Monday, July 13

09:00 - 10:00 Registration
10:00 - 10:30 Coffee break
10:30 - 12:00 Robert L. Jerrard: Dynamics of topological defects in nonlinear Hamiltonian PDEs. (part 1)
12:00 - 14:00 Lunch break
14:00 - 15:30 Stéphane Mischler: Semigroups in Banach spaces and applications to (nonlinear) evolution PDEs (part 1)
15:30 - 16:00 Coffee
16:00 - 17:30 Short talks: Esther Daus, Patrick van Meurs , Min-Gi Lee, Maxime Herda
17:30 - 19:00 Reception & Postersession

Esther Daus: Hypocoercivity for a linearized multi-species Boltzmann system

A new coercivity estimate on the spectral gap of the linearized Boltzmann collision operator for multiple species is proved. The assumptions on the collision kernels include hard and Maxwellian potentials under Grad's angular cut-off condition. Two proofs are given: a non-constructive one, based on the decomposition of the collision operator into a compact and a coercive part, and a constructive one, which exploits the "cross-effects'' coming from collisions between different species and which yields explicit constants. Furthermore, the essential spectra of the linearized collision operator and the linearized Boltzmann operator are calculated. Based on the spectral-gap estimate, the exponential convergence towards global equilibrium with explicit rate is shown for solutions to the linearized multi-species Boltzmann system on the torus. The convergence is achieved by the interplay between the dissipative collision operator and the conservative transport operator and is proved by using the hypocoercivity method of Mouhot and Neumann.

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Helge Dietert: Stability and bifurcation for the Kuramoto model

We study the mean-field limit of the Kuramoto model of globally coupled oscillators. By studying the evolution in Fourier space and understanding the domain of dependence, we show a global stability result. Moreover, we can identify function norms to show damping of the order parameter for velocity distributions and perturbations in W^{n,1} for n > 1. Finally, for sufficiently regular velocity distributions we can identify exponential decay in the stable case and otherwise identify finitely many eigenmodes. For these eigenmodes we can show a center-unstable manifold reduction, which gives a rigorous tool to obtain the bifurcation behaviour. The damping is similar to Landau damping for the Vlasov equation.

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Amit Einav: Local Existence and Uniqueness for the Boltzmann-Nordheim Equation for Bosons.

One of the most influential equations in the kinetic theory of gases is the so-called Boltzmann equation, describing the time evolution of the probability density of a particle in a classical dilute gas. The irrefutable appearance of Quantum Mechanics, however, required a modification to this celebrated kinetic equation, resulting in the celebrated Boltzmann-Nordheim equation.

In our talk we will present a newly found local Cauchy Theory for the spatially homogeneous bosonic Boltzmann-Nordheim equation in any dimension d \geq 3 under mild restrictions on the collision kernel, extending previous studies that only dealt with the isotropic settings of the problem. Interestingly enough, the locality of this result is quite sharp due to the so-called Bose-Einstein condensation.

The methods used to achieve this theory are similar to those available for the classical Boltzmann equation, yet are entangled with L^\infty control that dominates the difference between the classical and quantum kinetic equation. Time permitting we will discuss some details about the existence of a global solution to the equation.

This is a joint work with Marc Briant.

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Mohammad El Smaily: Influence of large advection on reactive-diffusive fronts

Reaction-advection-diffusion equations form an important domain that is both rich and challenging mathematically. These equations are related to numerous applications in population dynamics, biology and chemistry. In a particular homogeneous framework, the first results about the existence of traveling-wave solutions date back to 1937--the work of Kolmogorov, Petrovsky and Piskunov (KPP). In heterogeneous frameworks, nontrivial hurdles face deterministic analysis of these nonlinear evolution equations. In 2005, Berestycki-Hamel-Nadirashvilli proved that KPP reaction-advection-diffusion equations admit traveling-wave-like solutions (or pulsating traveling fronts) in a complex periodic unbounded domain. These fronts propagate with a minimal speed c* that occurs to be the most interesting in the spectrum of speeds. In this talk, I will discuss the asymptotic behaviors of c* with respect to the large underlying advection field in any spatial dimension. Then I will give a sharp description of the flows that lead to an optimal speed-up of the fronts in the two dimensional case. In dimensions higher than 2, we will show that there are flows that exhibit ergodic behaviors yet succeed to enhance the propagation linearly in M. The proofs rely on a mixture of techniques from deterministic PDEs, ergodic theory, variational calculus and measure theory.

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Maxime Herda: On massless electron limit for a multispecies kinetic system with external magnetic eld

We consider a three-dimensional kinetic model for a two species plasma consisting of electrons and ions con fined by an external nonconstant magnetic eld. Then we derive a kinetic-fluid model when the mass ratio me/mi tends to zero.

Each species initially obeys a Vlasov-type equation and the electrostatic coupling follows from a Poisson equation. In our modeling, ions are assumed non-collisional while a Fokker-Planck collision operator is taken into account in the electron equation. As the mass ratio tends to zero we show convergence to a new system where the macroscopic electron density satis es an anisotropic drift-di usion equation. To achieve this task, we overcome some speci c technical issues of our model such as the strong e ect of the magnetic eld on electrons and the lack of regularity at the limit. With methods usually adapted to di usion limit of collisional kinetic equations and including renormalized solutions, relative entropy dissipation and velocity averages, we establish the rigorous derivation of the limit model.

