### Journées 2021 du projet EFI: Titles and abstracts

##### 13-15 octobre 2021, Paris-Dauphine

Homepage of the event

**Franz Achleitner**
*Short- and long-time behavior in (hypo)coercive ODE-systems and kinetic equations*

We will discuss hypocoercivity on the level of ODEs and devise a new way to construct strict Lyapunov functionals: Systems of ODEs dx/dt = Ax with semi-dissipative matrix A (i.e. the Hermitian part of matrix A is negative semi-definite) are Lyapunov stable but not necessarily asymptotically stable. There exist many equivalent conditions, to decide if the ODE system is asymptotically stable or not. Some conditions allow to construct a strict Lyapunov functional in a natural way. We will review these classical conditions/approaches and identify a "hypocoercivity index" which e.g. characterizes the short-time asymptotics of the propagator norm for semi-dissipative ODEs. Finally, using these results, we revisit the analysis of the long-time behavior of (hypocoercive) nonlinear BGK-type model with constant collision frequency, and (kinetic) Fokker-Planck equations. In particular, we will compare our strict Lyapunov functionals for the linear(ized) kinetic equations with other classical approaches. References [1] F. Achleitner, A. Arnold, E. A. Carlen, On linear hypocoercive BGK models, in Springer Proceedings in Mathematics & Statistics, Vol. 126, (2016) 1-37. [2] F. Achleitner, A. Arnold, E. A. Carlen, On multi-dimensional hypocoercive BGK models, Kinetic Rel. Models 11[4] (2018). [3] F. Achleitner, A. Arnold, E. A. Carlen, The hypocoercivity index for the short and large time behavior of ODEs, preprint (2020). [4] F. Achleitner, A. Arnold, V. Mehrmann, Hypocoercivity and controllability in linear dissipative ODEs and DAEs, preprint (2021). Further references [K90] S. Kawashima, The Boltzmann Equation and thirteen moments, Japan J. Appl. Math. 7 (1990) 301-320. [BZ11] K. Beauchard, E. Zuazua, Large Time Asymptotics for Partially Dissipative Hyperbolic Systems, Arch. Rational Mech. Anal. 199 (2011) 177–227.

**Esther Bou Dagher**
*Coercive Inequalities and U-Bounds*

In the setting of step-two Carnot groups, we prove Poincaré and $\beta$-Logarithmic Sobolev inequalities for probability measures as a function of various homogeneous norms. To do that, the key idea is to obtain an intermediate inequality called the U-Bound inequality (based on joint work with B. Zegarlinski). Using this U-Bound inequality, we show that certain infinite dimensional Gibbs measures- with unbounded interaction potentials as a function of homogeneous norms- on an infinite product of Carnot groups satisfy the Poincaré inequality (based on joint work with Y. Qiu, B. Zegarlinski, and M. Zhang). We also enlarge the class of measures as a function of the Carnot-Carathéodory distance that gives us the q−Logarithmic Sobolev inequality in the setting of Carnot groups. As an application, we use the Hamilton-Jacobi equation in that setting to prove the p−Talagrand inequality and hypercontractivity.

**Giovanni Brigati**
*Time averages for kinetic Fokker-Planck equations*

Kinetic Fokker-Planck equations are a paradigmatic example of degenerate diffusion, corresponding to Langevin dynamics in probability. Unlike coercive diffusion, the convergence of the solutions towards equilibrium is not immediate in standard norms. Hence, hypocoercivity techniques are required. After a brief review of the main contributions in the theory, we introduce the study of weak norms of solutions, started by Armstrong and Mourrat in 2019. Then, we derive careful quantitative estimates for the decay of time averages of the solutions via generalised Poincaré inequalities. Results are extended to the case of non-Maxwellian local equilibria.

**Kleber Carrapatoso**
*Équations cinétiques linéaires avec potentiel de confinement*

Je présenterai des résultats sur le comportement en temps long des solutions d'équations cinétiques linéaires dans tout l'espace, où l'opérateur de collision satisfait les lois de conservation physiques (masse, quantité de mouvement et énergie) et les particules sont confinées via un potentiel extérieur. Travail en commun avec J. Dolbeault, F. Hérau, S. Mischler, C. Mouhot et C. Schmeiser.

**Gauthier Clerc**
*Comportement en temps long des interpolations entropiques*

Dans cette exposé on s'intéresse aux interpolations entropiques. Ces courbes à valeurs dans l'ensemble des mesures de probabilités d'un espace donné décrivent les minimiseurs du problème de Schrödinger. Il est bien connu qu'en temps court les interpolations entropiques convergent vers les géodésiques du transport optimal. Ici nous nous intéressons à la convergence en temps long de ces interpolations.

