Talks of D4 Applications

Weizhu Bao, National University of Singapore, Singapore

MULTISCALE METHODS AND ANALYSIS FOR THE DIRAC EQUATION IN THE NONRELATIVISTIC LIMIT REGIME

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

Abstract

In this talk, I will review our recent works on numerical methods and analysis for solving the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite, which bring significant difficulty in analysis and heavy burden in numerical computation [4]. We begin with four frequently used finite difference time domain (FDTD) methods and the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter [3]. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the Dirac equation [3]. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic limit regime [3]. We find that the time-splitting spectral method performs super-resolution in temporal discretization when the Dirac equation has no magnetic potential [5]. Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish its error bound which uniformly accurate in term of the small dimensionless parameter [1]. Numerical results demonstrate that our error estimates are sharp and optimal. Finally, these methods and results are then extended to the nonlinear Dirac equation in the nonrelativistic limit regime [2]. This is a joint work with Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin.

References

1. W. Bao, Y. Cai, X. Jia and Q. Tang, A uniformly accurate multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime, SIAM J. Numer. Anal. 54 (2016), 1785-1812.
2. W. Bao, Y. Cai, X. Jia and J. Yin, Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime, Sci. China Math. 59 (2016), 1461-1494.
3. W. Bao, Y. Cai, X. Jia and Q. Tang, Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime, J. Sci. Comput. 71 (2017), 1094-1134.
4. W. Bao and J. Yin, A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation, Res. Math. Sci. 6 (2019), article 11.
5. W. Bao, Y. Cai and J. Yin, Improved stability of optimal traffic paths, Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic limit regime, arXiv: 1811.02174.

Russel Caflisch, New York University, USA

ACCELERATED SIMULATION FOR PLASMA KINETICS

Joint work with Denis Silantyev and Bokai Yann

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

Abstract

This presentation will discuss the kinetics of Coulomb collisions in plasmas, as described by the Landau-Fokker-Planck equation, and its numerical solution using a Direct Simulation Monte Carlo (DSMC) method. Acceleration of this method is achieved by coupling the particle method to a continuum fluid description. Efficiency of the resulting hybrid method is greatly increased by inclusion of particles with negative weights. This complicates the simulation, and introduces difficulties have plagued earlier efforts to use negatively weighted particles. This talk will describe significant progress that has been made in overcoming those difficulties.

Albert Cohen, Sorbonne Université, France

OPTIMAL SAMPLING AND RECONSTRUCTION IN HIGH DIMENSION

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

Abstract

Motivated by non-intrusive approaches for high-dimensional parametric PDEs, we consider the general problem of approximating an unknown arbitrary function in any dimension from the data of point samples. The approximants are picked from given or adaptively chosen finite dimensional spaces. One principal objective is to obtain an approximation which performs as good as the best possible using a sampling budget that is linear in the dimension of the approximating space. We will show that this object if can is met by taking a random sample distributed according to a well chosen probability measure, and reconstructing by appropriate least-squares or pseudo-spectral methods.

Hugo Duminil-Copin, IHÉS, France & University of Geneva, Switzerland

MARGINAL TRIVIALITY OF THE SCALING LIMITS OF CRITICAL ISING AND \(\varphi^4\) MODELS IN 4D

Joint work with Michael Aizenman

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

Abstract

The question of constructing a non-Gaussian field theory, i.e. a field with non-zero Ursell functions, is at the heart of Euclidean (quantum) field theory. While non-triviality results in \(d<4\) and triviality results in \(d>4\) were obtained in famous papers by Glimm, Jaffe, Aizenman, Frohlich and others, the crucial case of dimension 4 remained open. In this talk, we show that any continuum \(\varphi^4\) theory constructed from Reflection Positive lattice \(\varphi^4\) or Ising models is inevitably free in dimension \(4\). The proof is based on a delicate study of intersection properties of a non-Markovian random walk appearing in the random current representation of the model.

László Erdős, Institute of Science and Technology, Austria

FROM WIGNER-DYSON TO PEARCEY: UNIVERSAL EIGENVALUE STATISTICS OF RANDOM MATRICES

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

Abstract

E. Wigner's revolutionary vision postulated that the local eigenvalue statistics of large random matrices are independent of the details of the matrix ensemble apart from its basic symmetry class. There have recently been a substantial development to prove Wigner's conjecture for larger and larger classes of matrix ensembles motivated by applications. They include matrices with entries with a general correlation structure and addition of deterministic matrices in a random relative basis. We also report on three types of universality, commonly known as the bulk, edge and cusp universality, referring to the behaviour of the density of states in the corresponding energy regime. While bulk and edge universalities have been subject to intensive research, the cusp universality has been studied only in very special cases before. Our recent work settles the question of this third and last type of universality in full generality.

Eduard Feireisl, Czech Academy of Sciences, Czech Republic

DISSIPATIVE SOLUTIONS TO THE COMPRESSIBLE EULER SYSTEM

Joint work with Dominic Breit and Martina Hofmanová

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

Abstract

We introduce the concept of (generalized) dissipative solutions to the compressible Euler system and review their basic properties:

\(\bullet\) Existence. Dissipative solutions exist globally in time for any finite energy initial data.

\(\bullet\) Maximal dissipation, semigroup selection. One can select a solution semigroup among dissipative solutions. The selected solution maximizes the energy dissipation (entropy production), see [1].

\(\bullet\) Weak-strong uniqueness. A dissipative and a weak solution emanating from the same initial data coincide as soon as the weak solution belongs to certain Besov class and its velocity gradient satisfies a one sided Lipschitz condition, see [2].

\(\bullet\) Convergence of numerical schemes. Cesaro avarages produced by suitable numerical schemes converge strongly to a dissipative solution, see [3].

References

1. D. Breit, E. Feireisl, M. Hofmanová, Solution semiflow to the isentropic Euler system, Arxive Preprint Series, arXiv 1901.04798, 2019.
2. E. Feireisl, S.S. Ghoshal, A. Jana, On uniqueness of dissipative solutions to the isentropic Euler system, Arxive Preprint Series, arXiv 1903.11687, 2019.
3. E. Feireisl, M. Lukáčová–Medviďová, H. Mizerová, \(\mathcal{K}\)-convergence as a new tool in numerical analysis, Arxive Preprint Series, arXiv 1904.00297, 2019.

Mats Gyllenberg, University of Helsinki, Finland

FINITE DIMENSIONAL STATE REPRESENTATION OF STRUCTURED POPULATION MODELS

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

Abstract

Structured population models can be formulated as delay systems. We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. The ideas and results are illustrated by models for infectious diseases and physiologically structured populations.

Clément Mouhot, University of Cambridge, UK

QUANTITATIVE LINEAR STABILITY (HYPOCOERCIVITY) FOR CHARGED PARTICLES IN A CONFINING FIELD

Joint work with K. Carrapatoso, J. Dolbeault, F. Hérau, S. Mischler, and C. Schmeiser

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

Abstract

We report on recent joint results in which we develop quantitative methods for proving the existence of a spectral gap and estimating the gap, for hypocoercive kinetic equations that combine the local conservation laws of fluid mechanics and a confining potential force. The proofs involve a cascade of correctors and global commutator estimates, as well as new quantitative inequalities of Korn type. The latter extend to the case of the whole space with a potential force the classical Korn inequality in a bounded domain of elasticity theory. These results are a step towards constructing global solutions near equilibrium to the full nonlinear Boltzmann equation for charged particles subject to a confining potential.

Thaleia Zariphopoulou, University of Texas at Austin, USA

STOCHASTIC MODELING AND OPTIMIZATION IN HUMAN-MACHINE INTERACTION SYSTEMS

Joint work with Agostino Capponi and Svein Olefsson

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

Abstract

I will introduce a family of human-machine interaction (HMI) models in optimal asset allocation, risk management and portfolio choice (robo-advising). Modeling difficulties stem from the limited ability to quantify the human's risk preferences and describe their evolution, but also from the fact that the stochastic environment, in which the machine optimizes, itself adapts to real-time incoming information that is exogenous to the human. Furthermore, the human's risk preferences and the machine's states may evolve at different scales. This interaction creates an adaptive cooperative game with asymmetric and incomplete information exchange between the two parties.

As a result, challenging questions arise on, among others, how frequently the two parties should communicate, what information can the machine accurately detect, infer and predict, how the human reacts to exogenous events and what are the effects on the machine's actions, how to improve the inter-linked reliability between the human and the machine, and others.

Such HMI models give rise to new, non-standard optimization problems that include well-posed and ill-posed sub-problems, and combine adaptive stochastic control, stochastic differential games, optimal stopping, multi-scales and learning.

References

1. A. Capponi, S. Olefsson and T. Zariphopoulou, Personalized robo-advising, Preprint, 2019.

Niels Benedikter, Institute of Science and Technology, Austria

CORRELATION ENERGY OF THE MEAN-FIELD FERMI GAS BY THE METHOD OF COLLECTIVE BOSONIZATION

Joint work with Phan Thành Nam, Marcello Porta, Benjamin Schlein, and Robert Seiringer

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

Abstract

Quantum correlations play an important role in interacting systems; however, their mathematical description is a highly non-trivial task. I explain how correlations in fermionic systems can be described by bosonizing collective pair excitations. This leads us to an effective quadratic bosonic Hamiltonian. We establish a theory of approximate bosonized Bogoliubov transformations by which we derive a Gell-Mann-Brueckner-type formula for the fermionic ground state energy.

References

1. N. Benedikter, P.T. Nam, M. Porta, B. Schlein, R. Seiringer, Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime, to appear in Communications in Mathematical Physics, 2019.

Dongho Chae, Chung-Ang University, South Korea

ON THE TYPE I BLOW-UP FOR THE INCOMPRESSIBLE EULER EQUATIONS

Joint work with Jörg Wolf

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

Abstract

In this talk we discuss the Type I blow up and the related problems in the 3D Euler equations. We say a solution \(v\) to the Euler equations satisfies Type I condition at possible blow up time \(T_*\) if \(\lim\sup_{t\nearrow T_*} (T_*-t) \|\nabla v(t)\|_{L^\infty} <+\infty\). The scenario of Type I blow up is a natural generalization of the self-similar(or discretely self-similar) blow up. We present some recent progresses of our study regarding this. We first localize previous result that "small Type I blow up" is absent. After that we show that the atomic concentration of energy is excluded under the Type I condition. This result, in particular, solves the problem of removing discretely self-similar blow up in the energy conserving scale, since one point energy concentration is necessarily accompanied with such blow up. We also localize the Beale-Kato-Majda type blow up criterion. Using similar local blow up criterion for the 2D Boussinesq equations, we can show that Type I and some of Type II blow up in a region off the axis can be excluded in the axisymmetric Euler equations.

Michele Coti Zelati, Imperial College London, UK

SUFFICIENT CONDITIONS FOR TURBULENCE SCALING LAWS IN 2D AND 3D

Joint work with Jacob Bedrossian, Sam Punshon-Smith, and Franziska Weber

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

Abstract

We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws for both 2d [1] and 3d [2] stochastically forced Navier-Stokes equations. These conditions, which we name weak anomalous dissipation, replace the classical anomalous dissipation condition. For statistically stationary solutions, weak anomalous dissipation appear to be very effective and not too far from being necessary as well.

References

1. J. Bedrossian, M. Coti Zelati, S. Punshon-Smith, F. Weber, Sufficient conditions for dual cascade flux laws in the stochastic 2d Navier-Stokes equations, arXiv 1905.03299 (2019).
2. J. Bedrossian, M. Coti Zelati, S. Punshon-Smith, F. Weber, A Sufficient Condition for the Kolmogorov 4/5 Law for Stationary Martingale Solutions to the 3D Navier-Stokes Equations, Comm. Math. Phys. 367 (2019), 1045-1075.

Jan Dereziński, University of Warsaw, Poland

PROPAGATORS ON CURVED SPACETIMES

Joint work with Daniel Siemssen and Adam Latosiński

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

Abstract

Quantum Field Theory on curved spacetimes has many interesting links to various branches of mathematics, such as differential geometry, symplectic dynamics, partial differential equations, pseudodifferential calculus, symmetric spaces and operator theory. I will discuss some of these links.

Petr Dunin-Barkowski, National Research University Higher School of Economics, Russia

LOOP EQUATIONS AND A PROOF OF ZVONKINE’S \(qr\)-ELSV FORMULA

Joint work with Reinier Kramer, Alexandr Popolitov, and Sergey Shadrin

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

Abstract

The talk is devoted to the outlining of the proof of the 2006 Zvonkine's conjecture that expresses Hurwitz numbers with completed cycles in terms of intersection numbers with the Chiodo classes via the so-called \(r\)-ELSV formula. In fact, this proof works in even a bit more general setting, namely it works for the \(qr\)-ELSV formula (which is the orbifold generalization of the \(r\)-ELSV formula), conjectured recently in [1]. The proof relies on expressing both the aforementioned Hurwitz and intersection numbers in terms of expansions of multi-point functions resulting from the application of the spectral curve topological recursion procedure on a particular spectral curve.

The talk is based on [2].

References

1. R. Kramer, D. Lewanski, A. Popolitov, S. Shadrin, Towards an orbifold generalization of Zvonkine’s \(r\)-ELSV formula, arXiv:1703.06725, 1–20.
2. P. Dunin-Barkowski, R. Kramer, A. Popolitov, S. Shadrin, Loop equations and a proof of Zvonkine’s \(qr\)-ELSV formula, arXiv:1905.04524, 1–17.

Semyon Dyatlov, University of California, Berkeley & Massachusetts Institute of Technology, USA

CONTROL OF EIGENFUNCTIONS ON NEGATIVELY CURVED SURFACES

Joint work with Jean Bourgain, Long Jin, and Stéphane Nonnenmacher

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

Abstract

Given an \(L^2\)-normalized eigenfunction with eigenvalue \(\lambda^2\) on a compact Riemannian manifold \((M,g)\) and a nonempty open set \(\Omega\subset M\), what lower bound can we prove on the \(L^2\)-mass of the eigenfunction on \(\Omega\)? The unique continuation principle gives a bound for any \(\Omega\) which is exponentially small as \(\lambda\to\infty\). On the other hand, microlocal analysis gives a \(\lambda\)-independent lower bound if \(\Omega\) is large enough, i.e. it satisfies the geometric control condition.

