Analysis, modeling and numerical method for kinetic and related models

Bordeaux, 14-15 November 2023

Organizing commitee

Stéphane Brull Stephane.Brull@math.u-bordeaux.fr



Presentation of the workshop

This workshop is devoted to kinetic and related models. Kinetic models have merged as a powerful tool in order to describe out of equilibrium systems. They constitute a bridge between macroscopic and microscopic descriptions. They play vital roles in the modeling of physical systems ranging from gas dynamics, plasmas, semiconductor devices to biological and social sciences, radiative transfer,...
For example, when a space vehicle (shuttle) enters the atmosphere, its environment is extremely severe. Its velocity can reach several kilometers per seconds (hypersonic) and the temperature of its surroundings can be very high (more than 10000 K). At these temperatures, the molecules of the rarefied air are dissociated and complex phenomena happen. Therefore precise computations are needed to estimate for example heat fluxes on the wall of a shuttle and the modelling rests on kinetic equations that have to describe rarefied, dissociated gas mixtures.
For this conference, we aim to bring together junior and senior experts in all aspects of kinetic theory.

The goal of this workshop is to gather researchers in the area of modelling, analysis and simulation of kinetic equations in order to review recent advances and explore open issues of this field.

The language of this meeting is english

Registration

Registration is mandatory even if there is no registration fee for the conference. You can register by sending an email to Stéphane Brull > until 1st of November.


Sébastien Guisset (CEA-CESTA) : Chapman-Enskog derivation of multicomponent Navier-Stokes equations
Abstract: There are several reasons to extend the presentation of Navier-Stokes equations to multicomponent systems. Many technological applications are based on physical phenomena that are present in neither pure elements nor in binary mixtures. Whereas Fourier's law must already be generalized in binaries, it is only with more than two components that Fick's law breaks down in its simple form. The emergence of dissipative phenomena also affects the inertial confinement fusion configurations, designed as prototypes for the future fusion nuclear plants hopefully replacing the fission ones. This important topic can be described in much simpler terms than it is in many textbooks since the publication of the formalism put forward recently by Snider [Phys. Rev. E 82, 051201 (2010)]. In a very natural way, it replaces the linearly dependent atomic fractions by the independent set of partial densities. Then, the Chapman-Enskog procedure is hardly more complicated for multicomponent mixtures than for pure elements. In our comprehensive presentation, we emphasize the physical arguments behind Chapman-Enskog derivation and keep the mathematics as simple as possible. This excludes, as a technical non-essential aspect, the solution of the linearized Boltzmann equation through an expansion in Hermite polynomials. We discuss the link with the second principle of thermodynamics of entropy increase, and what can be learned from this exposition.

François Golse (Ecole polytechnique) : Particle Deceleration in Random Media talk.pdf
Abstract:Consider a distribution of point particles moving on straight lines and being strongly slowed down whenever crossing small spherical inhomogeneities whose centers follow Poisson's distribution in the d-dimensional Euclidean space. The purpose of this talk is to obtain a governing equation for the limiting distribution function averaged over obstacle configuration. The analysis of this problem follows in part Gallavotti's 1969 derivation of the kinetic equation for the Lorentz gas. (Work in collaboration with A.-J. Costa Soares and V. Ricci)

Nathalie Bonamy Parrilla (IMB-Bordeaux) : Kinetic model and simulation for electrons in glow discharge plasmas talk.pdf
Abstract: In order to describe electrons behaviour in glow discharge plasmas submitted to an electric field, we consider a drift-diffusion system. By proceeding as in [1], we recover the corresponding drift-diffusion system, from a rescaled kinetic system taking into account electron-neutrals, elecron-electron and ionization collisions. We obtain in particular an explicit expression of the mobility and diffusion coefficients that satisfies Einstein relations. Taking advantage of this asymptotic analysis, we aim to solve numerically the fluid system using lattice Boltzmann methods (LBM). Such methods have proven to be advantageous for fluid equations but are not broadly used for cold plasmas. By following the same asymptotic procedure on the lattice Boltzmann equation we obtained an approached fluid system. Finally we derived a numerical scheme with the D1Q2 lattice.

