation.
2. A. Moinat, H. Weber, Space-time localisation for the dynamic $$\Phi^ 4_3$$ model, arXiv preprint arXiv:1811.05764 (2018).

Tusheng Zhang, University of Manchester, UK

LARGE DEVIATION PRINCIPLES FOR FIRST-ORDER SCALAR CONSERVATION LAWS WITH STOCHASTIC FORCING

Joint work with Zhao Dong, Jianglun Wu, and Rangrang Zhang

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 2.18.

Abstract

In this paper, we established the Freidlin-Wentzell type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.

References

1. M. Boué, P. Dupuis, A variational representation for certain functionals of Brownian motion, Ann. Probab. 26(4) (1998), 1641-1659.
2. A. Debussche, M. Hofmanová, and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, Ann. Probab. 44(3) (2016), 1916-1955.
3. A. Matoussi, W. Sabbagh, T. Zhang, Large Deviation Principles of Obstacle Problems for Quasilinear Stochastic PDEs, To appear in Applied Mathematics and Optimization, arXiv:1712.02169.
4. A. Debussche, J. Vovelle, Scalar conservation laws with stochastic forcing (revised version). http://math.univ-lyon1.fr/vovelle/DebusscheVovelleRevised, J. Funct. Anal. 259(4) (2010), 1014-1042.

Huaizhong Zhao, Loughborough University, UK

RANDOM PERIODICITY: THEORY AND MODELLING

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 2.18.

Abstract

Random periodicity is ubiquitous in the real world. In this talk, I will provide the concepts of random periodic paths and periodic measures to mathematically describe random periodicity. It is proved that these two different notions are "equivalent". Existence and uniqueness of random periodic paths and periodic measures for certain stochastic differential equations are proved. An ergodic theory is established. It is proved that for a Markovian random dynamical system, in the random periodic case, the infinitesimal generator of its Markovian semigroup has infinite number of equally placed simple eigenvalues including 0 on the imaginary axis. This is in contrast to the mixing stationary case in which the Koopman-von Neumann Theorem says there is only one eigenvalue 0, which is simple, on the imaginary axis. Geometric ergodicity for some stochastic systems is obtained. Possible applications e.g. in stochastic resonance will be discussed.

This talk is based on a series of work with Chunrong Feng et al.