This talk presents a \(\lambda\)-independent lower bound for any set \(\Omega\) in the case when \(M\) is a negatively curved surface, or more generally a surface with Anosov geodesic flow. The proof uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from harmonic analysis called the Fractal Uncertainty Principle. Applications include control for Schrödinger equation and exponential decay of damped waves.

Pierre Germain, New York University, USA

ON THE DERIVATION OF KINETIC WAVE EQUATIONS

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

Abstract

Consider a nonlinear dispersive equation with random initial data. In the appropriate regime, it is conjectured that its dynamics are described, after averaging over the random data, by a kinetic wave equation. I will present recent progress towards the proof of this conjecture.

Leszek Hadasz, Jagiellonian University in Kraków, Poland

FROM CFT TO QUANTUM CURVES AND SUPER-AIRY STRUCTURES

Joint work with Vincent Bouchard, Paweł Ciosmak, Zbigniew Jaskólski, Masahide Manabe, Kento Osuga, Błażej Ruba, and Piotr Sułkowski

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

Abstract

During the talk I will discuss how the methods and notions developed in the area of two-dimensional, quantum conformal field theory allow to solve efficiently some problems related to matrix models: finding families of quantum curves related to a given classical algebraic curve and constructing interesting examples of super-Airy structures (algebras of differential operators engineered to solve topological recursion relations).

References

1. P. Ciosmak, L. Hadasz, Z. Jakólski, M. Manabe and P. Sułkowski, From CFT to Ramond super-quantum curves, JHEP 1805 133 (2018).
2. V. Bouchard, P. Ciosmak, L. Hadasz, K. Osuga, B. Ruba and P. Sułkowski, Super Airy Structures, work in progress.

Christian Hainzl, Universität Tübingen, Germany

LOWER BOUND ON THE HARTREE-FOCK ENERGY OF THE ELECTRON GAS

Joint work with David Gontier and Mathieu Lewin

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

Abstract

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the free Fermi Gas is always unstable under the formation of spin or charge density waves. I present the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas.

Gustav Holzegel, Imperial College London, UK

NON-LINEAR STABILITY OF THE SCHWARZSCHILD FAMILY OF BLACK HOLES

Joint work with Mihalis Dafermos, Igor Rodnianski, and Martin Taylor

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

Abstract

I will discuss the statement and the proof of the finite-codimension non-linear stability of the Schwarzschild family as solutions to the vacuum Einstein equations. The proof relies crucially on our previous work [1] on the linear stability of the Schwarzschild family and makes use of many analytical techniques developed over the years in the analysis of hyperbolic equations on black hole spacetimes, including control of the non-linearities of the Einstein equations in the radiation zone.

References

1. M. Dafermos, G. Holzegel, I. Rodnianski, The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Mathematica 222 (2019), 1-214.

Chun-Hsiung Hsia, National Taiwan University, Taiwan

ON THE MATHEMATICAL ANALYSIS OF SYNCHRONIZATION FOR THE TIME-DELAYED KURAMOTO OSCILLATORS

Joint work with Chang-Yeol Jung, Bongsuk Kwon, and Yoshihiro Ueda

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

Abstract

We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase and frequency synchronization are in view. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complete and partial phase synchronization for both models with the uniform time-delay and phase lag. Our results show that the Kuramoto models incorporated with small variation of time-delays and/or phase lag effect still exhibit the synchronization. These support that the original Kuramoto model (i.e., no time-delays/phase lags) is qualitatively robust in the perturbation of time-delay and phase lag effects. We also present several numerical experiments supporting our main results.

References

1. C.-H. Hsia, C.-Y. Jung, B. Kwon, On the synchronization theory of Kuramoto oscillators under the effect of inertia, Journal of Differential Equations 267 (2019), 742-775.
2. C.-H. Hsia, C.-Y. Jung, B. Kwon, Y. Ueda, Synchronization of Kuramoto oscillators with time-delayed interactions and phase lag effect, preprint.

Jacek Jendrej, CNRS & Université Paris 13, France

MULTI-BUBBLES FOR A CRITICAL WAVE EQUATION

Joint work with Yvan Martel

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 0.10b.

Abstract

If a solution of an evolution partial differential equation resembles, near some point in space, a rescaled copy of a fixed profile, with the scale tending to zero in finite or infinite time, we say that a bubble is created at this point. Such behavior can be possible only if the rescaling preserves the energy of the profile, which is called the energy-critical setting.

We consider the focusing nonlinear wave equation in the energy-critical case, in space dimension \(5\). Given any finite set of \(K\) points in space, we construct a solution for which a bubble is created at each of these points in infinite time. The energy of the solution is equal to the energy of the profile multiplied by \(K\), which means that no energy is radiated in the process. To our knowledge, we provide the first construction of bubbling at multiple points for a wave equation.

Petr Kaplický, Charles University, Czech Republic

ON UNIQUENESS OF GENERALIZED NEWTONIAN FLOWS

Joint work with Miroslav Bulíček, Frank Ettwein, and Dalibor Pražák

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

Abstract

In [3] a new model for fluid dynamics was suggested by O.A. Ladyzhenskaya. The inner properties of the fluid were described by the constitutive relation \[ \mathbb S = ( 1 + |\mathbb D|^{p-2}) \mathbb D \] where \(p\gt 1\) was a given parameter - power-law index, \(\mathbb S\) was the stress tensor and \(\mathbb D\) the symmetric part of the velocity gradient. If \(p=2\) the model reduces to the Navier-Stokes model. If \(p \gt 2\) one expects that the model exhibits better properties. It is indeed so. In particular, if \( p\ge 11/5\), one is allowed to test weak formulation of the equations with the weak solution itself. Consequently, any weak solution satisfies energy equality. A further motivation for this model was uniqueness of weak solutions. Already in [3], it is established provided \(p\ge 5/2\) or in case of smooth initial condition for \(p\ge 12/5\). The range \(p\in [11/5, 12/5)\) however remained untouched except the case of spatial periodic condition, for which one can improve spatial regularity.

I will present results on uniqueness of the weak solutions to a class of systems, including the one mentioned above, in three-dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index \(p\ge 11/5\), some regularity of a solution with respect to time variable was established. Consequently, this information can be used for showing the uniqueness of the solution provided that the initial data are good enough for all power–law indices \(p\ge 11/5\), see [1, 2]. Such a result was available for \(p\ge 12/5\) and therefore the result extends the uniqueness to the whole range of \(p\)’s for which the energy equality holds.

References

1. M. Bulíček, F. Ettwein, P. Kaplický, and D. Pražák, The dimension of the attractor for the 3D flow of a non-Newtonian fluid, Commun. Pure Appl. Anal. 8(5) (2009), 1503–1520.
2. M. Bulíček, P. Kaplický, D. Pražák, Uniqueness and regularity of flows of non-Newtonian fluids with critical power-law growth, Mathematical Models and Methods in Applied Sciences 29 (2019), 1207–1225.
3. O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

Marcin Napiórkowski, University of Warsaw, Poland

NORM APPROXIMATION FOR MANY-BOSON QUANTUM DYNAMICS

Joint work with Christian Brennecke, Phan Thành Nam, and Benjamin Schlein

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

Abstract

Because of the complexity of the many-body Schrödinger equation, to gain insight into the properties of many-body quantum systems it is necessary to use effective theories.

In my talk, I will review recent advances [1, 2] in the derivation of effective equations that govern the dynamics of Bose-Einstein condensates.

References

1. C. Brennecke, P.T. Nam, M. Napiórkowski, B. Schlein, Fluctuations of N-particle quantum dynamics around the nonlinear Schrödinger equation, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, in press.
2. P.T. Nam, M. Napiórkowski, Norm approximation for many-body quantum dynamics: focusing case in low dimensions, Adv. Math. 350, 547-587.

Marcello Porta, Universität Tübingen, Germany

ON THE CORRELATION ENERGY OF INTERACTING FERMI GASES IN THE MEAN-FIELD REGIME

Joint work with Christian Hainzl, Felix Rexze, Niels Benedikter, Phan Thành Nam, Benjamin Schlein, and Robert Seiringer

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

Abstract

In this talk I will discuss the ground state properties of a homogeneous, interacting Fermi gas, in the mean-field regime. I will focus on the correlation energy, defined as the difference between many-body and Hartree-Fock ground state energies. It is a long-standing open problem in mathematical physics to rigorously compute this quantity, for large quantum systems. I will present upper and lower bounds for the correlation energy, that are optimal in their dependence on the number of particles, and that agree for small interactions. The lower bound captures the corrections to the energy predicted by second-order perturbation theory; it is based on the combination of Bogoliubov theory and on correlation inequalities for the many-body interaction. The upper bound establishes the validity of the random-phase approximation as a rigorous upper bound to the ground state energy; it is based on a suitable choice of the trial state, and on a rigorous bosonization scheme.

Bruno Premoselli, Université libre de Bruxelles, Belgium

COMPACTNESS OF SIGN-CHANGING SOLUTIONS TO SCALAR CURVATURE-TYPE EQUATIONS WITH BOUNDED NEGATIVE PART

Joint work with Jérôme Vétois

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

Abstract

We consider the equation \(\Delta_g u+hu=|u|^{2^*-2}u\) in a closed Riemannian manifold \((M,g)\), where \(h\in C^{0,\theta}(M)\), \(\theta \in (0,1)\) and \(2^* = \frac{2n}{n-2}\), \(n:=\dim(M)\ge3\). We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is \(\textit{a priori}\) bounded. We obtain this result under the conditions that \(n\ge7\) and \(h<\frac{n-2}{4 (n-1) }\textrm{Scal}_g\) in \(M\), where \(\textrm{Scal}_g\) is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function \(h\).

Jan Sbierski, University of Oxford, UK

GENERIC BLOW-UP RESULTS FOR LINEAR WAVES IN THE INTERIOR OF A SCHWARZSCHILD BLACK HOLE

Joint work with Grigorios Fournodavlos

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

Abstract

I will discuss recent work [1], joint with Grigorios Fournodavlos, on the behaviour of generic solutions to the wave equation in the interior of a Schwarzschild black hole. We derive an asymptotic expansion of a general solution near the singularity at \(r=0\) and show that it is characterised by its first two leading order terms in \(r\), a principal logarithmic term and a bounded second order term. Based on results [2], [3], [4] by Angelopoulos, Aretakis, and Gajic on the late time asymptotics of generic solutions to the wave equation in the exterior of a Schwarzschild black hole we then show that the principal logarithmic term is non-vanishing in a neighbourhood of the asymptotic endpoints of the singular hypersurface \(r=0\).

References

1. G. Fournodavlos, J. Sbierski, Generic blow-up results for the wave equation in the interior of a Schwarzschild black hole, J. Arch. Rational Mech. Anal. (2019), https://doi.org/10.1007/s00205-019-01434-0.
2. Y. Angelopoulos, S. Aretakis, D. Gajic, A vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary spacetimes, Ann. PDE, 4(2) (2018), Art. 15, 120 p.
3. Y. Angelopoulos, S. Aretakis, D. Gajic, Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes, Adv. Math. 323 (2018), 529–621.
4. Y. Angelopoulos, S. Aretakis, D. Gajic, A proof of Price’s late-time asymptotics for all angular frequencies, in preparation.

Benjamin Schlein, Universität Zürich, Switzerland

EXCITATION SPECTRUM OF TRAPPED BOSE-EINSTEIN CONDENSATES

Joint work with C. Boccato, C. Brennecke, S. Cenatiempo, and S. Schraven

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

Abstract

In this talk, we will discuss some recent results [1, 2, 3] concerning the ground state energy and the low-energy excitation spectrum of gases of \(N\) bosons trapped in a volume of order one and interacting through a repulsive potential with scattering length of the order \(1/N\) (Gross-Pitaevskii regime). Our results confirm the validity of Bogoliubov theory.

References

1. C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein, Bogoliubov theory in the Gross-Pitaevskii limit, Acta Math. 222 (2019), 219–335.
2. C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein, Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime, Preprint arXiv:1812.03086.
3. C. Brennecke, B. Schlein, S. Schraven, Excitation spectrum of trapped Bose-Einstein condensates, In preparation.

Volker Schlue, University of Melbourne, Australia

STABILITY OF EXPANDING BLACK HOLE COSMOLOGIES

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

Abstract

In general relativity, an explicit family of solutions to the Einstein equations \(\textrm{Ric}(g)=\Lambda g\) with positive cosmological constant \(\Lambda>0\), the so-called Kerr-de Sitter space-times, describe the equilibrium states of a black hole in an expanding universe. The black hole exterior falls into two components, the stationary (near) zone, and the expanding (far) zone, separated by the cosmological horizon of the black hole. While the near region (bounded by the event and cosmological horizons) was recently proven to be dynamically stable [3], this talk reports on the dynamics of the far region (beyond the cosmological horizon) [5, 6]. Unlike in [3] (or the black hole stability problem for the Kerr solutions with \(\Lambda=0\) [1, 4]), the solution does not globally converge to an explicit family of solutions, but displays genuine asymptotic degrees of freedom; this was first observed for closed de Sitter cosmologies in [2]. Nonetheless we can prove that the conformal Weyl curvature decays [6] due to the expansion of the space-time geometry, which forms an essential part of the analysis of the cosmological region.

References

1. M. Dafermos, G. Holzegel and I. Rodnianski, The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Math. 222 (2019), 1–214.
2. H. Friedrich, On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Comm. Math. Phys. 107 (1986), 587–609.
3. P. Hintz and A. Vasy, The global non-linear stability of the Kerr–de Sitter family of black holes, Acta Math. 220 (2018), 1–206.
4. S. Klainerman and J. Szeftel, Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations, arXiv:1711.07597 [gr-qc], 2019.
5. V. Schlue, Global Results for Linear Waves on Expanding Kerr and Schwarzschild de Sitter Cosmologies, Communications in Mathematical Physics 334 (2015), 977–1023.
6. V. Schlue, Decay of the Weyl curvature in expanding black hole cosmologies, arXiv:1610.04172v1 [math.AP], 2016.

