We derive the expressions of the local Maxwellians that solve the Boltzmann equation in the interior of an open domain. We determine which of these local Maxwellians satisfy the Boltzmann equation in a regular domain with boundary, without assuming the boundedness of the domain. We investigate separately, on the one hand, the case of the bounce-back boundary condition in any dimension, and on the other hand the case of the specular reflection boundary condition, in dimension $d = 2$ and $d = 3$. In the case of the bounce-back boundary condition, we prove that the only local Maxwellians solving the Boltzmann equation with boundary condition are the global Maxwellians. In the case of the specular reflection, we provide a complete classification of the domains for which only the global Maxwellians solve the Boltzmann equation with boundary condition, and we describe all the local Maxwellians that solve the equation for the domains presenting symmetries.
We prove that Gevrey regularity is propagated by the Boltzmann equation with Maxwellian molecules, with or without angular cut-off. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey regularity by the Fourier transform.
We study the dynamics defined by the Boltzmann equation set in the Euclidean space $\mathbb{R}^D$ in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering …
We consider the non-relativistic quantum Boltzmann equation for fermions and bosons. Using the nonlinear energy method and mild formulation, we justify the global well-posedness when the density function is near the global Maxwellian and vacuum. This work is a generalization and adaptation of the classical Boltzmann theory. Our main contribution is a detailed analysis of the nonlinear operator $Q$ in the quantum context. This is the first piece of a long-term project on the quantum kinetic equations.
In this paper we study the Gevrey regularity for the weak solutions to the Cauchy problem of the non-cutoff spatially homogeneous Botlzmann equation for the Maxwellian molecules model with the singularity exponent $s\in (0,1)$. We establish that any weak solution belongs to the Gevrey spaces for any positive time.
In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we proved the global existence and uniqueness for the cutoff Boltzmann equation in polynomial weighted space for all $γ\in (-3, 1]$. We also proved initially polynomial decay for the large velocity in $L^2$ space will induce polynomial decay rate, while initially exponential decay will induce exponential rate for the convergence. Our proof is based on newly established inequalities for the cutoff Boltzmann equation and semigroup techniques. Moreover, by generalizing the $L_x^\infty L^1_v \cap L^\infty_{x, v}$ approach, w…
The purpose of this paper is to show how the combination of the well-known results for convergence to equilibrium and conditional regularity, in addition to a short-time existence result, lead to a quick proof of the existence of global smooth solutions for the non cutoff Boltzmann equation when the initial data is close to equilibrium. We include a short-time existence result for polynomially-weighted $L^\infty$ initial data. From this, we deduce that if the initial data is sufficiently close to a Maxwellian in this norm, then a smooth solution exists globally in time.
Since the work [13] by Guo [Invent. Math. 153 (2003), no. 3, 593--630], how to establish the global existence of perturbative classical solutions around a global Maxwellian to the Vlasov-Maxwell-Boltzmann system with the whole range of soft potentials has been an open problem. This is mainly due to the complex structure of the system, in particular, the degenerate dissipation at large velocity, the velocity-growth of the nonlinear term induced by the Lorentz force, and the regularity-loss of the electromagnetic fields. This paper aims to resolve this problem in the whole space provided that in…
In this paper, we are interested in the $L^p$-estimates of the Boltzmann equation in the case that the distribution function stays around a travelling local Maxwellian. For this, we divide both sides of the Boltzmann equation by the velocity distribution function with a fractional exponent and reformulate the Boltzmann equation into a regularized one. This amounts to endowing additional integrability on the collision kernel, which in turn enables us to apply simple Hölder type inequalities. Our results cover the whole range of Lebesgue exponents: $0<p\leq \infty$.
The Boltzmann H-theorem implies that the solution to the Boltzmann equation tends to an equilibrium, that is, a Maxwellian when time tends to infinity. This has been proved in varies settings when the initial energy is finite. However, when the initial energy is infinite, the time asymptotic state is no longer described by a Maxwellian, but a self-similar solution obtained by Bobylev-Cercignani. The purpose of this paper is to rigorously justify this for the spatially homogeneous problem with Maxwellian molecule type cross section without angular cutoff.
