The stability of the Maxwellian of the Boltzmann equation with a large amplitude external potential $Φ$ has been an important open problem. In this paper, we resolve this problem with a large $C3-$potential in a periodic box $\mathbb{T}^d$, $d \geq 3$. We use [1] in $L^p-L^{\infty}$ framework to establish the well-posedness and the $L^{\infty}-$stability of the Maxwellian $μ_E(x,v)=\exp\{-\frac{|v|^2}{2}-Φ(x)\}$.
In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with diffuse reflection boundary conditions. We consider a system of $N$ hard spheres of diameter $ε$ in a box $Λ:= [0, 1] \times (R/Z)^2$. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, and conserving kinetic energy. We prove that the first marginal of the process converges in the scaling $Nε^2 = 1$, $ε\rightarrow 0$ to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.
This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For th…
We prove the appearance of an explicit lower bound on the solution to the full Boltzmann equation in the torus for a broad family of collision kernels including in particular long-range interaction models, under the assumption of some uniform bounds on some hydrodynamic quantities. This lower bound is independent of time and space. When the collision kernel satisfies Grad's cutoff assumption, the lower bound is a global Maxwellian and its asymptotic behavior in velocity is optimal, whereas for non-cutoff collision kernels the lower bound we obtain decreases exponentially but faster than the Ma…
Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed 1. Assuming that the holes are absorbing -- i.e. that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distrib…
The two-dimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle and the obstacles, this dynamical system is studied in the Boltzmann-Grad limit, assuming that the obstacle radius $r$ and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle's distribution function is slowly varying in the space variable. In this limit, the period…
We study the diffusive expansion for solutions around Maxwellian equilibrium and in a periodic box to the Vlasov-Maxwell-Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDE's, the incompressible Vlasov-Navier-Stokes-Fourier system and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a unified nonlinear energy method and it guarantees the global in time validity of such a…
We study the acoustic limit from the Boltzmann equation in the framework of classical solutions. For a solution $F_\varepsilon=μ+\varepsilon \sqrtμf_\varepsilon$ to the rescaled Boltzmann equation in the acoustic time scaling \partial_t F_\varepsilon +\vgrad F_\varepsilon =\frac{1}{\varepsilon} \Q(F_\varepsilon,F_\varepsilon), inside a periodic box $\mathbb{T}^3$, we establish the global-in-time uniform energy estimates of $f_\varepsilon$ in $\varepsilon$ and prove that $f_\varepsilon$ converges strongly to $f$ whose dynamics is governed by the acoustic system. The collision kernel $\Q$ includ…
Time-periodic solutions of the Boltzmann equation with soft potentials in $\mathbb{R}^3$
Huazhong University of Science and Technology（华中科技大学）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing（工程建模与科学计算湖北省重点实验室）；Central China Normal University（华中师范大学）；Hubei Key Laboratory of Mathematical Sciences（数学科学湖北省重点实验室）；Wuhan University（武汉大学）；Computational Science Hubei Key Laboratory（计算科学湖北省重点实验室）
In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^\infty_\ell$. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of (Dudy{ń}ski and Ekiel-Je{ż}ewska, Comm. Math. Phys., 1988); this resolves the open question of global existence for the soft potentials.
In the study of solutions to the relativistic Boltzmann equation, their regularity with respect to the momentum variables has been an outstanding question, even local in time, due to the initially unexpected growth in the post-collisional momentum variables which was discovered in 1991 by Glassey & Strauss \cite{MR1105532}. We establish momentum regularity within energy spaces via a new splitting technique and interplay between the Glassey-Strauss frame and the center of mass frame of the relativistic collision operator. In a periodic box, these new momentum regularity estimates lead to a proo…
We prove that the Boltzmann-Grad limit of the Lorentz gas with periodic distribution of scatterers cannot be described with a linear Boltzmann equation. This is at variance with the case of a Poisson distribution of scatterers, for which the convergence to the linear Boltzmann equation has been proved by Gallavotti [Phys. Rev. (2) 185 (1969), p. 308].
We consider the spatially inhomogeneous quantum Boltzmann equation for bosons with a singular collision kernel, the weak-coupling limit of a large system of Bose-Einstein particles interacting through inverse power law. Global well-posedness of the corresponding Cauchy problem is proved in a periodic box near equilibrium for initial data satisfying high temperature condition.
We use some new nonlinear estimates found in \cite {LM} to improve the results of \cite{GL} that establish the acoustic limit for DiPerna-Lions solutions of Boltzmann equation in three ways. First, we enlarge the class of collision kernels treated to that found in \cite{LM}, thereby treating all classical collision kernels to which the DiPerna-Lions theory applies. Second, we improve the scaling of the kinetic density fluctuations with Knudsen number from $O(ε^m)$ for some $m>\frac12$ to $O(ε^\frac12)$. Third, we extend the results from periodic domains to bounded domains with impermeable boun…
Bedrossian and Masmoudi (2015) showed nonlinear stability of 2D inviscid parallel shear flows in the infinite stripe $(\mathbb{R}/2\mathbb{Z})\times\mathbb{R}$. Inspired by their result, combining a diffeomorphism group result by Misiołek (1993), in this paper we give a lower bound of perturbations in 2D inviscid shear flow in a periodic box.
We prove the existence of global renormalized solutions to the Boltzmann equation in bounded domain with incoming boundary condition, with non-cutoff collision kernels. Thus we extend the results of \cite{villani2002noncutoff} for whole spaces or periodic domain to bounded domains endorsed with incoming boundary condition.
