Webinar Kinetic and fluid equations for collective behavior

Stéphane Brull (Bordeaux), Seung-Yeal Ha (SNU)
Philippe Thieullen (Bordeaux)

• Every second and fourth week on Friday: a 40 mn talk
• Winter French time: 11:00-12:00,   Korean time: 19:00-20:00
UTC/GMT offset France = +1; UTC/GMT offset Korea = +9
Summer time: additional offset in France (2023/03/27 → 2022/10/27) = +1
Winter time: additional offset in France (2022/10/30 → 2022/03/30) = +0

Forthcoming speakers
• 2022/12/09: Yonggeun Cho (Jeonbuk National University)
Title: Global well-posedness of Hartree type Dirac equations at critical regularity

Abstract: In this talk I will introduce a recent result on the global well-posedness of classical Dirac equation with Hartree type nonlinearity in $\mathbb R^{1+3}$. The equation is essentially $L^2$-critical. A standard argument is to utilise spinorial null structure inside the equations. However, the null structure is not enough to attain the global well-posedness at critical regularity. I will impose an extra regularity assumption with respect to the angular variable to prove global well-posedness and scattering of Dirac equations for small $L^2_x$-data with additional angular regularity. This talk is based on the joint works with S. Hong and T. Ozawa, and S. Hong and K. Lee.

• 2022/12/16: ?? (??)
Title: ??

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Past speakers 2021-2022
• 2022/11/25: Daniel Han-Kwan (Ecole polytechnique)
PDF ZOOM Title: On the hydrodynamic description of aerosols

Abstract: We are interested in the dynamics of an aerosol, that is a cloud of fine particles immersed in an ambient fluid. Starting with a description by the Vlasov-Navier-Stokes system, we will show that in a certain high friction limit, the aerosol can be well described by the inhomogeneous incompressible Navier-Stokes system. This is a joint work with David Michel.

• 2022/11/18: Doctor Gi-Chan Bae (Seoul National University)
PDF ZOOM Title: The quantum Boltzmann and BGK model near a global equilibrium

Abstract: This talk considers the existence and asymptotic behavior of two quantum kinetic equations, the quantum BGK model and the relativistic quantum Boltzmann equation. More precisely, we establish the existence of unique classical solutions and their exponentially fast stabilization when the initial data starts sufficiently close to a global quantum equilibrium based on the nonlinear energy method. The two models have different difficulties. For the quantum BGK model, the difficulty is to extract dissipation from the highly nonlinear quantum local equilibrium. For the relativistic quantum Boltzmann equation, we should control the nonlinear part on the energy-momentum 4-vector space to close the argument on $L^2$ space.

• 2022/10/28: Marwa Shahine (University of Bordeaux)
PDF ZOOM Title: Fredholm Property of the Linearized Boltzmann Operator for a Mixture of Polyatomic Gases

Abstract: In this talk, we consider the Boltzmann equation that models a mixture of polyatomic gases assuming the internal energy to be continuous. Under some convenient assumptions on the collision cross-section, we prove that the linearized Boltzmann operator L is a Fredholm operator. For this, we write L as a perturbation of the collision frequency multiplication operator. We prove that the collision frequency is coercive and that the perturbation operator is Hilbert-Schmidt integral operator.

• 2022/10/14: Dr. Jae Yong Lee (KIAS, Korea)
PDF ZOOM Title: Deep learning approach for solving kinetic equations

Abstract: Recently, deep learning-based methods have been developed to solve PDEs with many advantages. In this talk, I introduce our recent results on the deep neural network solutions to the kinetic equation. We study Vlasov-Poisson-Fokker-Planck equation and its diffusion limit via the deep learning approach. Also, we propose a new framework to approximate the solution to Fokker-Planck-Landau equation which has a nonlinearity and a high dimensionality of variables.