For the spatially homogeneous Boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above, uniformly in time, by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.
In rotationally symmetric domains, the Boltzmann equation with specular reflection boundary condition has a special type of equilibrium states called the rotational local Maxwellian which, unlike the uniform Maxwellian, has an additional term related to the angular momentum of the gas. In this paper, we consider the initial boundary value problem of the Boltzmann equation near the rotational local Maxwellian. Based on the L2-L1 framework of [12], we establish the global well-posedness and the convergence toward such equilibrium states.
In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK model for a general class of collision frequencies. We prove that the Boltzmann-BGK model linearized around a global Maxwellian admits a unique global smooth solution if the initial perturbation is sufficiently small in a high order energy norm. We also establish an asymptotic decay estimate and uniform $L^2$-stability for nonlinear perturbations.
This is the first one of two papers on the global dynamics of the original Boltzmann equations without angular cutoff on the torus. We address the problem for the hard potentials and Maxwellian molecules in the present paper. The case of soft potentials is left to a forthcoming paper. The key to solve the problem is the energy-entropy method which characterizes the propagation of the regularity, $H$-theorem and the interplay between the energy and the entropy. Our main results are as follows: (i) We present a unified framework to prove the well-posedness for the original Boltzmann equation for…
We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.
As for the spatially homogeneous Boltzmann equation of Maxwellian molecules with the fractional Fokker-Planck diffusion term, we consider the Cauchy problem for its Fourier-transformed version, which can be viewed as a kinetic model for the stochastic time-evolution of characteristic functions associated with the symmetric stable Levy process and the Maxwellian collision dynamics. Under a non-cutoff assumption on the kernel, we establish a global existence theorem with maximum growth estimate, uniqueness and stability of solutions.
We consider the nonlinear boundary layers of the Boltzmann equation in a three-dimensional half-space by perturbing around a Maxwellian, under the assumption that the Mach number of the Maxwellian satisfies ${\cal M}_{\infty} < -1$. In preceding works, nonlinear boundary layers of the Boltzmann equation in a half-line are considered, with stationary solutions obtained and nonlinear stability confirmed. In this paper, we establish the unique existence of stationary solutions for the three-dimensional half-space model, and show that the stationary solution is asymptotic stable.
We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to get spectral information. They were inspired by former works on the Fokker-Planck equation and the main feature of this work is that they are relevant although the equation itself has no regularizing properties.
It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^d)\cap L\log L(\mathbb{R}^d)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molec…
Using a weighted $H^s$-contraction mapping argument based on the macro-micro decomposition of Liu and Yu, we give an elementary proof of existence, with sharp rates of decay and distance from the Chapman--Enskog approximation, of small-amplitude shock profiles of the Boltzmann equation with hard-sphere potential, recovering and slightly sharpening results obtained by Caflisch and Nicolaenko using different techniques. A key technical point in both analyses is that the linearized collision operator $L$ is negative definite on its range, not only in the standard square-root Maxwellian weighted n…
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 well-known Rutherford differential cross section, denoted by $ dΩ/dσ$, corresponds to a two body interaction with Coulomb potential. It leads to the logarithmically divergence of the momentum transfer (or the transport cross section) which is described by $$\int_{{\mathbb S}^2} (1-\cosθ) \frac{dΩ}{dσ} dσ\sim \int_0^π θ^{-1}dθ. $$ Here $θ$ is the deviation angle in the scattering event. Due to screening effect, physically one can assume that $θ_{\min}$ is the order of magnitude of the smallest angles for which the scattering can still be regarded as Coulomb scattering. Under ad hoc cutoff $…
We consider the $n$-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of $L^1$-Maxwellian weighted estimates, and consequently, the propagation $L^\infty$-Maxwellian weighted estimates to all derivatives of the initial value problem associated to the afore mentioned problem. More specifically, we extend to all derivatives of the initial value problem associated to this class of Boltzmann equations corresponding sharp moment (Povzner) inequalities and time propagati…
We prove the well-posedness for the non-cutoff Boltzmann equation with soft potentials when the initial datum is close to the {\it global Maxwellian} and has only polynomial decay at the large velocities in $L^2$ space. As a result, we get the {\it propagation of the exponential moments} and the {\it sharp rates} of the convergence to the {\it global Maxwellian} which seems the first results for the original equation with soft potentials. The new ingredients of the proof lie in localized techniques, the semigroup method as well as the propagation of the polynomial and exponential moments in $L…
We investigate the long time behavior of a system of viscoelastic particles modeled with the homogeneous Boltzmann equation. We prove the existence of a universal Maxwellian intermediate asymptotic state and explicit the rate of convergence towards it. Exponential lower pointwise bounds and propagation of regularity are also studied. These results can be seen as the generalization of several classical facts holding for the pseudo-Maxwellian and constant normal restitution models.