[1]: Choquet I., Degond P., Lucquin-Desreux B.: A hierarchy of diffusion models for partially ionized plasmas. Discrete and Continuous Dynamical Systems - B 8(4): 735-772, (2007).

Lois Delande (IMB-Bordeaux) : Sharp spectral gap of adaptative Langevin dynamics talk.pdf
Abstract: We consider a Fokker-Planck type differential operator associated to an adaptive Langevin dynamic. Under a few assumptions, we prove Eyring-Kramers formulas for the bottom of the spectrum of this operator in the low temperature regime. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method.

Laurent Desvillettes (Université Paris Cité) : Some novelties on Cercignani's conjecture talk.pdf
Abstract: Cercignani's conjecture concerns the link between the relative entropy and the entropy dissipation of Boltzmann's equation. It has been extensively studied in the 90s and the 2000s. We present here some new results in the case of Landau's equation (which can be seen as a limit of Boltzmann's equation, when the collisions become grazing).

Thomas Normand (Nantes) : Small eigenvalues of semiclassical Boltzmann operators talk.pdf
abstract We consider some inhomogeneous linear Boltzmann equations in a low temperature regime and in the presence of an external force deriving from a potential. We provide a sharp description of the spectrum near 0 of the associated operators. It enables us to obtain some precise information on the long time behavior of the solutions with in particular some quantitative results of return to equilibrium and metastability. This type of work usually requires two steps. First, we establish some resolvent estimates thanks to hypocoercive methods. We can then use and adapt some constructions of "gaussian quasimodes" (approximated eigenfunctions) involving some tools from semiclassical microlocal analysis which will provide the desired sharp localization of the small eigenvalues.

Mathieu Rigal (IMB-Bordeaux) : Implicit kinetic schemes for the shallow water system talk.pdf
Abstract: In this presentation, we are interested in the design of an entropy-stable, positive and well-balanced scheme for the shallow water system. To get the latter property, a well known strategy consists to perform a hydrostatic reconstruction step before computing the numerical fluxes. When combined with an explicit kinetic scheme, this reconstruction technique has been shown to satisfy a discrete entropy inequality with a positive error term which in some cases prevents the energy from being dissipated. We propose to improve the stability of this kinetic scheme by studying its implicit version. First, the case of a flat bottom is considered, where we obtain a fully implicit kinetic solver satisfying a discrete entropy inequality and keeping the water height positive without any restriction on the time step. Under a simplification, it is also possible to write the update analytically. Then we discuss the case of a varying bottom and propose an implicit kinetic scheme with hydrostatic reconstruction whose update is approximated by a fixed point method. Unlike its fully explicit counterpart, we can ensure that our scheme verifies a discrete entropy inequality without positive error term. These results will be illustrated through numerical experiments.

Marzia Bisi (Parma) : A Boltzmann-BGK model for gas mixtures and its hydrodynamic limits talk.pdf
Abstract: We present a consistent mixed Boltzmann-BGK model for an inert mixture, which combines the positive features of both models, namely the detailed description of collisions given by the Boltzmann integral operators with the simplicity and the numerical manageability of the BGK relaxation operators. This kinetic model has the same structure of the full Boltzmann equations, with the collision term of each constituent given by a sum of bi-species bilinear operators, that may be chosen of Boltzmann or of BGK type. The presence of a collision operator for any pair of gaseous components allows for a consistent derivation of evolution equations for the main macroscopic fields in different hydrodynamic regimes, according to the dominant collision process. Specifically, we consider the regime dominated by the whole collision phenomena and the one dominated by intra-species collisions only, suitable to describe the two-scale process appearing in mixtures of gases with disparate masses. A Chapman-Enskog procedure allows to obtain an explicit closure of macroscopic equations at Navier-Stokes level, with transport coefficients in agreement with physical expectations.

