Webinar Kinetic and fluid equations for collective behavior
Stéphane Brull (Bordeaux), Seung-Yeal Ha (SNU)
Philippe Thieullen (Bordeaux)
Link:
Forthcoming speakers
-
2021/05/28: Benoit Perthame (Sorbone Université)
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.
-
2021/06/25: Isabelle Tristani (ENS Ulm Paris)
Title: (not yet submitted)
Abstract:
Past speakers
-
2021/05/14: Donghyun Lee (Postech)
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: Prof. 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: Prof. 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.