The spectrum structure of the linearized relativistic Boltzmann equation around a global Maxwellian is studied in this paper. Based on the spectrum analysis, we establish the optimal time-convergence rates of the global solution to the Cauchy problem for the relativistic Boltzmann equation.
For the non cutoff radially symmetric homogeneous Boltzmann equation with Maxwellian molecules, we give the numerical solutions using symbolic manipulations and spectral decomposition of Hermit functions. The initial data can belong to some measure space.
Anderson-Witting model is a relaxational model equation of the relativistic Boltzmann equation, which sees a wide application in physics. In this paper, we study the existence of classical solutions and its asymptotic behavior when the solution starts sufficiently close to a global relativistic Maxwellian.
We consider the rate of convergence of solutions of spatially inhomogenous Boltzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians. Maxwellians are spatially homogenous static Maxwell velocity distributions with different temperatures and mean velocities. We study solutions in weighted space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3)$. We prove a conjecture of C. Villani: assume the solution is sufficiently localized and sufficiently smooth, then the solution, in $L^{1}$-space, converges to a Maxwellian, exponentially fast in time.
In many works, the linearized non-cutoff Boltzmann operator is considered to behave essentially as a fractional Laplacian. In the present work, we prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
We study the Cauchy problem for the relativistic Boltzmann equation near relativistic Maxwellians in the whole space. The purpose of this article is to handle hard potentials, and for initial data with finite $L^\infty$ norm, to construct global in time mild solutions. We also prove the optimal time decay rates of the solutions.
We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region, we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region, we obtain sub-exponential decay in the space variable. The singular wave esti…
We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the $L^r_v L^\infty_t L^\infty_x $ spaces where $r \in (1,\infty]$ by using the excess conservation laws and entropy inequality introduced in [5].
For the relativistic Boltzmann equation in $\R^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988 by Dudy{ń}ski and Ekiel-Je{ż}ewska.
We study the stationary Boltzmann equation in a thin slab for a rarefied gas for which the molecular mean free path is comparable to the film thickness. We prove that there exists a solution which converges, in the hydrodynamic limit, to a density Maxwellian -- this density is obtained as the solution to the generalized Reynolds equation. The convergence is proved using a truncated Hilbert expansion by carefully estimating the remainder.
This work concerns the Vlasov-Poisson-Boltzmann system without angular cutoff and Vlasov-Poisson-Landau system including Coulomb interaction in bounded domain, namely union of cubes. We establish the global stability, exponential large-time decay with specular-reflection boundary condition when an initial datum is near Maxwellian equilibrium. We provide the compatible specular boundary condition for high-order derivatives and a velocity weighted energy estimate.
We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient $α\in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove uniqueness of the stationary solution (with given mass) in the weakly inelastic regime; i.e., for any inelasticity parameter $α\in (α_0,1)$, with some constructive $α_0 \in [0, 1)$. Our analysis is based on a perturbative argument which uses the knowledge of the stationary solution in the elastic limit and quantitative estimates of the convergence of stationary so…
The paper is devoted to constructing the global solutions around global Maxwellians to the initial-boundary value problem on the Boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries. We allow a class of non-negative initial data which have arbitrary large amplitude and even contain vacuum. The result shows that the oscillation of solutions away from global Maxwellians becomes small after some positive time provided that they are initially close to each other in $L^2$. This yields the disappearance of any initial vacuum and the exponential convergence of l…
This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. Together with the results of Parts I and II about the well posedness of the Cauchy problem around Maxwellian, we conclude this series with a satisfactory mathematical theory for Boltzmann equation without angular cutoff.