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Robert L. Jerrad: Dynamics of topological defects in nonlinear Hamiltonian PDEs.

There are numerous nonlinear evolution equations, arising in diverse applications, whose solutions typically exhibit topological defects --- features such as interfaces, quantized vortices (in 2d) or vortex fialments (in 3d) around which energy concentrates. The goal of these lectures is to study, largely by consideration of concrete model problems, techniques for establishing rigorous descriptions of the dynamics of these defects in certain limits. We will focus equations of wave and Schrödinger type, which are less well-understood than their elliptic and parabolic couterparts.

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Tim Bastian Laux: Multi-phase mean-curvature flow

In the talk I consider the thresholding scheme, a time discretization for mean curvature flow introduced by Merriman, Bence and Osher. I will present a convergence result in the multi-phase case. The result establishes convergence towards a weak formulation of mean-curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movement scheme by Esedoglu and Otto. I will also present a similar convergence result for a variant of the algorithm for (two-phase) volume-preserving mean-curvature flow.

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Min-Gi Lee: Emergence of Coherent Localized Structures in Shear Deformations of Materials

Shear localization is a formation of a singularity of shear deformation in narrow region. This occurs in various instances of solid mechanical dynamics. Material instability is typically associated with Hadamard-instability, which can be attributed to a certain constitutive law of stress in an underlying model. Hadamard-instability results in catastrophic growth of oscillations, while what typically observed in localization process is organized coherent structures. We propose a model in which we explain the latter by non-linear effect. The constitutive law of stress in the model is supposed to be in a high-strain regime, in which the stress is monotonically decreasing with respect to the increase of the strain, but the second order regularizing effect is also counted together. This is a simplification of the previous work of. We construct a special localizing type solution of the model as a realization of an emergence of coherent localized structures.

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Stéphane Mischler: Semigroups in Banach spaces and applications to (nonlinear) evolution PDEs

The aim of the course is to present some recent progresses in the theory of semigroups which have been motivated by the application to several classes of PDE coming from the kinetic theory of gases and the biological modeling.

We present efficient versions of classical results such as the spectral mapping theorem, Weyl’s theorems, Krein-Rutman theorem, stability under perturbation theorem. The proofs are based on a factorization approach both for the spectral analysis of semigroup generators and semigroup growth estimates. More precisely, we systematically use iterated Duhamel formula and its resolvent counterpart in the spirit of Dyson series approach.

The abstract theory is motivated and illustrated by the applications to several PDEs. We will present an uniform treatment of the convergence to the equilibrium for the discrete and classical Fokker-Planck equation. We will also investigate the long-time asymptotic of solutions to the Keller-Segel equation for chemotaxis as well as of solutions to a time elapsed neuron network model.

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Alessia Nota: From microscopic dynamics to macroscopic equations. Scaling limits for the Lorentz gas

In this talk we will focus on some recent results concerning the derivation of the diffusion equation and the validation of Fick’s law for the microscopic model given by the random Lorentz Gas. These results are achieved by using a linear kinetic equation as an intermediate level of description between our original mechanical system and the diffusion equation. Moreover we will present some recent progress on the rigorous derivation of the linear Landau equation and the linear Boltzmann equation starting from the random Lorentz Gas in a magnetic field.

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Diana Stan: The Fisher-KPP equation with nonlinear fractional di usion

We study the propagation properties of nonnegative and bounded solutions of the class of reaction-di usion equations with nonlinear fractional di usion:

                                        ut + (-Δ)s(um)= f(u).

For all 0 < s < 1 and m > mc = (N - 2s)+=N, we prove that the level sets of the solution of the initial-value problem with suitable initial data propagate exponentially fast in time, in contradiction to the traveling wave behaviour of the standard KPP case, which corresponds to s = 1, m = 1 and f(u) = u(1 - u). The proof of this fact uses as an essential ingredient the recently established decay properties of the self-similar solutions of the purely
di usive equation, ut + (-Δ)sum = 0, combined with the construction of suitable sub- and supersolutions.

This problem is a joint work with Professor Juan Luis Vazquez (UAM).

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Patrick van Meurs: Discrete-to-continuum upscaling of the dynamics of dislocation walls

Plastic deformation of metals is the result of the collective behaviour of many microscopic defects (called dislocations) in the atomic lattice. These dislocations can be characterized by particular singularities in the stress eld within the metal. The dynamics of dislocations is governed by the non-local and singular interaction between them.

A large field of ongoing research addresses the question about how plasticity can be described by upscaling these microscopic dynamics (i.e. passing to the limit of the number of dislocations n to in nity). We contribute to this fi eld by upscaling the dynamics of a simpli ed, one-dimensional, interacting particle system, in which the particles represent dislocation walls (i.e. vertically periodic dislocation structures.) In our proof we use variational techniques such as Γ-convergence and the upscaling of gradient flows for convex energies.

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