Jan Philip Solovej, University of Copenhagen, Denmark

THE DILUTE LIMIT OF INTERACTING BOSE GASES

Joint work with Birger Brietzke and Søren Fournais

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

Abstract

I will discuss the interacting many-body Bose gase and, in particular, recent progress [2, 3] in understanding the asymptotics of the ground state energy in the dilute limit. The ground state energy of two bosons confined in a large box can be expressed in terms of the zero energy scattering length of the interacting potential. It has been a general belief in the physics literature [1, 4, 5] that the ground state energy in the dilute limit has a two term asymptotic expansion which is universal in the sense that the terms still depend only on the scattering length of the interaction potential. The asymptotics gives the celebrated Lee-Huang-Yang formula [4]. I will discuss recent progress of understanding this formula and the universality.

References

1. N.N. Bogolyubov, On the theory of superfluidity, Proc. Inst. Math. Kiev 9 (1947), 89-103.
2. B. Brietzke, S. Fournais, and J.P. Solovej, A simple 2nd order lower bound to the energy of dilute Bose gases, arXiv:1901.00539.
3. S. Fournais and J.P. Solovej, The energy of dilute Bose gases, arXiv:1904.06164.
4. T.D. Lee, K. Huang, and C.N. Yang, Eigenvalues and eigenfunctions of a bose system of hard spheres and its low-temperature properties, Physical Review 106 (1957), 1135-1145.
5. W. Lenz, Die Wellenfunktion und Geschwindigkeitsverteilung des entarteten Gases, Z. Phys. 56 (1929), 778–789.

Piotr Sułkowski, University of Warsaw, Poland & California Institute of Technology, USA

RANDOM MATRIX MODELS AND TOPOLOGICAL RECURSIONS

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

Abstract

Within the past few decades we have witnessed great progress in the theory of random matrices. In particular, in the last one-and-a-half decade a powerful formalism of topological recursions - which can be interpreted as generalization of Ward identities for matrix models - has been developed. Topological recursions have already found a lot of applications in various branches of mathematics (in particular algebraic geometry, enumerative geometry, knot theory), high energy physics, statistical physics, and various other fields. In this talk I will explain what topological recursions are and summarize some of their applications in research areas mentioned above.

Claude Warnick, University of Cambridge, UK

QUASINORMAL MODES OF BLACK HOLES

Joint work with Dejan Gajic

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 0.10b.

Abstract

In recent years the problem of defining the quasinormal modes of subextremal black holes has been satisfactorily resolved for asymptotically de Sitter [1] and anti-de Sitter black holes [2]. The quasinormal frequencies may be realised as eigenvalues of a Fredholm operator resulting from a natural choice of coordinates on the black hole background. I will report on recent work with Dejan Gajic to extend this approach to treat extremal and asymptotically flat black hole spacetimes.

References

1. A. Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces, Invent. Math 194 (2013), 381-513.
2. C. Warnick, On quasinormal modes of asymptotically anti-de Sitter black holes, Comm. Math. Phys. 333 (2015), 959-1035.

Klaus Widmayer, Ecole Polytechnique Fédérale de Lausanne, Switzerland

LONG TIME DYNAMICS IN THE ROTATING EULER EQUATIONS

Joint work with Yan Guo and Benoit Pausader

Date: 2019-09-19 (Thursday); Time: 12:05-12:25; Location: building B-8, room 0.10b.

Abstract

We investigate long time dynamics of solutions to the rotating Euler equations in three spatial dimensions. We develop a framework that is adapted to the symmetries and the dispersive properties of this problem and show how it can be used to understand the behavior of small data solutions, uniformly in the parameter of rotation.

The key idea is to use the available symmetries as much as possible, rather than to pursue a more brute force approach. While this streamlines the deduction of some energy type estimates, it also requires a fresh look at the (linear) dispersive estimates, deviating from the classical stationary phase intuition.

Ellen Baake, Universität Bielefeld, Germany

A PROBABILISTIC VIEW ON THE DETERMINISTIC MUTATION-SELECTION EQUATION

Joint work with Fernando Cordero and Sebastian Hummel

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 1.26.

Abstract

We reconsider the prototype version of the mutation-selection differential equation of population genetics. It describes the dynamics of the type composition of an infinite-size population of individuals with two possible types that undergo selection and recurrent mutation. We develop the genealogical point of view to this equation by tracing back the lines of descent of individuals from the population at some (forward) time \(t\). We revisit the ancestral selection graph, which is the standard tool to trace back ancestral lines in populations under selection. Based on this, we introduce the killed ancestral selection graph, which yields the type of a random individual, based on the individual’s potential ancestry and the mutations that define the individual’s type. The result is a stochastic representation of the solution of the (deterministic) differential equation, which is formulated in terms of a duality theorem. This way, the well-known stationary behaviour of the differential equation (and its transcritial bifurcation) translates into the asymptotic behaviour of a (killed) birth-death process.

References

1. E. Baake, F. Cordero, S. Hummel, A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent, J. Math. Biol. 77 (2018), 795-820.
2. E. Baake, A. Wakolbinger, Lines of descent under selection, J. Stat. Phys. 172 (2018), 156-174.

Piotr Biler, University of Wrocław, Poland

GLOBAL EXISTENCE VS FINITE TIME BLOWUP IN KELLER--SEGEL MODEL OF CHEMOTAXIS

Joint work with Grzegorz Karch and Jacek Zienkiewicz

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 1.26.

Abstract

We consider the parabolic-elliptic model for the chemotaxis with fractional (anomalous) diffusion (\(\alpha\in(0,2)\)) in several space dimensions (\(d\ge 3\)) \begin{align} u_t+(-\Delta)^{\alpha/2} u+\nabla\cdot(u\nabla v)&= 0,\ \ &x\in {\mathbb R}^d,\ t>0,\nonumber\\ \Delta v+u &= 0,\ \ & x\in {\mathbb R}^d,\ t>0,\label{eqv}\\ u(x,0)&= u_0(x)\ge0,\ \ &x\in {\mathbb R}^d.\nonumber \end{align} Global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of radial solutions in terms of suitable Morrey spaces norms are derived.

References

1. P. Biler, G. Karch, J. Zienkiewicz, Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math. 330 (2018), 834-875.
2. P. Biler, J. Zienkiewicz, Blowing up radial solutions in the minimal Keller-Segel model of chemotaxis, J. Evol. Equ. 19 (2019), 71-90.
3. P. Biler, Singularities of solutions in chemotaxis systems, De Gruyter, Berlin, 2019, ISBN 978-3-11-059789-9.

Barbara Boldin, University of Primorska, Slovenia

ECO-EVOLUTIONARY CYCLES OF VIRULENCE UNDER SELECTIVE PREDATION

Joint work with Eva Kisdi and Stefan A.H. Geritz

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 1.26.

Abstract

In nature, diseased animals do not die peacefully on their deathbeds. Rather, they are often more likely to fall victim to predators, who selectively prey upon the weak. Such disease-induced selective predation brings about changes in prey populations, which may in turn alter the evolutionary dynamics of pathogens. The resulting eco-evolutionary system may exhibit rich dynamics with multiple attractors and limit cycles. We introduce a novel technique to provide a constructive proof that the system can exhibit stable limit cycles under realistic assumptions about model ingredients and perform a detailed bifurcation analysis for a concrete example of a predator-prey-pathogen system.

References

1. E. Kisdi, S.A.H. Geritz, B. Boldin, Evolution of pathogen virulence under selective predation: A construction method to find eco-evolutionary cycles, Journal of Theoretical Biology 339 (2013), 140-150.

Tom Britton, Stockholm University, Sweden

EPIDEMICS IN STRUCTURED COMMUNITIES WITH SOCIAL DISTANCING

Joint work with Frank Ball, KaYin Leung, and Dave Sirl

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 1.26.

Abstract

Consider a large community, structured as a network, in which an epidemic spreads. Infectious individuals spread the disease to each of their susceptible neighbors, independently, at rate \(\lambda\), and each infectious individual recovers and becomes immune at rate \(\gamma\). The social distancing is modeled by each susceptible who has an infectious neighbor rewires away from this individual to a randomly chosen individual at rate \(\omega\). Our main result is surprising and says: the rewiring is rational from an individual perspective since it reduces the risk of being infected, but at the same time it may be harmful for the community at large in that the outbreak may get bigger compared to no rewiring (\(\omega=0\)).

References

1. F. Ball, T. Britton, KY. Leung, and D. Sirl, Individual preventive measures during an epidemic may have negative population-level outcomes, Journal of Royal Society: Interface. 15 (2018), 20180296.
2. F. Ball, T. Britton, KY. Leung, and D. Sirl, A stochastic SIR epidemic model with preventive dropping of edges, J. Math. Biol. 75 (2019), 1875-1951.

Andreas Deutsch, Technische Universität Dresden, Germany

BIOLOGICAL LATTICE-GAS CELLULAR AUTOMATON MODELS FOR THE ANALYSIS OF COLLECTIVE EFFECTS IN CANCER INVASION

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 1.11.

Abstract

Cancer invasion may be viewed as collective phenomenon emerging from the interplay of individual biological cells with their environment. Cell-based mathematical models can be used to decipher the rules of interaction. In these models cells are regarded as separate movable units. Here, we introduce an integrative modelling approach based on mesoscopic biological lattice-gas cellular automata (BIO-LGCA) to analyse collective effects in cancer invasion. This approach is rule- and cell-based, computationally efficient, and integrates statistical and biophysical models for different levels of biological knowledge. In particular, we provide BIO-LGCA models to analyse mechanisms of invasion in glioma and breast cancer cell lines.

References

1. A. Deutsch and S. Dormann, Cellular automaton modeling of biological pattern formation: characterization, applications, and analysis, Birkhäuser, Boston, 2018.

Marie Doumic, CNRS & Sorbonne Université, France

ESTIMATING THE FRAGMENTATION CHARACTERISTICS IN GROWING AND DIVIDING POPULATIONS

Joint work with Miguel Escobedo, Magali Tournus and Wei-Feng Xue

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 1.26.

Abstract

Growing and dividing populations may be described either by stochastic branching processes or by integro-differential equations, both related by the fact that the integro-differential equation may be seen as the Kolmogorov equation of the branching process. Denoting \(u(t,x)\) the concentration of individuals of size \(x\) at time \(t,\) a typical growth-fragmentation equation may be written as \[\frac{\partial}{\partial t} u(t,x) + \frac{\partial}{\partial t} (g(x) u(t,x) ) + B(x) u(t,x) =\int\limits_0^1 B(\frac{x}{z}) u(t,\frac{x}{z}) \frac{dk_0 (z)}{z},\] where \(g(x)\) is the growth rate, \(B(x)\) the division rate, and \(k_0\) is called the (self-similar) fragmentation kernel, which characterizes the probability for a dividing particle of size \(\frac{x}{z}\) to give rise to an offspring of size \(x\). During the last decade, using the asymptotic behaviour of this equation or of the related stochastic process to estimate the division rate (\(B(x)\) in the equation) of a population has led to many interesting questions and results, in mathematics as well as in biology. In this talk, I will review some of them, and focus on the question of estimating the fragmentation kernel \(k_0\), which revealed a much more ill-posed problem than estimating the division rate \(B(x)\). We then applied our methods to fragmenting protein fibrils, following the experiments done by W.F. Xue's laboratory in the university of Kent.

References

1. D. Béal, M. Tournus, R. Marchante, T. Purton, D. Smith, M.F. Tuite, M. Doumic, and W.-F. Xue, The Division of Amyloid Fibrils, bioRxiv, 2018.
2. T. Bourgeron, M. Doumic, and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel, Inverse Problems 30(2) (2014).
3. M. Doumic, M. Escobedo, and M. Tournus, Estimating the division rate and kernel in the fragmentation equation, Annales de l’Institut Henri Poincaré (C) Non Linear Analysis 35(7) (2018).

William Fagan, University of Maryland, College Park, USA

IMPROVED FORAGING BY SWITCHING BETWEEN DIFFUSION AND ADVECTION: BENEFITS FROM MOVEMENT THAT DEPENDS ON SPATIAL CONTEXT

Joint work with Tyler Hoffman, Daisy Dahiya, Eliezer Gurarie, Robert Stephen Cantrell, and Chris Cosner

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 1.26.

Abstract

Animals use different modes of movement at different times, in different locations, and on different scales. Incorporating such context-dependence in mathematical models represents a substantial increase in complexity, but creates an opportunity to more fully integrate key biological features. Here we consider the spatial dynamics of a population of foragers with two subunits. In one subunit, foragers move via diffusion (random search) whereas in the other, foragers move via advection (gradient-following search). Foragers switch back and forth between the subunits as functions of their spatial context (i.e., depending on whether they are inside or outside of a patch, or depending on whether or not they can detect a gradient in resource density). We consider a one dimensional binary landscape of resource patches and non-habitat and gauge success in terms of how well the mobile foragers overlap with the distribution of resources. Actively switching between dispersal modes can sometimes greatly enhance this spatial overlap relative to the spatial overlap possible when foragers merely blend advection and diffusion modes at all times. Switching between movement modes is most beneficial when an organism’s gradient-following abilities are weak compared to its overall capacity for movement, but switching can actually be quite detrimental for organisms that can rapidly follow resource gradients. An organism’s perceptual range plays a critical role in determining the conditions under which switching movement modes benefits versus disadvantages foragers as they seek out resources.

Jan Haskovec, KAUST, Saudi Arabia

RIGOROUS CONTINUUM LIMIT FOR THE DISCRETE NETWORK FORMATION PROBLEM

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 1.26.

Abstract

Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their \(\Gamma\)-convergence towards a continuum energy functional.

Laura Kanzler, Universität Wien, Austria

KINETIC MODELING OF COLONIES OF MYXOBACTERIA

Joint work with Sabine Hittmeir, Gerhard Kitzler, Angelika Manhart, Christian Schmeiser, and Joachim Schöberl

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 1.26.