We establish the global-wellposedness and stability of the Boltzmann equation with the specular reflection boundary condition in general smooth convex domains when an initial datum is close to the Maxwellian with or without a small external potential. In particular, we have completely solved the long standing open problem after an announcement of [20] in 1977.
By identifying a norm capturing the effect of the forcing governed by the Poisson equation, we give a detailed spectrum analysis on the linearized Vlasov-Poisson-Boltzmann system around a global Maxwellian. It is shown that the electric field governed by the self-consistent Poisson equation plays a key role in the analysis so that the spectrum structure is genuinely different from the well-known one of the Boltzmann equation. Based on this, we give the optimal time decay rates of solutions to the equilibrium.
We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignani's conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, both with and without a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain operator due to Matthes and Toscani (201…
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 $μ_E(x,v)=\exp\{-\frac{|v|^2}{2}-Φ(x)\}$.
The existence and stability of collisional kinetic equation, especially non-cutoff Boltzmann equation, in bounded domain with physical boundary condition is longstanding open problem. This work proves the global stability of the Landau equation and non-cutoff Boltzmann equation in union of cubes with the specular reflection boundary condition when an initial datum is near Maxwellian. Moreover, the solution enjoys exponential large-time decay in bounded domain. Our method is based on that fact that normal derivatives in cubes is also derivatives along axis, which allows us to obtain high-order …
We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.
We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary solutions solutions are Hölder continuous with order $\frac1{2}^-$ away from the boundary provided the incoming data have the same regularity. The key idea is to partially transfer the regularity in velocity obtained by collision to space through transport and collision.
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…
We study the Boltzmann equation near a global Maxwellian in the case of bounded domains. We consider the boundary conditions to be either specular reflections or Maxwellian diffusion. Starting from the reference work of Guo in $L^\infty_{x,v}\left(\left(1+|v|\right)^βe^{|v|^2/4}\right)$, we prove existence, uniqueness, continuity and positivity of solutions for less restrictive weights in the velocity variable; namely, polynomials and stretch exponentials. The methods developed here are constructive.
In this paper, we investigate the stability of Boltzmann equation with large external potential in $\mathbb{T}^3$. For a class of initial data with large oscillations in $L^\infty_{x,v}$ around the local Maxwellian, we prove the existence of a global solution to the Boltzmann equation provided the initial perturbation is suitably small in $L^2$-norm. The large time behavior of the Boltzmann solution with exponential decay rate is also obtained. This seems to be the first result on the perturbation theory of large-amplitude non-constant equilibriums for large-amplitude initial data.
We start with some global Maxwellian function $M$, which is a stationary solution (with the constant total density $ρ$) of the Boltzmann equation, and we denote the number of the corresponding space variables by $n$. The notion of distance between the global Maxwellian function and an arbitrary solution $f$ (with the same total density $ρ$ at the fixed moment $t$) of the Boltzmann equation is introduced. In this way we essentially generalize the important Kullback-Leibler distance, which was used before. An extremal problem to find a solution of the Boltzmann equation, such that $\dist\{M,f\}$…
We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hard potentials, or hard spheres collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species operator $\mathbf{L}$ is known to possess an explicit spectral gap $λ_{\mathbf{L}}$, in the global equilibrium weighted $L^2$ space. We study a new operator $\mathbf{L^\varepsilon}$ obtained by linearizing the Boltzmann operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order $\varep…
We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3).$ Special solutions of these equations, called "Maxwellians," are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted $L^{1}$-space converge to a Maxwellian in $L^{1},$ exponentially fast in time.
