Webinar Kinetic and fluid equations for collective behavior
Stéphane Brull (Bordeaux), SeungYeal Ha (SNU)
Philippe Thieullen (Bordeaux)
Link:

Every second and fourth week on Friday: a 40 mn talk

Winter French time: 11:0012:00, Korean time: 19:0020:00
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

Zoom link: ask the organizers
Forthcoming speakers

2022/12/09: Yonggeun Cho (Jeonbuk National University)
Title: Global wellposedness of Hartree type Dirac equations at critical regularity
Abstract: In this talk I will introduce a recent result on the global wellposedness 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 wellposedness at
critical regularity. I will impose an extra regularity assumption
with respect to the angular variable to prove global wellposedness
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: ??
Abstract: ??
Past speakers 20212022

2022/11/25: Daniel HanKwan (Ecole polytechnique)
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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
VlasovNavierStokes system, we will show that in a certain high friction limit,
the aerosol can be well described by the inhomogeneous incompressible NavierStokes
system. This is a joint work with David Michel.

2022/11/18: Doctor GiChan Bae (Seoul National University)
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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 energymomentum
4vector space to close the argument on $L^2$ space.

2022/10/28: Marwa Shahine (University of Bordeaux)
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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 crosssection, 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 HilbertSchmidt
integral operator.

2022/10/14: Dr. Jae Yong Lee (KIAS, Korea)
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Title: Deep learning approach for solving kinetic equations
Abstract: Recently, deep learningbased 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
VlasovPoissonFokkerPlanck equation and its diffusion limit via the deep
learning approach. Also, we propose a new framework to approximate the
solution to FokkerPlanckLandau equation which has a nonlinearity and a
high dimensionality of variables.

2022/09/23: Francis Filbet (Université Paul Sabatier  Toulouse)
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Title: On the stability of Hermite spectral methods for the VlasovPoisson system and FokkerPlanck equation
Abstract: We study a class of spatial discretizations for the VlasovPoisson
system and FokkerPlanck 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 VlasovPoisson 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 FokkerPlanck 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)
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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 nonlinear 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 steadystate. The
analysis is based on Wasserstein distance inequalities.

2022/06/10: Dr. Jinwook Jung (Department of Statistics, Seoul National University)
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Title: Large time behavior of solutions to the pressureless EulerNavierStokes system in the whole space
Abstract: In this talk, we present a refined framework for the large time behavior estimates for the
pressureless EulerNavierStokes system. Specifically, under a suitable assumption on the density of
the pressureless Euler fluid flow, we show that the decay rate of the higherorder derivatives of
fluid velocities, whose order is smaller than the dimension, is faster than that of lowerorder
derivatives. As a byproduct, we establish the globalintime existence and uniqueness of classical
solutions to our main system in the twodimensional case. This talk is based on the joint work with
YoungPil Choi (Yonsei University).

2022/05/20: Filippo Santambrogio (Université ClaudeBernard Lyon 1)
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ZOOMTitle: The FokkerPlanck equation as a gradientflow in the Wasserstein space : estimates in the timediscrete 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
FokkerPlanck 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 timediscretization
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
continuoustime 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 MongeAmpère equation.

2022/05/13: Prof. Chanwoo Kim (Univ. of Wisconsin)
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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 nonLipschitz 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)
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Title: Hydrodynamic limit for granular gases: from Boltzmann equation to some modified
NavierStokesFourier 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 NavierStokesFourier
system with selfconsistent 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)
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Title: Convergence of the discrete consensusbased optimization algorithm with heterogeneous noises
Abstract: We present stochastic convergence analysis of the discrete consensusbased
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 discretetime
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
meansquare and almostsure sense under small noise assumption.

2022/03/25: Anne Nouri (AixMarseille University)
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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)
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Title: Metastability of random processes and solution of resolvent equations
Abstract: A random process exhibits the socalled 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 secondorder 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 selfcontained 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)
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Title: Phasespace and optimaltransport 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 matrixvalued optimal transportation problem.
This goes through a kind of "kinetic" or "phasespace" formulation of
the concept of Ricci curvature

2021/12/10: Woojoo Shim (Seoul National University)
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Title: CuckerSmale 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)
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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: 1a family of methods based on quadrature formulae (QMOM methods); 2a family of
models based on the entropy dissipation of the kinetic equation ; 3a 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)
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Title: LTE and NonLTE 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 Eulerlike 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 (nonLTE). We understand that the structure of the equations
describing the gasradiation system is very different in the LTE and nonLTE cases.
We prove the existence of stationary solutions to the resulting limit models in the
LTE case. Lastly, we will also prove the nonexistence of stationary solutions with
zero velocities in a nonLTE situation. This is a joint work with Juan J. L. Velazquez
at Bonn.