The incompressible Navier-Stokes-Fourier system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos-Levermore-Ukai-Yang (2008). The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in $L^2\cap L^\infty$ setting in a periodic box. Based on an odd-even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the incompressible Navier-Stokes-Fourier system with viscous heating and the super Burnett functions. More impo…
The Boltzmann equation without an angular cutoff in a three-dimensional periodic domain is considered. The global-in-time existence of solutions in a function space $ W_k^{ζ,p}L^\infty_TL^2_v $ with $p>1$ and $ζ>3(1-\frac{1}{p})$ is established in the perturbation framework and the long-time behavior of solutions is also obtained for both hard and soft potentials. The proof is based on several norm estimates.
In this work, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. To justify this connection, our main idea is to use information on the limit system (for instance the fact that the Navier-Stokes equations are globally wellposed in two space dimensions or when the data are small). In particular we prove that the life span of the solutions to the rescaled Boltzmann equation is bounded from below by that of the Navier-Stokes system. We deal with general initial data in the whole space in dimensions 2 and 3, and also with well-prepared data in …
A nonlinear Poisson--Boltzmann equation with transmission boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic multiphase medium with dilute solid particles. The key issue is that the interfacial transfer allows jumps and thus discontinuous solutions of the problem. Based on variational techniques, we derive the homogenisation of the discontinuous problem subject to inhomogeneous transmission interface conditions. Moreover, we establish a rigorous residual error esti…
The hydrodynamic limit of the Vlasov-Maxwell-Boltzmann equations is considered for weak solutions. Using relative entropy estimate about an absolute Maxwellian, an incompressible Electron-Magnetohydrodynamics-Fourier limit for solutions of the Vlasov-Maxwell-Blotzmann equations over any periodic spatial domain in $\R^3$ is studied. It is shown that any properly scaled sequence of renormalized solutions of the Vlasov-Maxwell-Boltzmann equations has fluctuations that (in the weak $L^1$ topology) converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and B…
The study of positivity of solutions to the Boltzmann equation goes back to Carleman (1933), and the initial argument of Carleman was developed byPulvirenti-Wennberg (1997), the second author and Briant (2015). The appearance of a lower bound with Gaussian decay had however remained an open question for long-range interactions (the so-called non-cutoff collision kernels). We answer this question and establish such Gaussian lower bound for solutions to the Boltzmann equation without cutoff, in the case of hard and moderately soft potentials, with spatial periodic conditions, and under the sole …
We prove the unique existence and exponential decay of global in time classical solutions to the special relativistic Boltzmann equation without any angular cut-off assumptions with initial perturbations in some weighted Sobolev spaces. We consider perturbations of the relativistic Maxwellian equilibrium states. We work in the case of a spatially periodic box. We consider the general conditions on the collision kernel from Dudyński and Ekiel-Jeźewska (Commun Math Phys \textbf{115}(4):607--629, 1985). Additionally, we prove sharp constructive upper and coercive lower bounds for the linearized r…
In this paper, we are concerned with the Vlasov-Poisson-Boltzmann (VPB) system in three-dimensional spatial space without angular cutoff in a rectangular duct with or without physical boundary conditions. Near a local Maxwellian with macroscopic quantities given by rarefaction wave solution of one-dimensional compressible Euler equations, we establish the time-asymptotic stability of planar rarefaction wave solutions for the Cauchy problem to VPB system with periodic or specular-reflection boundary condition. In particular, we successfully introduce physical boundaries, namely, specular-reflec…
We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild…
This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness, and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Je$\dot{\text{z}}$ewska (Comm. Math. Phys. \textbf{115}(4):607--629, 1985), and our assumptions include the case of Israel particles (J. Math. Phys. \textbf{4}:1163--1181, 1963). In this physical s…
This article is on the simultaneous diffusion approximation and homogenization of the linear Boltzmann equation when both the mean free path $\varepsilon$ and the heterogeneity length scale $η$ vanish. No periodicity assumption is made on the scattering coefficient of the background material. There is an assumption made on the heterogeneity length scale $η$ that it scales as $\varepsilon^β$ for $β\in(0,\infty)$. In one space dimension, we prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation for any $β\le2$ in the $\varepsilon\to0$ limit. In…
We consider solutions $f=f(t,x,v)$ to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions $x \in \mathbb T^d$, for hard and moderately soft potentials \emph{without the angular cutoff assumption}, and under the \emph{a priori} assumption that the main hydrodynamic fields, namely the local mass $\int\_v f(t,x,v)$ and local energy $\int\_v f(t,x,v)|v|^2$ and local entropy $\int\_v f(t,x,v) \ln f(t,x,v)$, are controlled along time. We establish quantitative estimates of \emph{propagation} in time of "pointwise polynomial moments", i.e. $\sup\_{x,v} f(t,x,v) (1+|…
In this paper we consider the one-phase Stefan problem with surface tension, set in a two-dimensional strip-like geometry, with periodic boundary conditions respect to the horizontal direction $x_1\in\mathbb{T}$. We prove that the system is locally null-controllable in any positive time, by means of a control supported within an arbitrary open and non-empty subset. We proceed by a linear test and duality, but quickly find that the linearized system is not symmetric and the adjoint has a dynamic coupling between the two states through the (fixed) boundary. Hence, motivated by a Fourier decompos…