**Pierre Gervais**
*From Boltzmann to Incompressible Navier-Stokes with general initial data* pdf

Hilbert’s sixth problem, stated in 1900 during the International Congress of Mathematicians, consists in the axiomatization of physics. In the case of fluid dynamics, this issue reduces to the derivation of hydrodynamic equations (a macroscopic description) from kinetic equations (a mesoscopic description), which would be themselves derived from Newton’s laws of motion applied to the particles making up the fluid (a microscopic description). In the special case of a gas close to a global thermodynamic equilibrium with constant density, temperature and velocity, the fluctuations of these two last quantities are driven by the Navier-Stokes equations. The problem of deriving this hydrodynamic model from this kinetic model is still partially open for strong solutions (the link between weak solutions being well understood thanks to the works of C. Bardos, F. Golse, D. Levermore and L. Saint-Raymond between 1989 and 2003). Most of the strong theory of hydrodynamic limits consists in constructing solutions to the Boltzmann equation close to the solution of some hydrodynamic equation and quantifying this “closeness”. However, they require that the initial statistical distribution for the velocity decays like a Gaussian, although the ideal decay assumption, suggested by physical a priori bounds, would be an algebraic decay. The so called Enlargement Theory (of functional spaces), initiated by C. Mouhot and developped with M.P. Gualdani and S. Mischler between 2005 and 2017, allowed to construct solutions to several kinetic equations for initial data having an algebraic decay in the velocity variable. In this talk, I will explain how this theory can be combined with previous approaches (à la Bardos-Ukai or Gallagher-Tristani) to construct solutions to the Boltzmann equation for any initial distribution with algebraic decay, and detail the modes of convergence to the incompressible Navier-Stokes limit depending on how well prepared it is.

**Frédéric Hérau**
*Hypocoercivity genesis and bridges*

Hypocoercivity is now a well established theory, used in many kinetic contexts, from theoretical results of trend to equilibrium, to numerical problems, passing by rigorous diffusion limit results. In this review talk, we shall try to give some ideas of the genesis of the theory, its links with hypoellipticity as well as its deep similitudes with other approaches in other fields in PDE or spectral analysis.

**Lucas Journel**
*Convergence of the kinetic annealing for general potentials* pdf

The goal of a simulated annealing is to find, via a stochastic process, the minimum of some function U : R^d → R^+. To this end we study the process : dX_t = Y_t dt , dY_t = - nabla U(X_t) dt- gamma_t Y_t dt+ 2 sqrt {gamma_t / beta_t} dB_t , where beta_t = ln(exp(c beta_0)+t) / c . Let c∗ be the largest energy barrier of U. We proved under mild assumptions on the potential U the convergence of the kinetic annealing towards the minimum of U for c > c∗, as well as the non-convergence for c < c∗.

**Laura Kanzler**
*Kinetic Model for Myxobacteria with Directional Diffusion* pdf

The topic of this talk will be a kinetic model inspired by dynamics of myxobacteria colonies moving on flat surfaces. The model is based on the assumption of hard binary collisions of two different types: alignment and reversal, which results in a Boltzmann-type collision operator in the equation of interest. Further, we assume Brownian forcing in the free flight phase of single bacteria that gives rise to a directional diffusion term at the level of the kinetic equation, which opposes the concentrating effect of the alignment operator. A global existence and uniqueness result as well as exponential decay to uniform equilibrium is proved in the case where the diffusion is large enough compared to the total bacteria mass. Further, the question whether in a small diffusion regime nonuniform stable equilibria exist is positively answered by performing a formal bifurcation analysis, which revealed the occurrence of a pitchfork bifurcation. These results are illustrated by numerical simulations.

**Pierre Le Bris**
*Uniform in time propagation of chaos for the 2D vortex model and other singular stochastic systems*

In this talk, we show how we may obtain uniform in time propagation of chaos for a class of singular interaction kernels, extending the results of Fournier-Hauray-Mischler (JEMS) and Jabin-Wang (Inv. Math.). In particular, our models contain the Biot-Savart kernel on the torus and thus the 2D vortex model. The strategy is to combine the relative entropy approach of Jabin-Wang with functional inequalities as well as uniform bounds on all the derivatives of the solution of the non linear limit equation, in order to control both the entropy dissipation and the constants appearing in the large deviation estimates.

**Xingyu Li**
*The Vlasov-Poisson-Boltzmann equation with polynomial perturbation near Maxwellian* pdf

We consider the Vlasov-Poisson-Boltzmann equation without cutoff near Maxwellian, and we prove the global existence, uniqueness, and large time behaviour for solutions in a polynomial weighted space $H^2_{x, v}( \langle v \rangle^k)$ for some constant $k >0$ large enough. In this talk, we extend former results in $H^N_{x, v}(\mu^{-1/2})$ to polynomial weighted space $H^2_{x, v}( \langle v \rangle^k)$. The proof combines works by Y. Guo and the semigroup method introduced by M.P. Gualdani, S. Mischler, and C. Mouhot. In fact, our proof can be also be used in the Landau type equation. This is a joint work with Chuqi Cao and Dingqun Deng (Tsinghua Univeristy).