Abstract

Myxobacteria are rod-shaped, social bacteria that are able to move on flat surfaces by ’gliding’ and form a fascinating example of how simple cell-cell interaction rules can lead to emergent, collective behavior. Observed movement patterns of individual bacteria in such a colony include straight runs with approximately constant velocity, alignment interactions and velocity reversals [1], [2], [3]. Experimental evidence shows that above mentioned behavior is a consequence of direct cell-contact interaction rather than diffusion of chemical signals, which indicates the suitability of kinetic modeling.

In this talk a new kinetic model of Boltzmann-type for such colonies of myxobacteria will be introduced and investigated. For the spatially homogeneous case an existence and uniqueness result will be shown, as well as exponential decay to an equilibrium for the Maxwellian collision operator. The methods used for the analysis combine several tools from kinetic theory, entropy methods as well as optimal transport. The talk will be concluded with numerical simulations confirming the analytical results.

References

1. A. Baskaran, M.C. Marchetti,, Nonequilibrium statistical mechanics of self propelled hard rods, J. Stat. Mech. 2010 (2010), P04019.
2. E. Bertin, M. Droz, G. Gregoire, Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis, J. Phys. A: Math. Theor. 42 (2009), 445001.
3. O.A. Igoshin, G. Oster, Improved stability of optimal traffic paths, Rippling of Myxobacteria 188 (2004), 221-233.

Marek Kimmel, Rice University, USA

EXPECTED SITE FREQUENCY SPECTRA OF CELLS: COALESCENT VERSUS BIRTH-AND-DEATH-PROCESS APPROACH

Joint work with Khanh Dinh, Roman Jaksik, Amaury Lambert, and Simon Tavaré

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 1.11.

Abstract

Recent years have produced a large amount of work on inference about cancer evolution from mutations identified in cancer samples. Much of the modeling work has been based on classical models of population genetics, generalized to accommodate time-varying cell population size [1]. Reverse-time, genealogical, views of such models, commonly known as coalescent theory, have been used to infer aspects of the past of growing populations [2]. Another approach is to use branching processes, the simplest scenario being the linear birth-death process (lbdp), a binary fission Markov age-dependent branching process. A genealogical view of such models is also available [3]. As will be seen in the sequel, the two approaches lead to similar but not identical results.

Inference from evolutionary models of DNA often exploits summary statistics of the sequence data, a common one being the so-called Site Frequency Spectrum. In a sequencing experiment with a known number of sequences, we can estimate for each site at which a novel somatic mutation has arisen, the number of cells that carry that mutation. These numbers are then grouped into sites which have the same number of copies of the mutant. Consider genealogy of a sample of \(n = 20\) cells that . includes 13 mutational events out of which 7 mutations are present in a single cell, 3 are present in 3 cells, 2 are present in 6, and 1 mutation is present in 17 cells. If we denote the numberof mutations present in \(k\) cells by \(\eta_k\), \(\eta_1=7\), \(\eta_3 = 3\), \(\eta_6 = 2\), and \(\eta_{17} = 1\), with all other \(\eta_k\) equal to 0. The vector \(\eta\) is called the Site Frequency Spectrum (SFS). It can be computed from the statistics of mutations in a sample of cells, in which DNA has been sequenced.

We examine how the SFS based on birth-death processes differ from those based on the coalescent model. This may stem from the different sampling mechanisms in the two approaches. However, we also show that despite this, they can be made quantitatively comparable at least for the range of parameters typical for tumor cell populations. We also touch upon the "singleton estimation problem" and the "self-renewal fraction versus proliferation rate" controversy.

References

1. M.J. Williams, B. Werner, T. Heide, C. Curtis, C.P. Barnes, A. Sottoriva and T.A. Graham, Quantification of subclonal selection in cancer from bulk sequencing data, Nature Genetics 50 (2018), 895-903.
2. R. Griffiths and S. Tavaré, The age of a mutation in a general coalescent tree, Stochastic Models 14 (1998), 273-295.
3. A. Lambert and T. Stadler, Birth-death models and coalescent point processes: The shape and probability of reconstructed phylogenies, Theoretical Population Biology 90 (2013), 113-128.

Eva Kisdi, University of Helsinki, Finland

ADAPTIVE DYNAMICS OF PATHOGENS AND THEIR HOSTS

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 1.26.

Abstract

This talk will review how pathogens evolve to exploit their hosts, and how hosts in turn evolve under selective pressure from their pathogens. Special attention is paid to nonlinear feedbacks, which are responsible for a wide range of dynamical phenomena including the evolution of diversity and the emergence of evolutionary cycles via Hopf bifurcations. I discuss whether pathogens can evolve so virulent that they drive their hosts extinct, and whether hosts can escape their pathogens by evolving such that the pathogen is no longer viable.

Toshikazu Kuniya, Kobe University, Japan

GLOBAL BEHAVIOR OF A MULTI-GROUP SIR EPIDEMIC MODEL WITH AGE STRUCTURE AND ESTIMATION OF \(\cal R_0\) FOR THE CHLAMYDIA EPIDEMIC IN JAPAN

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 1.26.

Abstract

In this talk, we study the global behavior of a multi-group SIR epidemic model with age structure. Under the assumptions that the disease transmission coefficient is independent of the age of infective individuals and the sum of the mortality and the recovery rate is constant, the model can be rewritten into a multi-group SIR model with age-dependent susceptiblity. We then define the basic reproduction number \(\cal{R}_0\) by the spectral radius of the next generation matrix and show that \(\cal{R}_0\) completely determines the global behavior of the model: if \(\cal{R}_0 \lt 1\), then the disease-free equilibrium is globally attractive, whereas if \(\cal{R}_0 \gt 1\), then the endemic equilibrium is globally attractive. In the application, we estimate \(\cal{R}_0\) for the chlamydia epidemic in Japan in 2015 by comparing four special cases of our model: a homogeneous model, an age-independent two-sex model, an age-dependent one-sex model and an age-dependent two-sex model. In conclusion, we see that \(\cal{R}_0\) is in the range \(1.0148\)-\(1.0535\), the age structure has more influence on the estimation result than the two-sex structure and disregarding the age structure could lead to the underestimation of \(\cal{R}_0\).

References

1. T. Kuniya, Global behavior of a multi-group SIR epidemic model with age structure and an application to the chlamydia epidemic in Japan, SIAM J. Appl. Math. 79 (2019), 321-340.
2. T. Kuniya, J. Wang, H. Inaba, A multi-group SIR epidemic model with age structure, Disc, Cont. Dyn. Syst. Series B 21 (2016), 3515-3550.

Junling Ma, University of Victoria, Canada

HOST CONTACT STRUCTURE IS IMPORTANT FOR THE RECURRENCE OF INFLUENZA A

Joint work with Juan M. Jaramillo, Pauline van den Driessche, Sanling Yuan

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 1.26.

Abstract

An important characteristic of influenza A is its ability to escape host immunity through antigenic drift. Individuals infected by a strain of influenza A during an epidemic have decreased immunity (estimated to be 12-25% in literature) to the drifted strains. In this talk, we compute the required decrease in immunity so that a drifted strain can invade after a pandemic. By comparing the model predictions of a homogeneously mixing mode, a heterogeneously mixing model, and a network model, we show that, for realistic loss of immunity, the establishment of a drift strain is only possible on contact networks. This suggests that stable contacts like classmates, coworkers and family members are a crucial path for the spread of influenza in human populations.

Dietmar Ölz, University of Queensland, Australia

MODELLING AND SIMULATION OF COLLECTIVE MIGRATION IN EPITHELIAL LAYERS

Joint work with Z. Neufeld, A. Czirok, and H. Khatae

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 1.26.

Abstract

The mechanochemical coupling between the cells aligned in epithelial layers impacts characteristic features of their collective migration. I will introduce a 1D model for collective cell migration in epithelial sheets in which the cytoskeletons of adjacent cells are coupled both, mechanically and through mechanochemical feedback. The travelling wave analysis of the mathematical model can be made explicit predicting a polarization wave and associated wave speed which we can be observed in experiments. Finally I will also talk about recent results based on particle simulations of collective cell migration.

Joan Saldaña, University of Girona, Spain

EPIDEMIC OSCILLATIONS AND THE SPREAD OF AWARENESS

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-8, room 1.26.

Abstract

In this talk we consider an epidemic model with a preventive behavioural response triggered by the risk perception of infection among individuals. The analysis of models combining disease and behaviour dynamics has mostly focused on the impact of the latter on the initial growth of an outbreak (computation of \(R_0\)) and the existence of endemic equilibria (see, for instance, [2, 3, 5] for mean-field models). Here we ask whether the interplay between behaviour and disease spreading is always able to prevent periodic re-emergence of a communicable disease when awareness decays over time.

In a recent work [4], it was shown that oscillations are ruled out in Susceptible-Aware-Infectious-Susceptible deterministic models with a single compartment of alerted hosts and no demographics, but they can occur when one considers two distinct compartments of aware hosts who differ in their willingness to alert other susceptible hosts. Progressive levels of awereness in epidemic modelling were considered in [1], whereas the introduction of an additionnal compartment of active alerted hosts in an SIS model (without awareness decay) was made in [6].

In a deterministic context, such epidemic flare-ups translate into solutions of ODE models showing sustained oscillations which approach zero when there is a high enough fraction of alerted individuals in the population (see [4]). The question we will address here is how robust these oscillations are when we move away from such a deterministic framework and consider stochastic simulations of the epidemic dynamics.

References

1. S. Funk, E. Gilad, C. Watkins, V.A.A. Jansen, The spread of awareness and its impact on epidemic outbreaks, PNAS 21 (2009), 6872–6877.
2. S. Funk, E. Gilad, V.A.A. Jansen, Endemic disease, awareness, and local behavioral response, Journal Theoretical Biology 264(5) (2010), 01–509.
3. D. Juher, I.Z. Kiss, J. Saldaña, Analysis of an epidemic model with awareness decay on regular random networks, Journal of Theoretical Biology 365 (2015), 457–468.
4. W. Just, J. Saldaña, Y. Xin, Oscillations in epidemic models with spread of awareness, Journal of Mathematical Biology 62 (2018), 1027–10574.
5. I.Z. Kiss, J. Cassell, M. Recker, P.L. Simon, The impact of information transmission on epidemic outbreaks, Mathematical Biosciences 225 (2010), 1–10.
6. F.D. Sahneh, F.N. Chowdhury, C.M. Scoglio, On the existence of a threshold for preventive behavioral responses to suppress epidemic spreading, Scientific Reports 2 (2012), 632.

Christian Schmeiser, Universität Wien, Austria

MATHEMATICAL MODELS OF ACTIN DRIVEN CELL MOTILITY

Joint work with Aaron Brunk, Stefanie Hirsch, Gaspard Jankowiak, Angelika Manhart, Dietmar Oelz, Diane Peurichard, Nikolaos Sfakianakis, and Christoph Winkler

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 1.26.

Abstract

Actin is one of the most important proteins occurring in practically all eukaryotic cells. It has the ability to polymerize and to form microfilaments, an essential part of the cytoskeleton. Among other functions, actin filaments contribute to the motility of crawling cells. In particular, a network of actin filaments supports the lamellipodium, a motility organelle of many cell types. The lamellipodium is a flat cell protrusion, and a first theoretical explanation of its flatness has been provided in [6]. Based on this flatness, a two-dimensional anisotropic two-phase continuum model, the Filament Based Lamellipodium Model, has been formulated and analyzed in [5] and then extended and used for simulations in [3] and [7]. The qualitative behaviour of submodels has been analyzed in [1] and in [4].

Recently, polymerization driven cortical flow has been modelled as an alternative mechanism for cell motility in [2]. The model is able to explain experimental results on adhesion-free motility in artificial micro-channels with structured walls.

References

1. S. Hirsch, A. Manhart, C. Schmeiser, Mathematical modeling of myosin induced bistability of lamellipodial fragments, J. Math. Biol. 74 (2017), 1-22.
2. G. Jankowiak, D. Peurichard, A. Reversat, M. Sixt, C. Schmeiser, Modelling adhesion-independent cell migration, preprint, 2019.
3. A. Manhart, D. Oelz, C. Schmeiser, N. Sfakianakis, An extended Filament Based Lamellipodium Model produces various moving cell shapes in the presence of chemotactic signals, J. Theor. Biol. 382 (2015), 244-258.
4. A. Manhart, C. Schmeiser, Existence of and decay to equilibrium of the filament end density along the leading edge of the lamellipodium, J. Math. Biol. 74 (2017), 169-193.
5. D. Oelz, C. Schmeiser, Derivation of a model for symmetric lamellipodia with instantaneous cross-link turnover, Archive Rat. Mech. Anal. 198 (2010), 963-980.
6. C. Schmeiser, C. Winkler, The flatness of lamellipodia explained by the interaction between actin dynamics and membrane deformation, J. Theor. Biol. 380 (2015), 144-155.
7. N. Sfakianakis, D. Peurichard, A. Brunk, C. Schmeiser, Modelling cell-cell collision and adhesion with the Filament Based Lamellipodium Model, Biomath 7 (2018), Article ID: 1811097.

Kieran Sharkey, University of Liverpool, UK

CAPTURING THE QUASI-STATIONARY DISTRIBUTION WITHIN A DETERMINISTIC FRAMEWORK FOR STOCHASTIC SIS DYNAMICS

Joint work with Christopher Overton and Robert Wilkinson

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 1.26.

Abstract

The stochastic suscetible-infectious-susceptibe (SIS) model represents an important class of epidemic dynamics, and is thought to represent processes such as the spread of sexually transmitted diseases and computer viruses. A feature of this model is the existence of a single absorbing state, corresponding to the disease free state, to which the system will always converge for finite population sizes and finite infection transmission parameters.

There has been a long history of deterministic representations of the SIS model. Relating these models to the stochastic dynamics frequently makes use of mean-field assumptions, which are derived from the infinite population limit [1]. These models provide useful theoretical insight but do not feature the absorbing state, and therefore it is hard to link the insights back to the stochastic model.

In this work we develop novel methods to account for the absorbing state of the stochastic model within a deterministic framework. We do this by obtaining a deterministic approximation to the quasi-stationary distribution (QSD) of the model; i.e. the long-term steady-state behaviour conditional on not having reached the absorbing state [2, 3]. In particular, we build a system of population level equations, which when solved provide an accurate and efficient approximate to the QSD of the Markovian network-based SIS model for a large range of networks and parameter sets.