• 2022/09/23: Francis Filbet (Université Paul Sabatier - Toulouse)
PDF ZOOM Title: On the stability of Hermite spectral methods for the Vlasov-Poisson system and Fokker-Planck equation

Abstract: We study a class of spatial discretizations for the Vlasov-Poisson system and Fokker-Planck equation written as an hyperbolic system using Hermite polynomials. To obtain L^2 stability properties, we introduce a new $L^2$ weighted space, with a time dependent weight. For the Hermite spectral form of the Vlasov-Poisson system, we prove conservation of mass, momentum and total energy, as well as global stability for the weighted $L^2$ norm. These properties are then discussed for several spatial discretizations. For the Fokker-Planck equation, this approach allows to investigate the long time behavior and the asymptotic limit of the discrete model.

• 2022/06/24: Gael Raoul (Ecole Polytechnique)
PDF ZOOM Title: Wasserstein estimates and convergence to equilibrium for an evolutionary biology model

Abstract: Titre : We are interested in the dynamics of a population structured by a phenotypic trait. Individuals reproduce sexually, which is represented by a non-linear integral operator close to an inelastic Boltzmann operator. This operator is combined to a multiplicative operator representing selection. When the strength of selection is small, we show that the dynamics of the population is governed by a simple macroscopic differential equation, and that solutions converge exponentially to a steady-state. The analysis is based on Wasserstein distance inequalities.

• 2022/06/10: Dr. Jinwook Jung (Department of Statistics, Seoul National University)
ZOOM Title: Large time behavior of solutions to the pressureless Euler-Navier-Stokes system in the whole space

Abstract: In this talk, we present a refined framework for the large time behavior estimates for the pressureless Euler-Navier-Stokes system. Specifically, under a suitable assumption on the density of the pressureless Euler fluid flow, we show that the decay rate of the higher-order derivatives of fluid velocities, whose order is smaller than the dimension, is faster than that of lower-order derivatives. As a byproduct, we establish the global-in-time existence and uniqueness of classical solutions to our main system in the two-dimensional case. This talk is based on the joint work with Young-Pil Choi (Yonsei University).

• 2022/05/20: Filippo Santambrogio (Université Claude-Bernard Lyon 1)
PDF ZOOMTitle: The Fokker-Planck equation as a gradient-flow in the Wasserstein space : estimates in the time-discrete scheme

Abstract: From the work by Jordan, Kinderlehrer and Otto it is known that some parabolic PDEs have a variational structure (of steepest descent type) in the Wassestein space of probability densities endowed with a distance coming from optimal transport. The most typical example is the Fokker-Planck equation $\partial_t \rho = \Delta \rho + \nabla\cdot (\rho \nabla V)$, associated with the energy $F(\rho):=\int \rho\log\rho+\rho V$. Due to this variational structure, a very natural time-discretization scheme can be built, known as the JKO scheme : at each time step the sum of $F$ plus a suitable transport cost from the previous density is minimized, thus obtaining a recursive sequence of densities which approximate the solution of the PDE. It is interesting to see which bounds and regularity properties known to be satisfied by the solutions of the continuous-time equation are also satisfied in the discrete scheme. In the present talk, after recalling the main ingredients to understand the JKO scheme, I will give $L^\infty$ estimates on the solution and on its gradient, based on some easy manipulations on the Monge-Ampère equation.

• 2022/05/13: Prof. Chanwoo Kim (Univ. of Wisconsin)
PDF ZOOM Title: Vorticity Convergence from Boltzmann to 2D incompressible Euler equations below Yudovich class

Abstract: It has been an open problem to prove the convergence of solutions of the Boltzmann equations to non-Lipschitz solutions of the incompressible Euler equations. We settle this question affirmatively for Lagrangian solutions of the 2D incompressible Euler equation when the vorticity belongs to $L^p$ for any $p\geq 1$.