**Angeliki Menegaki**
*Quantitative unified framework for hydrodynamic limits*

We will present a new approach to the problem of proving hydrodynamic limits from microscopic stochastic particle systems, namely the zero-range, the simple exclusion (jump) and the Ginzburg-Landau (diffusion) processes, to macroscopic partial differential equations. The qualitative behavior of the particle densities of such processes moving on a lattice according to a Markovian law, is well-known. Our method is simplified and unified that gives an explicit, uniform in time, rate of convergence to the limit PDE (joint work with Daniel Marahrens and Clément Mouhot).

**Marlies Pirner**
*Hypocoercivity of a model describing the thermalization of a rarefied gas with total energy conservation*

The thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law. In this talk, I will present results on the large-time behaviour for this coupled non-linear system of a kinetic and a heat equation. For this, we use hypocoercivity methods. This is joint work with Christian Schmeiser, Gianluca Favre (University of Vienna)

**Christophe Poquet**
*Periodicity and long time diffusion for mean-field systems*

We will study the long time behavior of mean-field system in cases when the limit Fokker-Planck PDE admits a limit cycle. We will see that in the time scale Nt the empirical mesure admits a dephasing from the periodic solution that is given at the limit by a diffusion process.

**Mohamad Rachid**
*Incompressible Navier-Stokes-Fourier limit from the Landau equation*

In this presentation, we provide a result on the derivation of the incompressible Navier-Stokes-Fourier system from the Landau equation for hard, Maxwellian and moderately soft potentials. To this end, we first investigate the Cauchy theory associated to the rescaled Landau equation for small initial data. Our approach is based on proving estimates of some adapted Sobolev norms of the solution that are uniform in the Knudsen number. These uniform estimates also allow us to obtain a result of weak convergence towards the fluid limit system.

**Nikita Simonov**
*Stability in Gagliardo-Nirenberg-Sobolev inequalities*

In this talk, I will discuss recent stability results for a family of Gagliardo-Nirenberg-Sobolev inequalities obtained in collaboration with M. Bonforte, J. Dolbeault, and B. Nazaret. Our strategy is based on entropy methods and the use of fast diffusion equation (FDE). Using the quantitative regularity results for the FDE, we are able to go beyond the non-constructive variational results and provide fully constructive estimates.

**Gabriel Stoltz**
*Hypocoercivity without changing the scalar product*

Hypocoercive methods allow to prove the longtime convergence of the law of degenerate stochastic processes such as Langevin dynamics or random time HMC. From a technical viewpoint, these approaches usually involve a change of scalar product (as in the H^1 approach à la Villani, or the L^2 approach of Hérau and Dolbeault/Mouhot/Schmeiser). I will discuss two approaches which do not require such a change of scalar product, and henceforth provide the current sharpest quantitative estimates on the resolvent of the generator, which makes it possible for instance to bound the variance of estimators based on time averages along a realization of the process. The first approach, originally due to Armstrong/Mourrat, was adapted to dynamics on the whole space by Cao/Lu/Wang, and is based on space-time Poincare inequalities. The second approach, developed with Etienne Bernard, Max Fathi and Antoine Levitt, is based on Schur complements and provides direct bounds on the resolvent using the specific saddle-point like structure of the generator under an appropriate orthogonal decomposition of L^2.

**Tobias Wöhrer**
*Explicit decay rates for discrete velocity BGK models*

In this talk we analyse the long-time behaviour of solutions to prototypical transport-relaxation systems of BGK-type. We present a hypocoercivity method that modifies the standard norm based on Lyapunov matrix inequalities. The method provides optimal decay rates for homogeneous relaxation, where the equation can be Fourier decomposed into finite ODEs. Expressing the Lyapunov functional through pseudo-differential operators allows us to go beyond the ODE approach and prove explicit decay rates for non-homogeneous relaxation. We further point out that the two-velocity example connects the presented method to the DMS methods.

**Simon Zugmeyer**
*Formulation duale de la conjecture de Mahler fonctionnelle avec l'entropie de Boltzmann*

La conjecture de Mahler, ou inégalité de Santaló inverse, admet une formulation fonctionnelle, formulée par Fradelizi et Meyer en 2008. J'aimerais présenter quelques idées autour du lien de dualité qui lie cette inégalité fonctionnelle à sa contrepartie dans le monde du transport optimal, qui s'avère être une minoration du déficit dans l'intégalité de Sobolev logarithmique.