References

1. I.Z. Kiss, J. Miller, P.L. Simon, Mathematics of Epidemics on Networks, Springer, Pub. Place, 2017.
2. J.N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous time finite Markov chains, Journal of Applied Probability: Series B 2 (1967), 88-100.
3. I. Nåssell, On the time to extinction in recurrent epidemics, Journal of the Royal Statistical Society: Series B 61 (1999), 309-330.

Zuzanna Szymańska, Polish Academy of Sciences, Poland

MICROSCOPIC DESCRIPTION OF DNA THERMAL DENATURATION

Joint work with Mateusz Dębowski and Mirosław Lachowicz

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 1.26.

Abstract

We propose a microscopic model describing the process of DNA thermal denaturation. The process consists of the splitting of DNA base pairs resulting in the separation of two complementary DNA strands. In contrast to the previous modelling attempts we take into account the states of all base pairs of DNA which in fact imposes the microscopic nature of the approach. The model is a linear integro-differential non-autonomous equation describing the dynamics of probability density which characterizes the distances between the bases within individual base pairs. We take into account not only the strength of double and triple hydrogen bonds between the complementary bases but also the stacking interactions between neighborhood base pairs. We show basic mathematical properties of the model and present numerical simulations that reproduce the sigmoid shape of DNA melting curves and reveal the appearance of experimentally observed denaturation bubbles.

References

1. M. Dębowski, M. Lachowicz, Z. Szymańska, Microscopic description of DNA thermal denaturation, Appl Math Comput. 361 (2019), 47-60.

Katarzyna Szymańska-Dębowska, Lodz University of Technology, Poland

CANARD SOLUTIONS IN EQUATIONS WITH BACKWARD BIFURCATIONS

Joint work with Jacek Banasiak and Milaine Seuneu Tchamga

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 1.26.

Abstract

We consider a singularly perturbed initial value problem in the case of intersecting quasi stationary manifolds. The main results are concerned with the asymptotic behavior of solutions as the small parameter tends to zero.

Our results are related to the Tikhonov approach. The main condition for the validity of the Tikhonov theorem is that the quasi steady states be isolated and attractive. In applications, however, we often encounter the situation when two or more quasi steady states intersect. It involves the so called exchange of stabilities: the branches of the quasi steady states change from being attractive to being repelling (or conversely) across the intersection. The assumptions of the Tikhonov theorem fail to hold in the neighbourhood of the intersection but it is natural to expect that any solution that passes close to it follows the attractive branches of the quasi steady states on either side of the intersection. However, in many cases an unexpected behaviour of the solution is observed - it follows the attracting part of a quasi steady state and, having passed the intersection, it continues along the now repelling branch of the former quasi steady state for some prescribed time and only then jumps to the attracting part of the other quasi steady state. Such a behaviour we call the delayed switch of stability. We shall focus on the so called backward bifurcation, in which two of three quasi steady states intersect and exchange stabilities at the intersection.

As an application, we consider two predator-prey models: the Rosenzweig-MacArthur model and the Leslie-Gowers/Holling model. In both these cases the quasi-steady manifolds intersect and a backward bifurcation occurs along their intersection. We give a proof of the existence of canards and provide an exact value of time at which the stability switch occurs.

Anja Voß-Böhme, Hochschule für Technik und Wirtschaft Dresden, Germany

MODEL-BASED INFERENCES ABOUT CELLULAR MECHANISMS OF TUMOR DEVELOPMENT FROM TISSUE-SCALE DATA

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 1.11.

Abstract

Cancer development is widely understood as a multistep process in which cells increase in malignancy through progressive genotypic and phenotypic alterations. Although there is an increasing knowledge about the biology of the involved cellular processes in vitro, the transfer of the results to in-vivo situations remains a challenge. This is due to the fact that the early phase of tumor development, which initially involves only a small number of cells, is hardly observable while the cellular basis of tissue-scale observations is difficult to decipher. Cell-based mathematical models provide a valuable tool to investigate in which way tissue-scale observables depend on cellular mechanisms and intercellular interaction. Here, we analyze the population dynamics of spatial and non-spatial Markov models which describe genetic and phenotypic cell changes and use these findings to calibrate the models by tissue-scale data. This allows to generate quantitative predictions about tumor initiation, progression and regression behavior.

References

1. T. Buder, A. Deutsch, B. Klink, A. Voss-Böhme, Patterns of Tumor Progression Predict Small and Tissue-Specific Tumor-Originating Niches, Frontiers in Oncology 8(668) (2019).
2. T. Buder, A. Deutsch, M. Seifert, A. Voss-Böhme, CellTrans: An R package to quantify stochastic cell state transitions, Bioinformatics and Biology Insights 11 (2017), 1-14.
3. T. Buder, A. Deutsch, B. Klink, A. Voss-Böhme, Model-Based Evaluation of Spontaneous Tumor Regression in Pilocytic Astrocytoma, PLoS Computational Biology 11(12) (2015), e1004662.

Hao Wang, University of Alberta, Canada

ANIMAL MOVEMENT WITH SPATIAL MEMORY

Joint work with Junping Shi, Chuncheng Wang, Qingyan Shi, Yongli Song, and Xiangping Yan

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 1.26.

Abstract

Animals often self-organize into territorial structure from movements and interactions of individual animals. Memory is one of the cognitive processes that may affect the movement and navigation of the animals. I will review several mathematical approaches to animal spatial movements, and then introduce our recent work via a modified Fick's law to model and simulate the memory-based movement. Results on bifurcation and pattern formation will be shown for these non-standard reaction-diffusion models.

References

1. J. Shi, C. Wang, H. Wang, X. Yan, Diffusive spatial movement with memory, Journal of Dynamics and Differential Equations, in press, 2019.
2. J. Shi, C. Wang, H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, in press, 2019
3. Q. Shi, J. Shi, H. Wang, Spatial Movement with distributed memory, preprint.

Carsten Wiuf, University of Copenhagen, Denmark

ON THE STABILITY OF THE STEADY STATES IN THE N-SITE FUTILE CYCLE

Joint work with Elisenda Feliu and Alan Rendall

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 1.26.

Abstract

The multiple or \(n\)-site futile cycle is a biological process that resides in the cell. Specifically, it is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially \(n\) times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. In its standard form it has \(3n+3\) variables (concentrations of species) and \(6n\) parameters. It is known that the system might have at least as many as \(2\lfloor \tfrac{n}{2}\rfloor+1\) steady states (where \(\lfloor x\rfloor\) is the integer part of \(x\)) for particular choices of parameters. Furthermore, for the simple futile cycle (\(n=1\)) there is only one steady state which is globally stable. For the dual futile cycle (\(n=2\)) the stability of the steady states has been determined in the following sense: There exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general \(n\), evidence that the possible number of asymptotically stable steady states increases with \(n\) has been given, which has led to the conjecture that parameter values can be chosen such that \(\lfloor\tfrac{n}{2}\rfloor+1\) out of \(2\lfloor\tfrac{n}{2}\rfloor+1\) steady states are asymptotically stable and the remaining steady states are unstable.

We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity \(2\lfloor\tfrac{n}{2}\rfloor+1\). Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result.

Falko Ziebert, Universität Heidelberg, Germany

PHASE FIELD APPROACH TO SUBSTRATE-BASED CELL MOTILITY

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 1.26.

Abstract

I will give an introduction to the substrate-based crawling motility of eukaryotic cells and survey our recent advances in its modeling. A modular approach, based on the phase field method to track the deformable and moving cells [1], allows us to describe, e.g., cell movement on structured substrates with modulated adhesion or stiffness [2], collective cell migration [3], as well as motion in 3D confinement [4]. I will also discuss the example of cellular shape waves [5], where the computational approach allows for additional insight via semi-analytic methods (employing asymptotic reduction and multiple scales).

References

1. F. Ziebert, S. Swaminathan, I. S. Aranson, Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface 9 (2012), 1084.
2. J. Löber, F. Ziebert, I. S. Aranson, Modeling crawling cell movement on soft engineered substrates, Soft Matt. 10 (2014), 1365.
3. J. Löber, F. Ziebert, I. S. Aranson, Collisions of deformable cells lead to collective migration, Sci. Rep. 5 (2015), 9172.
4. B. Winkler, I. S. Aranson, F. Ziebert, Confinement and substrate topography control cell migration in a 3D computational model, submitted (2018).
5. C. Reeves, B. Winkler, F. Ziebert, I. S. Aranson, Rotating lamellipodium waves in polarizing cells, Comms. Phys. 1 (2018), 73.

Bruno Bouchard, CEREMADE, Paris-Dauphine, PSL Research University, France

PERFECT REPLICATION WITH MARKET IMPACT: TOWARDS A DUAL FORMULATION FOR A CLASS OF SECOND ORDER COUPLED FBSDEs

Joint work with Grégoire Loeper, Halil Mete Soner, and Chao Zhou

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 2.19.

Abstract

We first extend the study of [2, 3] to stochastic target problems with general market impacts. The perfect hedging problem amounts to solving a second order coupled FBSDEs. Unlike [2, 3], the related fully non-linear PDE is not concave and the regularization/verification approach of [2] can not be applied. In place, we need to generalize the a priori estimates of [3] and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the super-hedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefficients are smooth enough. This representation suggests a dual formulation for the Brownian diffusion Markovian setting. We shall explain how this dual formulation can indeed be exploited to solve a general class of non-Markovian second order coupled FBSDEs driven by general continuous martingales.

References

1. B. Bouchard, G. Loeper, M. Soner and C. Zhou, Second order stochastic target problems with generalized market impact, arXiv:1806.08533, (2018).
2. B. Bouchard, G. Loeper and Y. Zou, Hedging of covered options with linear market impact and gamma constraint, SIAM Journal on Control and Optimization 55(5) (2017), 3319–3348.
3. G. Loeper, Option pricing with linear market impact and nonlinear Black-Scholes equations, The Annals of Applied Probability 28(5) (2018), 2664–2726.

Giulia Di Nunno, University of Oslo, Norway

OPTIMAL STRATEGIES IN A MARKET WITH MEMORY

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 2.19.

Abstract

We consider a market model driven by Volterra type dynamics driven by a time-changed Levy noise. These dynamics allow both for memory features and clustering effects in the trading times. In this framework, we study an optimal portfolio problem, which is then tackled via maximum principle. To produce such results we use different kind of information flows that take care of the time-change in adequate way and we rely on the non-anticipating stochastic derivative for random fields. Moreover, we study the solutions of Volterra type SDEs and Volterra type BSDEs driven by time-changed Levy noises.

Roxana Dumitrescu, King's College London, UK

MEAN-FIELD GAMES OF OPTIMAL STOPPING: A RELAXED SOLUTION APPROACH

Joint work with Géraldine Bouveret and Peter Tankov

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 2.19.

Abstract

We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and show its convergence.

Heiðar Eyjólfsson, Reykjavík University, Iceland

HILBERT SPACE-VALUED STOCHASTIC VOLATILITY MODELS AND AMBIT FIELDS

Joint work with Fred Espen Benth

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 2.19.

Abstract

We study Hilbert space-valued stochastic volatility models, and discuss representation and approximation of such processes. A typical application of such a model is the modelling of forward curves as an element in a given Hilbert space. Specifically, in a separable Hilbert space, a Lévy process driven variance process is introduced. We discuss ways of approximating the variance process in this setting. A problem of specific interest is how one obtains the square-root of the variance process. We discuss ways of obtaining and approximating the square root in an infinite dimensional Hilbert space. We moreover relate these models to the class Hilbert space- valued volatility modulated Volterra processes we call Hambit fiels [2]. Hambit fields are Hilbert space-valued analogues of ambit fields as introduced by Barndorff-Nielsen and Schmiegel [1]. Hambit fields can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes, for which Ornstein-Uhlenbeck process constitute a particular case. Hambit fields can moreover be interpreted as the boundary of the mild solution of a certain first order stochastic partial differential equation.

References

1. O.E. Barndorff-Nielsen and J. Schmiegel, Lévy-based tempo-spatial modelling; with applications to turbulence, Uspekhi Mat. NAUK 59 (2004), 65-91.
2. F.E. Benth and H. Eyjolfsson, Representation and approximation of ambit fields in Hilbert space, Stochastics 89 (2017), 311-347

Masaaki Fukasawa, Osaka University, Japan

3R HYBRID SCHEME FOR BROWNIAN SEMISTATIONARY PROCESSES

Joint work with Asuto Hirano

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 2.19.

Abstract

The Brownian semistationary process has attracted much attention recently in quantitative finance in the context of rough volatility modeling. We propose a new numerical approximation scheme, 3R hybrid scheme, which refines the hybrid scheme proposed by Bennedsen et al. [1] for Brownian semistationary processes. The mean squared error is shown to be significantly reduced while computational costs remain almost the same. The key idea is to reuse random variables through orthogonal projections.

References

1. M. Bennedsen, A. Lunde and M.S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance and Stochastics 21 (2017), 931-965.

Diogo Gomes, KAUST, Saudi Arabia

A MEAN-FIELD GAME PRICE MODEL

Joint work with João Saúde

Date: 2019-09-16 (Monday); Time: 10:40-11:00; Location: building B-8, room 2.19.

Abstract

Here, we introduce a price-formation model where a large number of small players can store and trade electricity. Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply vs. demand balance condition. We establish the existence of a unique solution using a fixed-point argument. In particular, we show that the price is well-defined and it is a Lipschitz function of time. Then, we study linear-quadratic models that can be solved explicitly and compare our model with real data.

Olivier Guéant, Université Paris 1 Panthéon-Sorbonne, France

IT’S ALL RELATIVE: MEAN FIELD GAME EXTENSIONS OF MERTON’S PROBLEM

Joint work with Alexis Bismuth

Date: 2019-09-16 (Monday); Time: 11:05-11:25; Location: building B-8, room 2.19.