• 2022/04/22: Bertrand Lods (Università degli studi di Torino)
PDF ZOOM Title: Hydrodynamic limit for granular gases: from Boltzmann equation to some modified Navier-Stokes-Fourier system

Abstract: In this talk, we aim to present recent results about the rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and, to our knowledge, it is the first hydrodynamic system that properly describes rapid granular flows consistent with the kinetic formulation in physical dimension d=3. For that purpose, one of the main mathematical difficulty is to understand the relation between the restitution coefficient, which quantifies the energy loss at the microscopic level, and the Knudsen number. This is achieved by identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. The talk is based on a joint work with Ricardo Alonso (Texas A&M University at Qatar) and Isabelle Tristani (ENS Paris, Université PSL).

• 2022/04/08: Dongnam Ko (Catholic University of Korea)
PDF ZOOM Title: Convergence of the discrete consensus-based optimization algorithm with heterogeneous noises

Abstract: We present stochastic convergence analysis of the discrete consensus-based optimization (CBO) algorithm with random batch interactions and heterogeneous external noises, which guarantees the termination of the CBO algorithm. Despite successful performance in many practical simulations and remarkable analysis on kinetic level, the termination of the CBO algorithm was not rigorously investigated in such a generality as a discrete-time numerical algorithm. For this, we generalize the CBO algorithm with an abstract representative point, and then derive stochastic convergence of the individuals to a common point in mean-square and almost-sure sense under small noise assumption.

• 2022/03/25: Anne Nouri (Aix-Marseille University)
PDF ZOOM Title: Discrete velocity Boltzmann equations in the plane: stationary solutions

Abstract: Existence of stationary mild solutions for discrete velocity Boltzmann equations in the plane with no pair of colinear interacting velocities and given ingoing boundary values will be proven. A key property is $L^1$ compactness of integrated collision frequency for a sequence of approximations. This replaces the $L^1$ compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete. This a joint work with Leif Arkeryd.

• 2022/03/18: Insuk Seo (Seoul National University)
PDF ZOOM Title: Metastability of random processes and solution of resolvent equations

Abstract: A random process exhibits the so-called metastability when the associated potential function has multiple local minima. The quantitative analysis of the metastability is based on the careful investigation of the generator (which is usually a second-order elliptic operator) associated with the random process. Recently, it has been observed that the analysis of solutions of a class of resolvent equations associated with the generator of the random process contain all the essential information regarding the quantitative feature of the metastability. In this lecture, we explain this relation and its application to the small random perturbations of dynamical system studied originally by Freidlin and Wentzell in 70s. The lecture is self-contained and will not assume any prior knowledge regarding the study of metastability. This presentation is based on the work arXiv:2102.00998 and an ongoing work with Claudio Landim and Jungkyoung Lee.

• 2021/12/17: Yann Brenier (Ecole Normale Supérieure)
PDF ZOOM Title: Phase-space and optimal-transport formulations of Einstein's gravitation in vacuum

Abstract: Einstein's theory of gravitation, at least in vacuum, can be treated as a kind of quadratic matrix-valued optimal transportation problem. This goes through a kind of "kinetic" or "phase-space" formulation of the concept of Ricci curvature

• 2021/12/10: Woojoo Shim (Seoul National University)
PDF ZOOM Title: Cucker-Smale inspired deterministic Mean Field Game with velocity interactions

Abstract: In this talk, I will present a mean field game model for agents moving in a given domain and choosing their trajectories so as to minimize a cost including a penalization on the difference between their own velocity and that of the other agents they meet. For the proposed model, we discuss the existence of an equilibrium in a Lagrangian setting by using its variational structure, and then study its properties including regularity.

• 2021/11/26: Teddy Pichard (Ecole polytechnique)
PDF ZOOM Title: Some recent advances in the theory of moment model

Abstract: In kinetic theory, the method of moments is used to reduce a scalar transport equation depending on multiple variables (typically Boltzmann equation involves seven variables) into a system involving less variables (typically Euler equations). In this talk, I will present some recent constructions of moment closures in velocity. I will focus on three approaches that will be compared: 1-a family of methods based on quadrature formulae (QMOM methods); 2-a family of models based on the entropy dissipation of the kinetic equation ; 3-a family of methods based on the study of the set of admissible moments and exploiting projection techniques on it.