Seok-Bae Yun (SunkyunKwan University) : Weak solutions to the stationary BGK model in a slab
Abstract:We consider stationary flows between two condensed phases in the framework of the stationary BGK model in a slab. Under the physically minimum conditions on the inflow functions, namely the finite mass flux, energy flux and entropy flux , the existence of weak solution is derived. The main difficulties are, among others, (1) the impossibility of truncation of the relaxation operator in the construction of the approximate scheme, and (2) the control of the velocity distribution functions using the macroscopic fields near vanishing velocity region. This is a joint work with Stephane Brull.

Marwa Shahine (IMB-Bordeaux) : Fredholm Property of the Linearized Boltzmann Operator for a Mixture of Polyatomic Gases talk.pdf
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.

Ludovic Godard-Cadillac (IMB-Bordeaux) : A bi-species kinetic model for cylindrical Langmuir probe : existence result and numerical analysis talk.pdf
Abstract: we study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists in a bi-species Vlasov-Poisson equation in a domain contained between two cylinders with prescribed boundary conditions. The interior cylinder models the probe while the exterior cylinder models the interaction with the plasma core. We prove the existence of a weak-strong solution for this model in the sense that we get a weak solution for the 2 Vlasov equations and a strong solution for the Poisson equation. The first parts of this work are devoted to explain the model and proceed to a detailed study of the Vlasov equations. This study then leads to a reformulation of the Poisson equation as a 1D non-linear and non-local equation and we prove it admits a strong solution using an iterative fixed-point procedure. Eventually we proceed to a qualitative description of the solution under the so-called "generalized Bohm condition" on the incomming fluxes and a numerical investigation of the obtained equation. Due to technical obstacles, we mainly focussed on the "quasi-radial" fluxes for the numerical analysis, which turns out to be enough to validate the model. Curves of the obtained trajectories of particles and curves of the collected current versus the applied voltage are presented.

Yoshihiro Ueda (Kobe University) : Stability theory for the linear symmetric hyperbolic system with general relaxation talk.pdf
Abstract: In this talk, we study the dissipative structure for the linear symmetric hyperbolic system with general relaxation. If the relaxation matrix of the system has symmetric properties, Shizuta and Kawashima (1985) introduced the suitable stability condition, and Umeda, Kawashima and Shizuta (1984) analyzed the dissipative structure. On the other hand, Ueda, Duan and Kawashima (2012, 2018) focused on the system with non-symmetric relaxation and got partial results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, this talk aims to extend the stability theory introduced by Shizuta and Kawashima (1985) and Umeda, Kawashima and Shizuta (1984) to our general system. Furthermore, we will consider the optimality of the dissipative structure. If we have time, I would like to discuss some physical models for its application and new dissipative structures.

Programm


Tuesday 14th of November

9h30-10h30 Laurent Desvillettes
10h30-11h00 Coffee break
11h00-12h00 Thomas Normand
12h00-12h30 Loïs Delande
12h30-14h00 Lunch
14h00-15h00 Mathieu Rigal
15h00-16h00 Marzia Bisi
16h00-16h30 Coffee break
16h30-17h30 Marwa Shahine
20h00 Dîner at "Le Plana"


Wednesday 15th of November

9h30-10h30 François Golse
10h30-11h00 Coffee break
11h00-12h00 Yoshihiro Ueda
12h00-12h30 Nathalie Bonamy Parilla
12h30-14h00 Lunch
14h00-15h00 Seok-Bae Yun
15h00-16h00 Sébastien Guisset
16h00 -16h20 Coffee break
16h20-17h20 Ludovic Godard-Cadillac


Participants:

Stéphane Brull Denise Aregba Marwa Shahine
Thomas Normand François Golse Laurent Desvillettes
Nathalie Bonamy Parrilla Damien Toussaint Corentin Prigent
François Rogier Mathieu Rigal Sébastien Guisset
Ludovic Godard-Cadillac Seok-Bae Yun Loïs Delande
Marzia Bisi Yoshihiro Ueda Franck Sueur
Florent Noisette Pierre Gervais Adrien Tendani Soler
Gyuyoung Hwang Philippe Thieullen Mouez Dimassi

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