Abstract

Merton's problem deals with the optimal investment and consumption choices of economic agents. The classical results of Merton have been extended to add many features, but never, as far as we know, to take account of jealousy. In this talk, we show how the introduction of jealousy modifies Merton's problem and results in a problem of the mean field game (MFG) type (a mean field game of controls in fact). Interestingly, many analytical results can be obtained and will be presented, along with applications.

Piotr Jaworski, University of Warsaw, Poland

FROM APPLICATIONS TO EQUATIONS

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 2.19.

Abstract

In mathematical finance, the standard approach is to model the logarithms of prices of financial assets as Wiener processes, which are correlated in a deterministic way. In my talk I will deal with a pair of Wiener processes \(W^1\) and \(W^2\) which are randomly correlated. Under the assumption that the quadratic covariation of \(W^1\) and \(W^2\) can be described by a deterministic function of \(W^1\) and \(W^2\), \[ d\langle W^1,W^2\rangle_t = A(t,W_t^1,W_t^2) dt,\] I will show that the joint distribution function \(F(t,x_1,x_2)\) and the copula \(C(t,u,v)\) of the pair \((W^1,W^2)\) are generalized weak solutions of parabolic partial differential equations.

Sigrid Källblad, Technische Universität Wien, Austria

STOCHASTIC CONTROL OF MEASURE-VALUED MARTINGALES WITH APPLICATIONS TO ROBUST FINANCE

Joint work with A. Cox, M. Larsson, and S. Svaluto.

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 2.19.

Abstract

Motivated by robust pricing problems in mathematical finance, we consider in this talk a specific constrained optimisation problem. Our approach is based on reformulating this problem as an optimisation problem over so-called measure-valued martingales (MVMs) enabling the problem to be addressed by use of dynamic programming methods. In the emerging stochastic control problem MVMs appear as weak solutions to a specific SDE for which we prove existence of solutions; we then show that our control problem satisfies the Dynamic Programming Principle and relate the value function to a certain HJB-type equation. A key motivation for the study of control problems featuring MVMs is that a number of interesting probabilistic problems can be formulated as such optimisation problems; we illustrate this by applying our results to optimal Skorokhod embedding problems as well as robust pricing problems.

Martin Larsson, Carnegie Mellon University, USA

THE EXPRESSIVENESS OF RANDOM DYNAMICAL SYSTEMS

Joint work with Christa Cuchiero and Josef Teichmann

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 2.19.

Abstract

Deep neural networks perform exceedingly well on a variety of learning tasks, in particular in finance where they are quickly gaining importance. Training a deep neural network amounts to optimizing a nonlinear objective over a very large space of parameters. This would seem a hopeless task if a globally optimal solution were required. The fact that this can succeed suggests that the result is largely insensitive to the detailed structure of the selected locally near-optimal solution, a perspective that is supported by empirical evidence. In this work we attempt a step toward a theoretical understanding of this phenomenon. In a model of deep neural networks as discretizations of controlled dynamical systems, we rigorously prove that any learning task can be accomplished even if a majority of the parameters are chosen at random

Cosimo Munari, Universität Zürich, Switzerland

DUALITY FOR RISK FUNCTIONALS ON ORLICZ SPACES

Joint work with Niushan Gao, Denny Leung, Foivos Xanthos

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 2.19.

Abstract

A well-known result by Delbaen states that every convex risk measure defined on the space of bounded positions is automatically weak-star lower semicontinuous whenever it satisfies the Fatou property. This allows to derive a nice dual representation where the constraint set of dual elements consists of countably-additive measures. This result is no longer true if one abandons the bounded setting. The objective of the talk is to show that a dual representation with countably-additive measures holds in a general Orlicz space if the risk measure is additionally assumed to be either law invariant (the risk measure depends only on the probability law of a risky position) or surplus invariant (the risk measure depends, in a suitable way, only on the downside of a risky position).

References

1. N. Gao, D. Leung, C. Munari, F. Xanthos, Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces, Finance and Stochastics 22 (2018), 395–415.
2. N. Gao, C. Munari, Surplus-invariant risk measures, Mathematics of Operations Research, to appear.

Andrea Pascucci, University of Bologna, Italy

ON STOCHASTIC LANGEVIN AND FOKKER-PLANCK EQUATIONS

Joint work with Antonello Pesce

Date: 2019-09-16 (Monday); Time: 11:30-11:50; Location: building B-8, room 2.19.

Abstract

We study existence, regularity in Hölder classes and estimates from above and below of the fundamental solution of a degenerate SPDE satisfying the weak Hörmander condition. Our method is based on a Wentzell's reduction of the SPDE to a PDE with random coeffcients to which we apply the parametrix technique to construct a fundamental solution. This approach avoids the use of the Duhamel's principle for the SPDE and the related measurability issues that appear in the stochastic framework. Applications to stochastic filtering are also discussed.

References

1. N.V. Krylov, Hörmander's theorem for stochastic partial differential equations, Algebra i Analiz 27(3) (2015), 157-182.
2. N.V. Krylov and A. Zatezalo, A direct approach to deriving filtering equations for diffusion processes, Appl. Math. Optim. 42(3) (2000), 315-332.
3. H. Kunita, Stochastic flows and stochastic differential equations, Vol. 24 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.
4. E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino 52(1) (1994), 29-63.
5. P.L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A 346 1679 (1994), 191-204.
6. A. Pascucci and A. Pesce, The parametrix method for parabolic SPDEs, (2019), arXiv:1803.06543v3.

Marcin Pitera, Jagiellonian University in Kraków, Poland

RISK SENSITIVE DYADIC IMPULSE CONTROL FOR UNBOUNDED PROCESSES

Joint work with Łukasz Stettner

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 2.19.

Abstract

Dyadic impulse control of continuous time Feller-Markov processes with risk-sensitive long-run average cost is considered. The uncontrolled process is assumed to be bounded in the weighted norm and to be ergodic; the process could be unbounded in the supremum norm and do not necessarily satisfy uniform ergodicity property. The existence of solution to suitable Bellman equation using local span contraction method is shown, and link to optimal problem solution is established with the help of Hölder’s (entropic) inequalities.

Dylan Possamaï, Columbia University, USA

A GENERAL APPROACH TO NON–MARKOVIAN TIME-INCONSISTENT STOCHASTIC CONTROL FOR SOPHISTICATED PLAYERS

Joint work with Camilo Hernández

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 2.19.

Abstract

This paper is the first attempt at a general non-Markovian theory of time--inconsistent stochastic control problems in continuous-time. We consider sophisticated agents who are aware of their time-inconsistency and take into account in future decisions. We prove here that equilibria in such a problem can be characterised through a new type of multi-dimensional system of backward SDEs, for which we obtain wellposedness. Unlike the existing literature, we can treat the case of non-Markovian dynamics, and our results go beyond verification type theorems, in the sense that we prove that any (strict) equilibrium must necessarily arise from our system of BSDEs. This is a joint work with Camilo Hernández, Columbia University.

David Prömel, University of Oxford, UK

MARTINGALE OPTIMAL TRANSPORT DUALITY AND ROBUST FINANCE

Joint work with Patrick Cheridito, Matti Kiiski, and H. Mete Soner

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 2.19.

Abstract

Without assuming any probabilistic price dynamics, we consider a frictionless financial market given by the Skorokhod space, on which some financial options are liquidly traded. In this model-free setting we show various pricing-hedging dualities and the analogue of the fundamental theorem of asset pricing. For this purpose we study the corresponding martingale optimal transport (MOT) problem: We obtain a dual representation of the Kantorovich functional (super-replication functional) defined for functions (financial derivatives) on the Skorokhod space using quotient sets (hedging sets). Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. As an immediate consequence of the duality result, we deduce a general robust fundamental theorem of asset pricing.

References

1. P. Cheridito, M. Kiiski, D. J. Prömel, H. M. Soner, Martingale Optimal Transport Duality, ArXiv Preprint arXiv:1904.04644.

Sergio Pulido, ENSIIE & Université Paris-Saclay, France

STOCHASTIC VOLTERRA EQUATIONS

Joint work with Eduardo Abi Jaber, Christa Cuchiero, and Martin Larsson

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 2.19.

Abstract

We obtain general weak existence and stability results for Stochastic Convolution Equations (SVEs) with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. The motivation to study SVEs comes from the literature on rough volatility models. Our approach relies on weak convergence in \(L^p\) spaces. The main tools are new a priori estimates on Sobolev-Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.

Emanuela Rosazza Gianin, University of Milano-Bicocca, Italy

RISK MEASURES AND PROGRESSIVE ENLARGEMENT OF FILTRATIONS: A BSDE APPROACH

Joint work with Alessandro Calvia

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 2.19.

Abstract

The aim of the talk is to investigate dynamic risk measures in the case of enlargement of filtration and its impact on the corresponding risk measure and on its properties. More precisely, we show how to induce a dynamic risk measure from a BSDE whose noise is given by a Brownian motion and a marked point process. In terms of the underlying information flow, this corresponds to a progressive enlargement of a Brownian filtration with information brought by the occurrence of random events at random times. This may describe the presence of defaults. The class of BSDEs with jumps considered was introduced in [1]. In the single jump case, we show that dynamic risk measures induced by these BSDEs admit a decomposition into two risk measures, one before and the other after the default. Furthermore, we prove that standard properties of dynamic risk measures are guaranteed by similar properties of the driver of these BSDEs and that time-consistency holds. From a financial point of view, the decomposition of the "global” risk measure into different "local” ones is reasonable. Before and after a default time, indeed, the risk measure should be updated in order to take into account the new information.

References

1. I. Kharroubi, T. Lim, Progressive enlargement of filtrations and backward stochastic differential equations with jumps, J. Theoret. Probab. 27 (2014), 683-724.

Walter Schachermayer, Universität Wien, Austria

ASYMPTOTIC SYNTHESIS OF CONTINGENT CLAIMS IN A SEQUENCE OF DISCRETE-TIME MARKETS

Joint work with M. Kreps

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 2.19.

Abstract

We prove a connection between discrete-time models of financial markets and the celebrated Black-Scholes-Merton continuous-time model in which "markets are complete." Specifically, we prove that if (a) the probability law of a sequence of discrete-time models converges (in the functional sense) to the probability law of the Black-Scholes-Merton model, and (b) the largest possible one-period step in the discrete-time models converges to zero, then every bounded and continuous contingent claim can be asymptotically synthesized with bounded risk: For any \(\epsilon > 0\), a consumer in the discrete-time economy far enough out in the sequence can synthesize a claim that is no more than \(\epsilon\) different from the target contingent claim \(x\) with probability at least \(1 -\epsilon\), and which, with probability 1, has norm less or equal to the norm of the target claim. This shows that, in terms of important economic properties, the Black-Scholes-Merton model, with its complete markets, idealizes many more discrete-time models than models based on binomial random walks.

Łukasz Stettner, Polish Academy of Sciences, Poland

PORTFOLIO OPTIMIZATION WITH TRANSACTION COSTS - DIRECT APPROACH

Joint work with Tomasz Rogala

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 2.19.

Abstract

We consider discrete time market with general bid and ask prices and will be interested in maximization of utility from terminal wealth. Our approach will be direct, we shall not use well developed duality theory (see for example [1] and references therein). The result is based on an analysis of general dynamic programming equation and work with suitable selectors. We show in particular the existence and form of shadow price i.e. the price on the market without proportional transaction costs for which the strategies and the value of the functional (utility from terminal wealth) are the same as in the case of market with transaction costs. Such results are obtained under general conditions on the utility function as well as on the bid and ask prices, both in one as well as in multidimensional case. Results presented are based on the papers [2], [3] and [4].

References

1. C. Czichowski, J. Muhle-Karbe, W. Schachermayer, ransaction costs, shadow prices, and duality in discrete time, SIAM J. Financial Math. 5 (2014), 258-277.
2. T. Rogala, Ł. Stettner, On Construction of Discrete Time Shadow Price, Appl. Math. Optim. 72(3) (2015), 391-433.
3. T. Rogala, Ł. Stettner, Optimal strategies for utility from terminal wealth with general bid and ask prices, Appl. Math. Optim. (2019) published online.
4. T. Rogala, Ł. Stettner, Multidimensional shadow price with general bid and ask prices, in preparation.

Michael Tehranchi, University of Cambridge, UK

DUALITY FOR HOMOGENEOUS OPTIMISATION PROBLEMS

Joint work with David Driver

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 2.19.

Abstract

This talk is concerned with stochastic optimal control problems with a certain homogeneity. For such problems, a novel dual problem is formulated. The results are applied to a stochastic volatility variant of the classical Merton problem. Another application of this duality is to the study the right-most particle of a branching Levy process.

Giovanni S. Alberti, University of Genoa, Italy

INFINITE-DIMENSIONAL INVERSE PROBLEMS WITH FINITE MEASUREMENTS

Joint work with Matteo Santacesaria

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 3.22.

Abstract

In this talk I will discuss how ideas from applied harmonic analysis, in particular sampling theory and compressed sensing, may be applied to inverse problems for partial differential equations. The focus will be on inverse boundary value problems for the conductivity and the Schrodinger equations, but the approach is very general and allows to handle many other classes of inverse problems. I will give uniqueness and stability results, both in the linearized and in the nonlinear case. These results make use of a recent general theory of infinite-dimensional compressed sensing for deterministic and non-isometric operators, which will be briefly surveyed.

References

1. G.S. Alberti, M. Santacesaria, Calderón’s Inverse Problem with a Finite Number of Measurements, arXiv:1803.04224, 2018.
2. G.S. Alberti, M. Santacesaria, Infinite Dimensional Compressed Sensing from Anisotropic Measurements and Applications to Inverse Problems in PDE, arXiv:1710.11093, 2017.
3. G.S. Alberti, M. Santacesaria, Infinite-Dimensional Inverse Problems with Finite Measurements, in preparation.

Martin Brokate, Technische Universität München, Germany

RATE INDEPENDENT EVOLUTIONS: DIFFERENTIAL SENSITIVITY

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 3.22.

Abstract

Rate independent evolutions are inherently nonsmooth. Mappings involving rate independent evolutions do not possess classical derivatives. However, basic rate independent processes like the scalar play or stop hysteresis operator have weak derivatives in the sense of Bouligand. Moreover, they are semismooth. As a consequence, the control-to-state mapping of certain evolutions which include rate independent elements also enjoy these properties. This work is published in [1, 2], based on the previous work [3].