• 2021/11/12: Jin Woo Jang (Department of Math. Pohang University of Science and Technology)
PDF ZOOM Title: LTE and Non-LTE Solutions in Gases Interacting with Radiation

Abstract: The goal of this talk is to discuss a class of kinetic equations describing radiative transfer in gases which include also the interaction of gas molecules with themselves. We first introduce a system of kinetic PDEs that describes the dynamics of gas molecules coupled with an equation for photons radiative transfer. We then discuss several scaling limits and introduce some Euler-like systems coupled with radiation as an aftermath of specific scaling limits. We consider scaling limits in which local thermal equilibrium (LTE) holds, as well as situations in which this assumption fails (non-LTE). We understand that the structure of the equations describing the gas-radiation system is very different in the LTE and non-LTE cases. We prove the existence of stationary solutions to the resulting limit models in the LTE case. Lastly, we will also prove the non-existence of stationary solutions with zero velocities in a non-LTE situation. This is a joint work with Juan J. L. Velazquez at Bonn.

• 2021/10/29: Amic Frouvelle (University of Dauphine)
PDF ZOOM Title: Body-attitude alignment : phase transition, link with suspensions of rodlike polymers and quaternions

Abstract: We present a model of alignment of individuals based on body attitude (birds aligning their heading and wings directions for instance). The kinetic model in consideration is a Fokker-Planck model for which the velocity variable is a rotation matrix of dimension 3. We present an interesting link between this model and a generalization of the Maier-Saupe model for alignment of diluted rodlike polymers in dimension 4, due to the fact that a rotation can be represented by a unit quaternion (or its opposite, which relates to the fact that a rodlike polymer is unoriented). We obtain the phase diagram of this model : when the alignment strength is low, the uniform distribution is the only equilibria, when the strength is sufficiently large, there exists a unique family of stable (concentrated) distributions, and in between, we have stability of both non-aligned and aligned states.

This comes from works in collaboration with Pierre Degond, Antoine Diez, Sara Merino-Aceituno and Ariane Trescases.

• 2021/10/15: Renjun Duan (The Chinese University of Hong Kong)
PDF ZOOM Title: The Boltzmann equation for plane Couette flow

Abstract: In the talk I will report a recent work in collaboration with Shuangqian Liu and Tong Yang on a study of the Couette flow for a rarefied gas between two parallel infinite plates moving relative to each other. We reformulate it as the boundary value problem on the Boltzmann equation with a shear force subject to the homogeneous non-moving diffuse reflection boundary, and then establish the existence and large time asymptotic stability of stationary solutions for any small enough shear strength.

• 2021/09/24: Laurent Desvillettes (Université de Paris)
PDF ZOOM Title: Some new use of the duality methods for parabolic systems

Abstract: Duality methods for L^2-estimates of solutions of reaction-diffusion systems were introduced by Pierre and Schmitt. They can also be used to obtain estimates for coagulation-fragmentation-diffusion systems, or cross diffusion systems. We present in this talk a new application of those methods for a system coming out of the modeling of cells/chemical species including chemotaxis terms.

• 2021/09/10: Jeongho Kim (Hanyang University)
PDF ZOOM Title: Hydrodynamic limits of the Schrodinger equation with gauge fields

Abstract: In this talk, we present the hydrodynamic limits of the Schrodinger equation, affected by different gauge fields. Precisely, we first present the hydrodynamic limit of the Schrodinger equation with the Chern-Simons gauge fields (Chern-Simons-Schrodinger equation), toward to the Euler-Chern-Simons equation on the two-dimensional state space. Then, we consider the hydrodynamic limit of the Schrodinger equation with the Maxwell gauge fields (Maxwell-Schrodinger equation), toward to the Euler-Maxwell equation on the three-dimensional state space. Both estimate use the estimate on the modulated energy functionals.

