L. Desvillettes, G. Furioli, E. Terraneo
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.
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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 study the Gevrey class regularity for solutions of the spatially homogeneous Landau equations in the hard potential case and the Maxwellian molecules case.
By using the energy-type inequality, we obtain, in this paper, the result on propagation of Gevrey regularity for the solution of the spatially homogeneous Landau equation in the cases of Maxwellian molecules and hard potential.
On the radius of analyticity and Gevrey regularity for the Boltzmann equation
本文研究了Boltzmann方程在硬势场中的分析半径和Gevrey正则性，通过宏微观分解和次椭圆估计获得短时和全局时间的正则性估计。
In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for t…
In this paper we show the Gevrey regularizing effect of solutions to the non-cutoff spatially homogeneous and inhomogeneous Boltzmann equation for a particular soft potential with critical singularity s=1/2.
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…
We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.
In this paper we consider the non-cutoff Boltzmann equation in spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order $1/(2s)$ in the velocity variable $v$ and order 1 in the space variable $x$. The strategy relies on our recent results for spatially homogeneous case (J. Diff. Equ. 253(4) (2012), 1172-1190. DOI: 10.1016/j.jde.2012.04.023). Rather, we need much more intricate analysis additionally in order to handle w…
In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for $C^\infty$ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto's recent paper (See Morimoto: J. Pseudo-Differ. Oper. Appl. (2010) 1: 139-159, DOI:10.1007/s11868-010-0008-z), but we extend the range of the index $γ$ satisfying $γ+ 2s \in (-1,1)$, $s\in (0,1/2)$ and in this case we consider the kinetic factor in the form of $Φ…
The Gevrey Gelfand-Shilov regularizing effect of the Landau equation with soft potential
本文针对麦克斯韦分布附近微扰框架下带软势的空间非齐次朗道方程柯西问题，在初始数据满足速度变量指数衰减小性假设时，建立了解的最优盖弗雷-盖尔范德-希洛夫正则化效应。
In this paper we study the Gevrey smoothing effect of solutions to the non-cutoff spatially homogeneous and inhomogeneous Boltzmann equation for soft potential. We consider the mild singularity case $s<1/2$ as we did in the previous work for spatially homogeneous case (J. Diff. Equ. 253(4) (2012), 1172-1190. DOI: 10.1016/j.jde.2012.04.023) and for spatially inhomogeneous case (arXiv:1304.2971), and try to extend the range of $γ$. We derive a new coercivity estimate for collision operator, using which we can obtain the Gevrey regularity for $γ\in (-5/2,0)$ improving the previous assumption $γ\i…
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.
On the local Maxwellians solving the Boltzmann equation with boundary condition
本文研究带边界 Boltzmann 方程的局部 Maxwell 分布，证明 bounce-back 条件下仅有全局解，并对二维、三维镜面反射条件下的区域及对称局部解进行完整分类。
In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.
In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.
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 Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian
本文研究无角截断的Vlasov-Poisson-Boltzmann系统与库仑势的Vlasov-Poisson-Landau系统，在多项式加权Sobolev空间中建立解的整体存在性、唯一性及大时间行为，证明依赖半群方法的额外耗散与静电场能量估计。
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.
The Vlasov-Maxwell-Boltzmann/Landau system with polynomial perturbation near Maxwellian
本文研究麦克斯韦分布附近多项式扰动下的Vlasov-Maxwell-Boltzmann/Landau系统，通过Duhamel原理和时间衰减分析，证明了多项式加权空间中解的整体存在性、唯一性及大时间行为。
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.
In the paper, for the Cauchy problem on the non-cutoff Boltzmann equation in torus, we establish the global-in-time Gevrey smoothness in velocity and space variables for a class of low-regularity mild solutions near Maxwellians with the Gevrey index depending only on the angular singularity. This together with [24] provides a self-contained well-posedness theory for both existence and regularity of global solutions for initial data of low regularity in the framework of perturbations. For the proof we treat in a subtle way the commutator between the regularization operators and the Boltzmann co…
Half-space problem on the Boltzmann equation with zero Mach number at infinity
研究Boltzmann方程在半空间中的长时间动力学，采用扩散反射边界条件和小扰动，证明了低正则性解和Gevrey正则性传播，克服了边界效应和非线性相互作用的挑战。
Polynomial tail solutions of the non-cutoff Boltzmann equation near local Maxwellians
本文将Caflisch分解融入玻尔兹曼宏微观分解，构造无截断玻尔兹曼方程在局部麦克斯韦分布附近的多项式尾解，设计新能量泛函得到ε^{1/2}量级扰动范数及常状态下全局存在性结果。
Global Regularity of the Vlasov-Poisson-Boltzmann System Near Maxwellian Without Angular Cutoff for Soft Potential
针对软势无角截断VPB系统，证明近Maxwellian初值下全空间Cauchy问题的全局正则性，覆盖$0<s<1$全范围，完善平滑效应，采用时间加权能量法与伪微分演算。
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…
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.
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.
Quantitative lower bound for solutions to the Boltzmann equation in non-convex domains
本文在非凸域中研究Boltzmann方程解的定量下界，建立了截断和非截断情况下的不同下界结果，扩展了此前要求域凸性的研究成果。
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.
We study local existence for the Boltzmann equation near a global Maxwellian.
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.
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.
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…
Landau damping, collisionless limit, and stability threshold for the Vlasov-Poisson equation with nonlinear Fokker-Planck collisions
本文研究小碰撞频率下 VPFP 方程，证明 Gevrey 类扰动可引发 Landau 阻尼与增强耗散，给出稳定性阈值上界，并证明短时间内解收敛到无碰撞 Vlasov-Poisson 方程的解。
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 $…
The non-isothermal Maxwell-Stefan asymptotics of the multi-species Boltzmann equations
本文严格证明了多组分Boltzmann方程在扩散尺度下收敛到非等温Maxwell-Stefan系统，建立了全局适定性与局部强制性，推广了等温结果。
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.
The $L^p$ estimate for the gain term of the Boltzmann collision operator and its application
本文证明了Boltzmann碰撞算子增益项在Maxwellian分子、硬势和硬球模型下的尖锐$L^p$估计，并利用不增加矩的新估计证明了小初值条件下Boltzmann方程全局温和解的存在性与散射性。
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.
Non Maxwellian long-time asymptotics for Homoenergetic Boltzmann flows under strong shear
利用匹配渐近展开，研究了在双曲主导区域、均匀碰撞核下，含剪切项的非线性Boltzmann方程同能解的长期渐近行为，发现解的质量集中在递增的二进尺度族上，导致与经典麦克斯韦分布截然不同的行为。
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 $n$-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 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.
On the Boltzmann equation with soft potentials: Existence, uniqueness and smoothing effect of mild solutions
本文研究了无角截断软势空间非齐次Boltzmann方程，在初始质量、能量和熵密度有界且质量远离真空的条件下，证明了温和解的局部时间存在性与唯一性，并首次给出了软势在解析或Gevrey空间中的光滑化效应结果。
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 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.
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.
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 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…
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