References

1. M. Brokate, Newton and Bouligand derivatives of the scalar play and stop operator, arXiv:1607.07344v2 [math.FA] (2019), 41 pp., submitted to Math. Model. Nat. Phenom.
2. M. Brokate, K. Fellner, M. Lang-Batsching, Weak differentiability of the control-to-state mapping in a parabolic equation with hysteresis, arXiv:1905.01863 [math.AP] (2019), 18 pp., submitted to Nonlinear Differ. Equ. Appl.
3. M. Brokate, P. Krejčí, Weak differentiability of scalar hysteresis operators, Discrete Contin. Dyn. Syst. 35 (2015), 2405-2421.

Yves Capdeboscq, Université de Paris, France

SOLVING HYBRID INVERSE PROBLEMS MOST OF THE TIME

Joint work with Giovanni S. Alberti

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 3.22.

Abstract

We show that combining a projection method attributed to H. Whitney in geometry, and a unique continuation principle for solution of 2nd order PDEs, a small number of boundary condition can be selected so that the vector space generated by the gradient fields of electrical potentials be of maximal rank everywhere in the domain. This result applies in particular when the conductivity of the medium varies in space. This provides a generalization of the so-called thermal coordinates beyond the two dimensional case.

Marco Caroccia, University of Florence, Italy

INTEGRAL REPRESENTATION OF LOCAL ENERGIES ON \(BD\)

Joint work with Matteo Focardi and Nicolas Van Goethem

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 3.22.

Abstract

In collaboration with Matteo Focardi and Nicolas Van Goethem we make use of the result on the structure of the \(\mathcal{A}\)-free measure provided in [1] to give a full integral representation result for local energies defined on the space of bounded deformation maps \(BD(\Omega)\) which accounts also for the Cantor part of the symmetric gradient measure. We employ the global method introduced in [2] and exploited also in [3] to give an integral representation Theorem for energies defined on \(SBD(\Omega)\). A full integral representation result for energy on \(BD(\Omega)\) has been missing until now due to a lack of information on the structure of the Cantor part of the symmetric gradient measure. This piece of information has been finally provided by Rindler, De Philippis in the celebrated result of 2016 [1] and it represents the crucial ingredient of our proof, together with a double blow-up procedure and a fine analysis of the blow-up limits around Cantor points. I will briefly describe the methodology of the proof and the consequences of our result in fracture mechanics and in damage models.

References

1. G. De Philippis and F. Rindler, On the structure of A-free measures and applications, Annals of Mathematics, (2016) pp. 1017–1039.
2. G. Bouchitté, I. Fonseca, and L. Mascarenhas, A global method for relaxation, Archive for rational mechanics and analysis 145(1) (1998) pp. 51-98.
3. F. Ebobisse and R. Toader, A note on the integral representation of functionals in the space SBD (O), Rend. Mat. Appl. 7 (2001).

Daan Crommelin, CWI & University of Amsterdam, Netherlands

UNRESOLVED SCALES AND STOCHASTIC PARAMETERIZATION IN ATMOSPHERE-OCEAN MODELING

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 3.22.

Abstract

The question how to represent or parameterize processes at unresolved scales remains an important issue in atmosphere-ocean modeling, posing both practical and theoretical challenges. For multiscale dynamical systems such as atmosphere, ocean and climate, the theoretical framework of the Mori-Zwanzig (MZ) formalism can help to guide the development of reduced (or coarse-grained) models in which processes at small/fast scales are no longer explicitly resolved but instead parametrized by stochastic models terms [1]. In practice it is often not feasible to derive these reduced models including their stochastic closure analytically following MZ. However, the formalism gives insight in suitable functional forms of the reduced models and in particular in the role of memory terms. This insight can be useful in data-based approaches, where model closures or parameterizations are extracted from available data. I will discuss work where such a data-based approach is developed and used in atmosphere-ocean modeling. As part of this approach, both resampling methods [2, 4] and discrete models [3] have been explored.

References

1. G. Gottwald, D. Crommelin, C. Franzke, Stochastic Climate Theory [in:] Nonlinear and Stochastic Climate Dynamics (ed. C. Franzke and T. O'Kane), 209-240, Cambridge University Press, 2017.
2. N. Verheul, D. Crommelin, Data-driven stochastic representations of unresolved features in multiscale models, Comm. Math. Sci. 14 (2016), 1213-1236.
3. J. Dorrestijn, D. Crommelin, P. Siebesma, H. Jonker, F. Selten, Stochastic convection parameterization with Markov chains in an intermediate complexity GCM, J. Atmos. Sci. 73 (2016), 1367-1382.
4. W. Edeling, D. Crommelin, Towards data-driven dynamics surrogate models for ocean flow, to appear in Proceedings of PASC 2019 conference (Zurich, 2019).

Jana de Wiljes, Universität Potsdam, Germany

ACCURACY OF NONLINEAR FILTERS

Joint work with Xin Tong

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 3.22.

Abstract

In the context of nonlinear high dimensional filtering problems ensemble based techniques such as the Ensemble Kalman Filter are still consider state of the art despite the lack of mathematical foundation in this setting. In a recent study long time stability and accuracy results for the deterministic Ensemble Kalman Bucy Filter were derived for a setting with fully observed state subject to small measurement noise [1]. Here these results are extended to the case where the ensemble size is larger than the state space for a localized deterministic Ensemble Kalman Bucy Filter. In contrast to the previously derived bounds the bounds for the localized filter are independent of the ensemble size. Further a dimension independent bound is obtained for the individual components of the error and a Laplace type condition holds.

References

1. J. de Wiljes, S. Reich, W. Stannat, Long-Time Stability and Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and Small Measurement Noise, SIAM J. APPLIED DYNAMICAL SYSTEMS 17 (2018), 1152-1181.

Emre Esenturk, University of Warwick, UK

MATHEMATICAL ANALYSIS OF EXCHANGE DRIVEN GROWTH: FUNDAMENTALS AND LARGE TIME BEHAVIOUR

Joint work with Juan Velazquez

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 3.22.

Abstract

Exchange-driven growth is a process (represented by an infinite set of ODEs) in which pairs of clusters interact and exchange single unit of mass at a time. In the recent years EDG has been used to model several natural and social phenomena The rate of exchange is given by an interaction kernel \(K(j,k)\) which depends on the masses of the two interacting clusters (of sizes \(j\) and \(k\)). Despite its wide use first mathematical analyses of this system were provided only recently [1].

In this talk we present results on the fundamentals properties (existence, uniqueness, nonexistence and etc) and the large time behaviour. For the existence, we show two different sets of results depending on whether \(K(j,k)\) is symmetric or not. In the case of non-symmetric kernels we present global existence and uniqueness of solutions for kernels satisfying \(K(j,k)\leq Cjk\). This result is optimal in the sense that for faster growing kernels the solutions cannot exist (up to some technical assumptions). On the other hand, in the case of symmetric kernels we show that global unique solutions exist for kernels satisfying \(K(j,k)\leq j^{\mu}k^{\nu}+j^{\nu}k^{\mu}\) (\(\mu +\nu\leq3\) and \(\mu,\nu\leq2)\) and that the nonexistence is also "delayed". For the large time behavior, we again show two sets of result for separable type kernels. Under some technical assumptions, we show that the system admits equilibrium solutions up to a critical mass above which there is no equilibrium. We show that if the system has an initial mass above the critical mass then the solutions converge to critical equilibrium distribution in a weak sense while strong convergence can be shown when initial mass is below.

References

1. E. Esenturk, Mathematical theory of exchange-driven growth, Nonlinearity 31 (2018), 3460-3483.

Michael Herrmann, Technische Universität Braunschweig, Germany

LONGTIME BEHAVIOR AND ASYMPTOTIC REGIMES FOR SMOLUCHOWSKI EQUATIONS

Joint work with Barbara Niethammer and Juan J.L. Velázquez

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 3.22.

Abstract

Smoluchowski's coagulation equation is the most fundamental dynamical model for mass aggregation and appears in many different branches of physics, chemistry, biology, and materials science. The mathematical properties of this nonlinear integral equation, however, are only partially understood and depend intimately on the chosen coagulation kernel.

In this talk we discuss several asymptotic regimes for kernel functions and investigate the longtime behavior of solutions by combining asymptotic analysis, heuristic arguments, and numerical simulations. In particular, we study traveling waves and self-similar profiles for near-diagonal kernels with homogeneity one and provide analytical or numerical evidence for the onset of instabilities and the formation of oscillations. We further sketch the challenges in the numerical computation of self-similar solutions and initial value problems.

Cartoon of the self-similar solutions in two different asymptotic regimes.

References

1. M. Herrmann, B. Niethammer, J.J.L. Velázquez, Instabilities and oscillations in coagulation equations with kernels of homogeneity one, Quart. Appl. Math. 75(1) (2016), 105–130.
2. M. Bonacini, B. Niethammer, J.J.L. Velázquez, Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one, Arch. Rational. Mech. Anal. 233(1) (2019) 1–43.
3. Ph. Laurençot, B. Niethammer, J.J.L. Velázquez, Oscillatory dynamics in Smoluchowski’s coagulation equation with diagonal kernel, Kinet. Relat. Models 11(4) (2018) 933–952.

Dietmar Hömberg, WIAS & Technische Universität Berlin, Germany

MATHEMATICS FOR STEEL PRODUCTION AND MANUFACTURING

Joint work with Manuel Arenas, Prerana Das, and Robert Lasarzik

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 3.22.

Abstract

In my presentation I will discuss some results from the European Industrial Doctorate project "MIMESIS - Mathematics and Materials Science for steel production and manufacturing". Applications cover tube welding, induction hardening and flame cutting.

The mathematical models considered combine a vector potential formulation of Maxwell's equations with a nonlinear heat equation and an evolution equation for the change of microstructure. In the presentation we analyse the well-posedness of these multi-field problems, discuss related optimal control problems and show some simulation results related to real industrial use cases.

References

1. D. Hömberg, T. Petzold, E. Rocca, Analysis and simulations of multifrequency induction hardening, Nonlinear Analysis: Real World Applications 22 (2015), 84-97.
2. D. Hömberg, Q. Liu, J. Montalvo-Urquizo, D. Nadolski, T. Petzold, A. Schmidt, A. Schulz, Simulation of multi-frequency-induction-hardening including phase transitions and mechanical effects, Finite Elements in Analysis and Design 121 (2016), 86-100.
3. J.I. Asperheim, P. Das, B. Grande, D. Hömberg, T. Petzold, Numerical simulation of high-frequency induction welding in longitudinal welded tubes, WIAS Preprint No. 2600, (2019).

Radu Ignat, Université Paul Sabatier, France

PROGRESS IN THE ANALYSIS OF DOMAIN WALLS IN THIN FERROMAGNETIC FILMS

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 3.22.

Abstract

Ferromagnetic materials are nowadays widely used as technological tools, especially for magnetic data storage. The modeling of small ferromagnets is based on the micromagnetic theory, an intriguing example of multiscale, nonconvex and nonlocal variational problems. One of the main challenges consists in understanding the pattern formation of the magnetization, in particular the domain walls. The aim of my talk is to present recent progress in the analysis of domain walls such as Bloch and Néel walls. I will present several results concerning their structure, their properties (stability, symmetry etc.) as well as the interaction energy of domain walls. The proof of these results is based on methods coming from geometric analysis and harmonic maps, elliptic regularity theory, variational methods and hyperbolic conservation laws.

References

1. L. Döring, R. Ignat, Asymmetric domain walls of small angle in soft ferromagnetic films, Arch. Ration. Mech. Anal. 220 (2016), 889-936.
2. L. Döring, R. Ignat, F. Otto, A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types, J. Eur. Math. Soc. (JEMS) 16 (2014), 1377-1422.
3. R. Ignat, Singularities of divergence-free vector fields with values into \(\mathbb{S}^1\) or \(\mathbb{S}^2\). Applications to micromagnetics, Confluentes Mathematici 4 (2012), 1-80.
4. R. Ignat, B. Merlet, Lower bound for the energy of Bloch walls in micromagnetics, Arch. Ration. Mech. Anal. 199 (2011), 369-406.
5. R. Ignat, A. Monteil, A DeGiorgi type conjecture for minimal solutions to a nonlinear Stokes equation, Comm. Pure Appl. Math., accepted (2019).
6. R. Ignat, R. Moser, A zigzag pattern in micromagnetics, J. Math. Pures Appl. 98 (2012), 139-159.
7. R. Ignat, R. Moser, Interaction energy of domain walls in a nonlocal Ginzburg-Landau type model from micromagnetics, Arch. Ration. Mech. Anal. 221 (2016), 419-485.
8. R. Ignat, R. Moser, Energy minimisers of prescribed winding number in an \(\mathbb{S}^1\)-valued nonlocal Allen-Cahn type model, preprint arXiv:1810.11427 (2018).
9. R. Ignat, F. Otto, The magnetization ripple: a nonlocal stochastic PDE perspective, J. Math. Pures Appl., online (2019).

Flaviana Iurlano, Sorbonne Université, France

CONCENTRATION VERSUS OSCILLATION EFFECTS IN BRITTLE DAMAGE

Joint work with Jean-François Babadjian and Filip Rindler

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 3.22.

Abstract

This work is concerned with an asymptotic analysis, in the sense of \(\Gamma\)-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero Lebesgue measure and, at the same time and to the same order \(\varepsilon\), the stiffness of the damaged material becomes small. Three main features make the analysis highly nontrivial: at \(\varepsilon\) fixed, minimizing sequences of each brittle damage model oscillate and develop microstructures; as \(\varepsilon\to 0\), concentration of damage and worsening of the elastic properties are favoured; and the competition of these phenomena translates into a degeneration of the growth of the elastic energy, which passes from being quadratic (at \(\varepsilon\) fixed) to being linear (in the limit). Consequently, homogenization effects interact with singularity formation in a nontrivial way, which requires new methods of analysis. We explicitly identify the \(\Gamma\)-limit in two and three dimensions for isotropic Hooke tensors. In the limit, a surprising expression for the bulk density appears involving now a continuum damage variable. We further consider the regime where the divergence is square-integrable, which in the limit leads to a Tresca-type plasticity model.