2021/10/29: Amic Frouvelle (University of Dauphine)
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Title: Bodyattitude 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
FokkerPlanck 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 MaierSaupe
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 nonaligned and
aligned states.
This comes from works in collaboration with Pierre Degond, Antoine Diez, Sara MerinoAceituno
and Ariane Trescases.

2021/10/15: Renjun Duan (The Chinese University of Hong Kong)
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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 nonmoving 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)
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Title: Some new use of the duality methods for parabolic systems
Abstract: Duality methods for L^2estimates of solutions of reactiondiffusion systems
were introduced by Pierre and Schmitt. They can also be used to obtain estimates for
coagulationfragmentationdiffusion 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)
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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 ChernSimons gauge fields (ChernSimonsSchrodinger
equation), toward to the EulerChernSimons equation on the twodimensional state space.
Then, we consider the hydrodynamic limit of the Schrodinger equation with the Maxwell
gauge fields (MaxwellSchrodinger equation), toward to the EulerMaxwell equation on the
threedimensional state space. Both estimate use the estimate on the modulated energy functionals.
Past speakers 20202021

2021/06/25: Isabelle Tristani (ENS Ulm Paris)
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Title: Incompressible NavierStokes limit of the Boltzmann equation
Abstract: In this talk, we are interested in the link between strong solutions of the
Boltzmann and the NavierStokes 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.)
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Title: Asymptotic emergent dynamics of the SchrödingerLohe model
Abstract: In this talk, we introduce a coupled system of nonlinear Schrödinger equations,
socalled the SchrödingerLohe (SL) model as a phenomenological model for quantum
synchronization. Then, we briefly review recent progress on the SL model from the perspective
of asymptotic emergent dynamics. For the analytic results, the twopoint 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 CrankNicolson
method to discretize the SL model.

2021/05/28: Benoit Perthame (Sorbone Université)
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Title: From voltageconductance kinetic models to integrate and fire equation for neural assemblies
Abstract: The voltageconductance 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 slowfast limit of the kinetic type
equation to an I\&F equation. After proving the weak convergence of the voltageconductance 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 voltageconductance kinetic equation shares some similar dynamics in the
correct range of connectivity.

2021/05/14: Donghyun Lee (Postech)
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Title: Large amplitude solution of the Boltzmann equation
Abstract: We study wellposedness theory the Boltzmann equation in low regularity L^{\infty} space.
After low regularity approach with L^2L^\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)
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Title: Quantitative De Giorgi Methods in Kinetic Theory
Abstract: We consider hypoelliptic equations of kinetic FokkerPlanck 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 intermediatevalue 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: YoungPil Choi (Yonsei Univ.)
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Title: Quantified overdamped limit for VlasovFokkerPlanck equations with singular interaction forces
Abstract: In this talk, I will discuss a quantified overdamped limit for kinetic VlasovFokkerPlanck
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 aggregationdiffusion equation or McKeanVlasov equation. Our strategy
only requires weak integrability of the interaction potentials, thus in particular it includes the
quantified overdamped limit of the kinetic VlasovPoissonFokkerPlanck system to the aggregationdiffusion
equation with either repulsive electrostatic or attractive gravitational interactions.

2021/03/26: Helge Dietert (Université Paris Diderot)
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Title: Hypocoercivity with spatial weight
Abstract: We will study linear kinetic equations for a density in phasespace 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, ChungAng Univ.)
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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 NavierStokes 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 threedimensional damped Boussinesq equations with strong stratification.
We find the globalintime 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(ChungAng Univ.).

2021/02/26: François Golse (Ecole Polytechnique)
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Title: Halfspace 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 halfspace problem for a rarefied gas whose behavior is described by the Boltzmann
equation (with slab symmetry). The gas is assumed to fill a halfspace on top of a liquid which is
its condensed phase, and the velocity distribution function of molecules entering the halfspace
from the condensed phase is the centered Maxwelllian parametrized by the temperature at the
gasliquid 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), 869997] have proposed a complete method for handling
this kind of problem. The purpose of this talk is to present an alternative, selfcontained 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
UkaiYangYu penalization method for halfspace problems in kinetic theory (suitably modified).
[Work with N. Bernhoff.]

2021/02/08: SeokBae Yun (Sunkyunkwan University (SKKU))
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Title: Ellipsoidal BGK model of the Boltzmann equation with the correct Prandtl number
Ellipsoidal BGK model (ESBGK) is a generalized version of the BoltzmannBGK 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 NavierStokes limit. In this talk,
we review some of the recent results on ESBGK model such as the existence (stationary or nonstationary)
theory and the entropyentropy 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.