Past speakers 2020-2021
• 2021/06/25: Isabelle Tristani (ENS Ulm Paris)
PDF ZOOM Title: Incompressible Navier-Stokes limit of the Boltzmann equation

Abstract: In this talk, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. The problem of justifying the connection between mesoscopic and macroscopic equations has been extensively studied. Here, we propose an approach that intertwines fluid mechanics and kinetic estimates. It enables us to prove convergence of smooth solutions of the Boltzmann equation to solutions to the fluid dynamics equations when the Knudsen number goes to zero. We do not require any smallness at initial time, and our result is valid for any initial data (well prepared or not) in the case of the whole space. We also prove that the time of existence of the solution to the Boltzmann equation is bounded from below by the existence time of the fluid equation as soon as the Knudsen number is small enough. This is a joint work with Isabelle Gallagher

• 2021/06/11: Dohyun Kim (Sungshin Women's Univ.)
PDF ZOOM Title: Asymptotic emergent dynamics of the Schrödinger-Lohe model

Abstract: In this talk, we introduce a coupled system of nonlinear Schrödinger equations, so-called the Schrödinger-Lohe (S-L) model as a phenomenological model for quantum synchronization. Then, we briefly review recent progress on the S-L model from the perspective of asymptotic emergent dynamics. For the analytic results, the two-point correlation function defined as the inner product of two wavefunctions is mainly used. On the other hand for the numerical result, we adopt the time splitting spectral method together with the Crank-Nicolson method to discretize the S-L model.

• 2021/05/28: Benoit Perthame (Sorbone Université)
PDF ZOOM Title: From voltage-conductance kinetic models to integrate and fire equation for neural assemblies

Abstract: The voltage-conductance systems for neural networks has been introduced by biophysicists for modeling the visual cortex. In terms of mathematical structure, it can be compared to a kinetic equations with a macroscopic limit which turns out to be the Integrate and Fire equation. This talk is devoted to a mathematical description of the slow-fast limit of the kinetic type equation to an I\&F equation. After proving the weak convergence of the voltage-conductance kinetic problem to potential only I\&F equation, we prove strong a priori bounds and we study the main qualitative properties of the solution of the I\&F model, with respect to the strength of interconnections of the network. In particular, we obtain asymptotic convergence to a unique stationary state for weak connectivity regimes. For intermediate connectivities, we prove linear instability and numerically exhibit periodic solutions. These results about the I\&F model suggest that the more complex voltage-conductance kinetic equation shares some similar dynamics in the correct range of connectivity.

• 2021/05/14: Donghyun Lee (Postech)
PDF ZOOM Title: Large amplitude solution of the Boltzmann equation

Abstract: We study well-posedness theory the Boltzmann equation in low regularity L^{\infty} space. After low regularity approach with L^2-L^\infty bootstrap argument was introduced, the method was widely used to solve many boundary condition problems in small perturbation framework. Moreover, these results have extended into more general L^\infty solution whose amplitude can be arbitrary large. We mainly discuss about recent developments in large amplitude Boltzmann theory including boundary condition problems. This is a joint work with R.Duan(CUHK) and G. Ko(POSTECH).

• 2021/04/23: Clément Mouhot (University of Cambridge)
ZOOM Title: Quantitative De Giorgi Methods in Kinetic Theory

Abstract: We consider hypoelliptic equations of kinetic Fokker-Planck type, also sometimes called of Kolmogorov or Langevin type, with rough coefficients in the diffusion matrix. We present novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities (which imply Holder continuity with quantitative estimates). This is a joint work with Jessica Guerand.