Hans Knüpfer, Universität Heidelberg, Germany

DOMAIN PATTERN FORMATION IN FERROMAGNETIC SAMPLES

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 3.22.

Abstract

We investigate the optimal shape and patterns for magnetic domains from the perspective of Calculus of Variations. These patterns are driven mainly by the competition of local energies such as interfacial energies and anisotropy energy and the nonlocal magnetostatic interaction. In particular, we are interested in pattern formation in thin ferromagnetic films with perpendicular anistropy and in the phase transformation for non-trivial states. For the analysis, we do not use any assumptions on the shape of the domain, rather the arguments are based on the derivation of suitable interpolation inequalities.

Kei Fong Lam, Chinese University of Hong Kong, Hong Kong

TOTAL VARIATION AND PHASE FIELD REGULARISATIONS OF AN INVERSE PROBLEM WITH QUASILINEAR MAGNETOSTATIC EQUATIONS

Joint work with Irwin Yousept

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-8, room 3.22.

Abstract

We tackle the inverse problem of reconstructing a discontinuous coefficient in magnetostatic equations from measurements in a subdomain. This problem is motivated from identifying the location of magnetic materials (e.g. iron) in a bounded domain containing also non-magnetic materials (e.g. copper), and can be viewed as an idealised problem for non-invasive/non-destructive testing based on electromagnetic phenomena. The magnetic material produces a stronger response compared to the non-magnetic material in the presence of an applied current field, and the situation can be well-described by quasilinear H(curl) magnetostatic equations. As the inverse problem is likely to be ill-posed, we reformulate it into a constraint minimisation problem with perimeter penalisation. Existence of minimisers, stability with respect to data perturbation, and consistency as the penalisation parameter tends to zero are discussed. We then introduce a further phase field approximation of the minimisation problem and derive the first order necessary optimality conditions. Then, we investigate the sharp interface limit to demonstrate the phase field approximation is a meaningful method to solve the inverse problem.

Ilaria Lucardesi, Université de Lorraine, France

ENERGY RELEASE RATE IN PLANAR ELASTICITY IN PRESENCE OF REGULAR CRACKS

Joint work with Stefano Almi and Giuliano Lazzaroni

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 3.22.

Abstract

In this talk, we first analyze the singular behavior of the displacement of a linearly elastic body in dimension 2 close to the tip of a \(C^\infty\) crack, extending the well-known results for straight fractures [4]. As conjectured by Griffith [3], the displacement behaves as the sum of an \(H^2\)-function and a linear combination of two singular functions, whose profile is similar to the square root of the distance from the tip. The coefficients of the linear combination are the so called stress intensity factors. Afterwards, we prove the differentiability of the elastic energy with respect to an infinitesimal fracture elongation and we compute the energy release rate, enlightening its dependence on the stress intensity factors [2]. In the last part of the talk we present the generalization to \(C^{1,1}\) fractures and an application to crack evolution [1].

References

1. S. Almi, G. Lazzaroni, I. Lucardesi, Paper in preparation.
2. S. Almi, I. Lucardesi, Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks, Nonlinear Differ. Equ. Appl. 25 (2018).
3. A.A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Royal Soc. A 221 (1921), 163-198.
4. P. Grisvard, Singularities in Boundary Value Problems, Research in Applied Mathematics 22 Springer, Berlin, 1992.

Boris Muha, University of Zagreb, Croatia

EXISTENCE AND REGULARITY FOR THE NON-LINEAR KOITER SHELL INTERACTING WITH THE \(3D\) INCOMPRESSIBLE FLUID

Joint work with Sebastian Schwarzacher

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 3.22.

Abstract

We study the unsteady Navier Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter Type. The latter one constitutes a moving part of the boundary of the physical domain of the fluid. This leads to a coupled system of non-linear PDEs with the moving boundary. We study weak solution to the corresponding fluid-structure interaction (FSI) problem. We introduce new methods that allow to prove higher regularity estimates for the shell. Due to the improved regularity estimates it is then possible to extend the known existence theory of weak solutions to the FSI problem with non-linear Koiter shell. The regularity result holds for arbitrary weak solution under certain geometric condition on the deformation of the boundary.

Barbara Niethammer, Universität Bonn, Germany

DYNAMIC SCALING IN COAGULATION EQUATIONS

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 3.22.

Abstract

In 1916 Smoluchowski derived a mean-field model for mass aggregation in order to develop a mathematical theory for coagulation processes. Since Smoluchowski's groundbreaking work his model and variants of it including fragmentation terms have been used in a diverse range of applications such as aerosol physics, polymerization, population dynamics, or astrophysics.

In this talk I will give an overview on recent work, studying the long-time behaviour of such equations. A key question is that of dynamic scaling, that is whether solutions develop a universal self-similar form for large times. This issue is only understood for some exactly solvable cases, while in the general case most questions are still completely open. I will give an overview of the main results in the past decades and explain why we believe that in general the scaling hypothesis is not true.

Elisabetta Rocca, University of Pavia, Italy

RECENT RESULTS ON ADDITIVE MANUFACTURING GRADED-MATERIAL DESIGN BASED ON PHASE-FIELD AND TOPOLOGY OPTIMIZATION

Joint work with Ferdinando Auricchio, Elena Bonetti, Massimo Carraturo, Dietmar Hoemberg, and Alessandro Reali

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 3.22.

Abstract

A novel graded-material design for additive manufacturing based on phase-field and topology optimization is introduced by means of an additional phase-field variable in the classical single-material phase-field topology optimization algorithm. This new variable is used to grade the material properties in a continuous fashion. Different numerical examples are discussed and first order optimality conditions are obtained including possible stress constraints in the objective functional. From the presented results we can observe that the proposed algorithm adds additional freedom in the design, exploiting the higher flexibility coming from additive manufacturing technology.

References

1. M. Carraturo, E. Rocca, E. Bonetti, D. Hoemberg, A. Reali, A. Auricchio, Additive Manufacturing Graded-material Design based on Phase-field and Topology Optimization, arXiv:1811.07205v2 (2018).

Lucjan Sapa, AGH University of Science and Technology, Poland

MATHEMATICAL MODELS OF INTERDIFFUSION WITH VEGARD RULE

Joint work with Bogusław Bożek and Marek Danielewski

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 3.22.

Abstract

We study the diffusional transport in an \(s\)-component solid solution. Let \(\Omega\subset\mathbb{R}^n\) be an open and bounded set with a piecewise smooth boundary \(\partial\Omega\) and let \(T>0\) be fixed. Moreover, let \(\Omega_i\), \(D_i\), \(j_i\) and \(c_{0i}\) mean the partial molar volumes, the diffusion coefficients, the evolution of a mass through \(\partial\Omega\) and the initial concentrations. The unknowns are the concentrations of the components of a mixture \(c_i\) and the potential \(F\) of a drift velocity.

The local mass conservation law for fluxes with the Darken drift term and the Vegard rule lead to the parabolic-elliptic system of strongly coupled nonlinear differential equations \[ \left\{ \begin{array} {lcc} \partial_tc_i+{\rm{div}}\bigl(-D_i(c_1,...,c_s)\nabla c_i+c_i\nabla F\bigl)=0 & \text{on} & [0,T]\times\Omega,\\ \triangle F={\rm{div}}\bigl(\sum_{k=1}^s\Omega_k D_k(c_1,...,c_s)\nabla c_k\bigl) & \text{on} & [0,T]\times\Omega,\\ \int_\Omega Fdx=0 & \text{on} & [0,T],\\ \end{array} \right. \tag{1}\] with the nonlinear coupled initial-boundary conditions \[ c_i(0,x)=c_{0i}(x) \quad\text{on}\quad \Omega, \tag{2}\] \[ \left\{ \begin{array} {lcc} -D_i(c_1,...,c_s)\frac{\partial c_i}{\partial{\bf{n}}}+c_i\frac{\partial F}{\partial{\bf{n}}}=j_i(t,x) & \text{on} & [0,T]\times\partial \Omega,\\ \frac{\partial F}{\partial{\bf{n}}}=\sum_{k=1}^s\Omega_k\bigl(D_k(c_1,...,c_s)\frac{\partial c_k}{\partial{\bf{n}}}+j_k(t,x)\bigl) & \text{on} & [0,T]\times\partial\Omega, \end{array} \right.\tag{3} \] for \(i=1,...,s\). This model was introduced in [3] and a some special case in [4]. In the one-dimensional case it can be transformed to the well-known model studied for example in [2, 5].

We will present theorems on existence, uniqueness and properties of weak solutions in the suitable Sobolev spaces. Moreover, finite implicit difference methods (FDM) and theorems concerned convergence and stability will be given. The agreement between the theoretical results, numerical simulations and experimental data will be shown.

References

1. B. Bożek, L. Sapa, M. Danielewski, Difference methods to one and multidimensional interdiffusion models with Vegard rule, Math. Model. Anal. 24 (2019), 276–296.
2. B. Bożek, L. Sapa, M. Danielewski Existence, uniqueness and properties of global weak solutions to interdiffusion with Vegard rule, Topol. Methods Nonlinear Anal. 52 (2018), 423–448.
3. L. Sapa, B. Bożek, M. Danielewski, Weak solutions to interdiffusion models with Vegard rule, 6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017) (Budapest, Hungary, 2017), 020039-1–020039-9, AIP Conference Proceedings 1926, American Institute of Physics, https://doi.org/10.1063/1.5020447, 2018.
4. B. Wierzba, M. Danielewski, The lattice shift generated by two dimensional diffusion process, Comp. Mater. Sci. 95 (2014), 192–197.
5. K. Holly, M. Danielewski, Interdiffusion and free-boundary problem for r-component \((r \ge 2)\) one- dimensional mixtures showing constant concentration, Phys. Rev. B 50 (1994), 13336–13346.

Francesco Solombrino, University of Naples Federico II, Italy

STABEL CONFIGURATIONS OF PRESTRAINED RODS

Joint work with Marco Cicalese and Matthias Ruf

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-8, room 3.22.

Abstract

We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By \(\Gamma\)-convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global and/or local stability of straight and helical configurations. We also show, by means of a bifurcation analysis, an exchange of stability between the straight configuration and a branch of local minimizers with so-called hemihelical shape, confirming experimental results.

References

1. M. Cicalese, M. Ruf, and F. Solombrino, On global and local minimizers of prestrained thin elastic rods, Calculus of Variations and Partial Differential Equations 56(4):115 (2017).
2. M. Cicalese, M. Ruf, and F. Solombrino, Hemihelical local minimizers in prestrained elastic bistrips, Zeitschrift für angewandte Mathematik und Physik ZAMP 68(6):122 (2017).

Samuel Stechmann, University of Wisconsin-Madison, USA

CHALLENGES IN DATA ASSIMILATION AND PREDICTION OF TROPICAL WEATHER AND CLIMATE

Joint work with Ying Li

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 3.22.

Abstract

Data assimilation is the process of combining (imperfect) observational data and (imperfect) model data in order to estimate the state of a complex system, such as the atmosphere. It is crucial to weather prediction as it provides the initial conditions for a forecast. In this talk, we describe the unique challenges that arise for tropical weather/climate, along with some recent results on estimating the intrinsic limits of predictability in the tropics using observational data.

Marita Thomas, WIAS, Germany

ANALYTICAL AND NUMERICAL ASPECTS OF RATE-INDEPENDENT GRADIENT-REGULARIZED DAMAGE MODELS

Date: 2019-09-19 (Thursday); Time: 12:05-12:25; Location: building B-8, room 3.22.

Abstract

This presentation deals with techniques for the spatial and temporal discretization of models for rate-independent damage featuring a gradient regularization and a non-smooth constraint due to the unidirectionality of the damage process. A suitable notion of solution for the non-smooth process is introduced and its corresponding discrete version is studied by combining a time-discrete scheme with finite element discretizations of the domain. Results and challenges on the convergence of the discrete problems in the sense of evolutionary Gamma-convergence in dependence of the choice of the gradient term and the mesh properties are discussed. Directions towards adaptive strategies are pointed out. The presented results are based on collaborations with S. Tornquist (WIAS), Ch. Kuhn and A. Schlüter (Kaiserslautern), S. Bartels and M. Milicevic (U Freiburg) and with M. Walloth and W. Wollner within the Priority Program SPP 1748 of the German Research Foundation.

Nicolas Van Goethem, University of Lisbon, Portugal

INTRINSIC VIEWS IN ELASTO-PLASTICITY

Joint work with Samuel Amstutz

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 3.22.

Abstract

In this talk, I will present a new model of elasto-plasticity based on the elastic strain incompatibility. Existence results for a novel boundary value problem will be reported as well as asymptotic results. Indeed, this new system for incompatible elasticity generalizes the classical linearized elasticity system to which it converges as the incompatibility modulus converges to \(-\infty\). I will also briefly introduce intrinsic views from a historical point of view.

Alex Viguerie, University of Pavia, Italy

A FAT BOUNDARY-TYPE METHOD FOR LOCALIZED NONHOMOGENEOUS MATERIAL PROBLEMS

Joint work with Ferdinando Auricchio and Silvia Bertoluzza

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 3.22.

Abstract

Problems with localized nonhomogeneous material properties arise frequently in many applications and are a well-known source of difficulty in numerical simulations. In certain applications (including additive manufacturing), the physics of the problem may be considerably more complicated in relatively small portions of the domain, requiring a significantly finer local mesh compared to elsewhere in the domain. This can make the use of a uniform mesh numerically unfeasible. While nonuniform meshes can be employed, they may be challenging to generate (particularly for regions with complex boundaries) and more difficult to precondition. To address the aforementioned challenges, we employ a technique related to the Fat boundary method [1] as a possible alternative. We analyze the proposed methodology from a mathematical point of view and validate our findings with a series of two-dimensional numerical tests.

References

1. A. Viguerie, S. Bertoluzza, F. Auricchio, A Fat Boundary-type Method for Localized Nonhomogeneous Material Problems, submitted.