We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator, in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all type of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.
In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Lévy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Lévy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heav…
A quadratic optimal transport metric on the set of probability measure over $\R^2$ is introduced. The quadratic cost is given by the euclidean norm on $\R^2$ associated to some well chosen symmetric positive matrix, which makes the metric equivalent to the usual Wasserstein-2 metric. The dissipation of the distance to the equilibrium along the kinetic Fokker-Planck flow, is bounded by below in terms of the distance itself. It enables to obtain some new type of trend to equilibrium estimate in Wasserstein-2 like metric, in the case of non-convex confinement potential.
In this note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter ca…
The long time behavior of a model for a first order, weakly reversible chemical reaction network is considered, where the movement of the reacting species is described by kinetic transport. The reactions are triggered by collisions with a nonmoving background with constant temperature, determining the post-reactional equilibrium velocity distributions. Species with different particle masses are considered, with a strong separation between two groups of light and heavy particles. As an approximation, the heavy species are modeled as nonmoving. Under the assumption of at least one moving species…
We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space $(x,v) \in \mathbb{R}^d \times \mathbb{R}^d$ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $L^1$ or weighted $L^1$ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we…
We study the kinetic Fokker-Planck equation on the whole space with a confining potential. We show quantitative rates of exponential convergence to equilibrium in a well chosen Wasserstein-1 distance. We use the Wasserstein-1 version of Harris's theorem introduced by Hairer and Mattingly. We make use of similarities between hypocoercivity and hypoellipticity in order to use Malliavin calculus to see hypocoercivity for this equation on the level of the SDE.
We propose and study a fully discrete finite volume scheme for the Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted L 2 relative entropy. Then, we adapt the arguments developed in [12] based the hypocoercivity method to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play.
We present the quantitative method of the recent work arXiv:2209.09340 in a simple setting, together with a compactness argument that was not included in arXiv:2209.09340 and has interest per se. We are concerned with the exponential stabilization (spectral gap) for linear kinetic equations with degenerate thermalization, i.e. when the collision operator vanishes on parts of the spatial domain. The method in arXiv:2209.09340 covers both scattering and Fokker-Planck type operators, and deals with external potential and boundary conditions, but in these notes we present only its core argument an…
The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir \cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$ setting, there is lack of coercivity in $H^k$ for the associated operator. To remedy this issue, we shall modify the usual $H^k$ norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on $k$. In parallel, a similar strategy but with coefficients depending on time (c.f. \cite{Herau}), usually referred as Hérau's method, can be employ…
Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs
The Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Department of Mathematics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校数学系）
This paper is dealing with two $L^2$ hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space. The main idea is to perturb the standard $L^2$ norm by a twist obtained either by a nonlocal perturbation build upon diffusive macroscopic dynamics, or by a change of the scalar product based on Lyapunov matrix inequalities. We explore various estimates for equations involving a Fokker-Planck and a linear relaxation operator. We review existing results in simp…
In this paper, hypocoercivity methods are applied to linear kinetic equations with mass conservation and without confinement, in order to prove that the solutions have an algebraic decay rate in the long-time range, which the same as the rate of the heat equation. Two alternative approaches are developed: an analysis based on decoupled Fourier modes and a direct approach where, instead of the Poincaré inequality for the Dirichlet form, Nash's inequality is employed. The first approach is also used to provide a simple proof of exponential decay to equilibrium on the flat torus. The results are …
On a class of nonlinear BGK-type kinetic equations with density dependent collision rates
Warwick Mathematics Institute, University of Warwick（华威大学华威数学研究所）；Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology（代尔夫特理工大学电气、数学与计算机科学学院应用数学研究所）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）
This paper studies convergence to equilibrium for the spatially inhomogeneous linear relaxation Boltzmann equation in Boltzmann entropy and related entropy functionals the $p$-entropies. Villani proved in \cite{V09} entropic hypocoercivity for a class of PDEs in a Hörmander sum of squares form. It was an open question to prove such a result for an operator which does not share this form. We show exponentially fast convergence to equilibrium with explicit rate in entropy for a linear relaxation Boltzmann equation. The key new idea appearing in our proof is the use of a total derivative of the e…
In this work, we mainly concern the limiting behavior of the electromagnetic field of two species Vlasov-Maxwell-Botlzmann system in diffusive limits. As knudsen numbers goes to zero, the electric magnetic and magnetic field may perserve or vanish. We verify rigorously Navier-Stokes, Navier-Stokes-Poisson and Navier-Stokes Maxwell limit of the two species Vlasov-Maxwell-Boltzmann system on the torus in three dimension. The justification is based on the unified and uniform estimates of solutions to the dimensionless Vlasov-Maxwell-Boltzmann. The uniform estimates of solutions are obtained by em…
We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian local equilibria. Using adapted Poincaré and Lions-type inequalities, we develop an explicit and constructive method for estimating the decay rate of time averages of norms of the solutions, which covers various regimes corresponding to subexponential, exponential and superexponential (including Maxwellian) local equilibria. As a consequence, we also derive…
We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 ($μ$) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.
In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial point is that the evolution operator L may have singular and nonsmooth coefficients. This allows the application of the method e.g. to degenerate and singular particle systems arising in Mathematical Physics. As far as we know in such singular cases the relaxation to equilibrium can't be discussed with the help of existing approaches using hypoellipticity, hypocoercivity or stochastic Lyapunov type te…
In this paper, we study the hydrodynamic limits of both the Landau equation and the Vlasov-Maxwell-Landau system in the whole space. Our main purpose is two-fold: the first one is to give a rigorous derivation of the compressible Euler equations from the Landau equation via the Hilbert expansion; while the second one is to prove, still in the setting of Hilbert expansion, that the unique classical solution of the Vlasov-Maxwell-Landau system converges, which is shown to be globally in time, to the resulting global smooth solution of the Euler-Maxwell system, as the Knudsen number goes to zero.…
We study a kinetic model for non-reactive mixtures of monatomic gases with hard-sphere cross-sections under isothermal condition. By considering a diffusive scaling of the kinetic model and using the method of moments, we formally obtain from the continuity and momentum balance equations of the species, in the limit as the scaling parameter goes to zero, the Maxwell-Stefan diffusion equations, with an explicit expression for the diffusion coefficients.
This paper aims to justify the Maxwell-Boltzmann approximation for electrons, preserving the dynamics of ions at the kinetic level. Under sufficient regularity assumption, we provide a precise scaling where the Maxwell-Boltzmann approximation is obtained. In addition, we prove that the reduced ions problem is well-posed globally in time.
We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman-Enskog picture of near-equilibrium ow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dyn…