• 2021/04/09: Young-Pil Choi (Yonsei Univ.)
ZOOM Title: Quantified overdamped limit for Vlasov-Fokker-Planck equations with singular interaction forces

Abstract: In this talk, I will discuss a quantified overdamped limit for kinetic Vlasov-Fokker-Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equation or McKean-Vlasov equation. Our strategy only requires weak integrability of the interaction potentials, thus in particular it includes the quantified overdamped limit of the kinetic Vlasov-Poisson-Fokker-Planck system to the aggregation-diffusion equation with either repulsive electrostatic or attractive gravitational interactions.

• 2021/03/26: Helge Dietert (Université Paris Diderot)
ZOOM Title: Hypocoercivity with spatial weight

Abstract: We will study linear kinetic equations for a density in phase-space evolving through Hamiltonian transport and a linear collision operator in the velocity variable. In this talk, we study the effect of a spatial weight for the collision operator which vanishes in part of the domain. In particular, I will present a quantitative proof for exponential relaxation under a geometric control condition.

• 2021/03/12: Jihoon Lee (Dept. of Math, Chung-Ang Univ.)
PDF ZOOM Title: Decay estimates of solutions to the fluid equations with rotation or stratification

Abstract: In this talk, we consider the incompressible fluid equations with rotation or stratification. First, we consider three dimensional incompressible Navier-Stokes equations with fractional dissipation and Coriolis force. We find Coriolis force gives extra temporal decay of the solutions under some conditions on the initial data. Next, we consider the three-dimensional damped Boussinesq equations with strong stratification. We find the global-in-time existence of solutions under some conditions of the initial data and the temporal decay of solutions.
This is based on the joint work with Jaewook Ahn(Dongguk Univ.) and Junha Kim(Chung-Ang Univ.).

• 2021/02/26: François Golse (Ecole Polytechnique)
PDF ZOOM Title: Half-space problem for the Boltzmann equation with phase transition at the boundary

Abstract: Y. Sone, K. Aoki and their group have studied numerically the existence of a solution to the steady half-space problem for a rarefied gas whose behavior is described by the Boltzmann equation (with slab symmetry). The gas is assumed to fill a half-space on top of a liquid which is its condensed phase, and the velocity distribution function of molecules entering the half-space from the condensed phase is the centered Maxwelllian parametrized by the temperature at the gas-liquid interface, and the saturating vapor pressure at this temperature. The state at infinity (i.e. far from the interface) is (another) Maxwellian. In a remarkable paper T.-P. Liu and S.-H. Yu [Arch. Rational Mech. Anal. 209 (2013), 869-997] have proposed a complete method for handling this kind of problem. The purpose of this talk is to present an alternative, self-contained proof of one of results in the work of Liu and Yu, specifically the existence and uniqueness of solutions that are decaying as the distance to the interface goes to infinity, uniformly in the Mach number of the Maxwellian at infinity. The proof uses a variant of the generalized eigenvalue problem studied by Nicolaenko in his work on the shock profile for the Boltzmann equation, and the Ukai-Yang-Yu penalization method for half-space problems in kinetic theory (suitably modified).
[Work with N. Bernhoff.]

• 2021/02/08: Seok-Bae Yun (Sunkyunkwan University (SKKU))
PDF Title: Ellipsoidal BGK model of the Boltzmann equation with the correct Prandtl number

Ellipsoidal BGK model (ES-BGK) is a generalized version of the Boltzmann-BGK model where the local Maxwellian in the relaxation operator of the BGK model is extended to an ellipsoidal Gaussian with a Prandtl parameter ν, so that the correct Prandtl number can be computed in the Navier-Stokes limit. In this talk, we review some of the recent results on ES-BGK model such as the existence (stationary or non-stationary) theory and the entropy-entropy production estimates. A dichotomy is observed between −1/2 < v < 1 and ν=-1/2. In the former case, an equivalence relation between the local temperature and the temperature tensor enables one to apply theories developed for the original BGK model in a modified form. In the critical case (ν=-1/2), where the correct Prandtl number is achieved, such equivalence break down, and the structure of the flow has to be incorporated to estimate the temperature tensor from below. This is from joint works with Stephane Brull and Doheon Kim.