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This paper aims to incorporate the Caflisch's decomposition into the macro-micro decomposition in Boltzmann theory for allowing the microscopic component to exhibit only the polynomial tail in large velocities. In particular, we treat the Cauchy problem on the non-cutoff Boltzmann equation under the compressible Euler scaling in case of three-dimensional whole space. Up to a finite time we construct the Boltzmann solution around a local Maxwellian corresponding to small-amplitude classical solutions of the full compressible Euler system around constant states. We design a new energy functional which can capture the convergence rate in the small Knudsen number $\varepsilon$ and allow the microscopic part of solutions to decay polynomially in large velocities. Moreover, the energy norm of perturbations can be of the order $\varepsilon^{1/2}$ which the usual method of Hilbert expansion fails to obtain. As a byproduct of the proof, our estimates immediately yield a global-in-time existence result when the Euler solutions are taken to be constant states., Comment: 58 pages. All comments are welcome

In this note we adopt an approach by Grenier, Nguyen and Rodnianski in \cite{GNR} for studying the nonlinear Landau damping of the two-species Vlasov-Poisson system in the phase space $\mathbb{T}^d_x \times \mathbb{R}^d_v$ with the dimension $d\geq 1$. The main goal is twofold: one is to extend the one-species case to the two-species case where the electron mass is finite and the ion mass is sufficiently large, and the other is to modify the $G$-functional such that it involves the norm in $L^{d+1}$ instead of $L^2$ as well as derivatives up to only the first order., Comment: 29 pages

In this paper, we consider the perturbed solutions with polynomial tail in large velocities for the non-cutoff Boltzmann equation near global Maxwellians in the whole space. The global in time existence is proved in the weighted Sobolev spaces and the almost optimal time decay is obtained in Fourier transform based low-regularity spaces. The result shows a time-velocity decay structure of solutions that can be decomposed into two parts. One part allows the slow polynomial tail in large velocities, carries the initial data and enjoys the exponential or arbitrarily large polynomial time decay. The other part, with zero initial data, is dominated by the non-negative definite symmetric dissipation and has the exponential velocity decay but only the slow polynomial time decay.

We study the Boltzmann equation in a smooth bounded domain featuring a mixed boundary condition. Specifically, gas particles experience specular reflection in two parallel plates, while diffusive reflection occurs in the remaining portion between these two specular regions. The boundary is assumed to be motionless and isothermal. Our main focus is on constructing global-in-time small-amplitude solutions around global Maxwellians for the corresponding initial-boundary value problem. The proof relies on the $L^2$ hypocoercivity at the linear level, utilizing the weak formulation and various functional inequalities on the test functions, such as Poincar\'e and Korn inequalities. It also extends to the linear problem involving Maxwell boundary conditions, where the accommodation coefficient can be a piecewise constant function on the boundary, allowing for more general bounded domains. Moreover, we develop a delicate application of the $L^2-L^\infty$ bootstrap argument, which relies on the specific geometry of our domains, to effectively handle this mixed-type boundary condition., Comment: 32 pages. All comments are welcome

Two fundamental models in plasma physics are given by the Vlasov-Poisson-Landau system and the compressible Euler-Poisson system which both capture the complex dynamics of plasmas under the self-consistent electric field interactions at the kinetic and fluid levels, respectively. Although there have been extensive studies on the long wave limit of the Euler-Poisson system towards Korteweg-de Vries equations, few results on this topic are known for the Vlasov-Poisson-Landau system due to the complexity of the system and its underlying multiscale feature. In this article, we derive and justify the Korteweg-de Vries equations from the Vlasov-Poisson-Landau system modelling the motion of ions under the Maxwell-Boltzmann relation. Specifically, under the Gardner-Morikawa transformation $$ (t,x,v)\to (\delta^{\frac{3}{2}}t,\delta^{\frac{1}{2}}(x-\sqrt{\frac{8}{3}}t),v) $$ with $ \varepsilon^{\frac{2}{3}}\leq \delta\leq \varepsilon^{\frac{2}{5}}$ and $\varepsilon>0$ being the Knudsen number, we construct smooth solutions of the rescaled Vlasov-Poisson-Landau system over an arbitrary finite time interval that can converge uniformly to smooth solutions of the Korteweg-de Vries equations as $\delta\to 0$. Moreover, the explicit rate of convergence in $\delta$ is also obtained. The proof is obtained by an appropriately chosen scaling and the intricate weighted energy method through the micro-macro decomposition around local Maxwellians., Comment: 68 pages. All comments are welcome. arXiv admin note: text overlap with arXiv:2212.07654

In the paper, we consider the Cauchy problem on the spatially one-dimensional Vlasov-Poisson-Landau system modelling the motion of ions under a generalized Boltzmann relation. Let the Knudsen number and the Debye length be given as $\varepsilon>0$ and $\varepsilon^{b}$ with $\frac{3}{5}\leq b\leq 1$, respectively. As $\varepsilon\to 0$ the formal Hilbert expansion gives the fluid limit to the quasineutral compressible Euler system. We start from the small-amplitude rarefaction wave of the Euler system that admits a smooth approximation with a parameter $\delta\sim\varepsilon^{\frac{3}{5}-\frac{2}{5}a}$, where the wave strength is independent of $\varepsilon$ and we take $\frac{2}{3}\leq a\leq 1$ if $\frac{2}{3}\leq b\leq 1$ and $4-5b\leq a\leq 1$ if $\frac{3}{5}\leq b< \frac{2}{3}$. Under the scaling $(t,x)\to (\varepsilon^{-a}t,\varepsilon^{-a}x)$, for well-prepared initial data we construct the unique global classical solution to the Vlasov-Poisson-Landau system around the rarefaction wave in the vanishing limit $\varepsilon\to 0$ and also obtain the global-in-time convergence of solutions toward the rarefaction wave with rate $\varepsilon^{\frac{3}{5}-\frac{2}{5}a}|\ln\varepsilon|$ in the $L^{\infty}_xL^2_v$ norm. The best rate is $\varepsilon^{\frac{1}{3}}|\ln\varepsilon|$ with the choice of $a=\frac{2}{3}$ and $\frac{2}{3}\leq b\leq 1$. Note that the nontrivial electric potential in the solution connects two fixed distinct states at the far fields $x=\pm\infty$ for all $t\geq 0$ and tends asymptotically as $\varepsilon\to 0$ toward a profile determined by the macro density function under the quasineutral assumption. Our strategy is based on an intricate weighted energy method capturing the quartic dissipation to give uniform bounds of the nonlinear dynamics around rarefaction waves., Comment: 95 pages. All comments are welcome

We are concerned with the Cauchy problem on the Boltzmann equation in the whole space. The goal is to construct global-in-time bounded mild solutions near Maxwellians with the perturbation admitting a polynomial tail in large velocities. The proof is based on the Caflisch's decomposition together with the $L^2- L^\infty$ interplay technique developed by Guo. The full range of both hard and soft potentials under the Grad's cutoff assumption can be covered. The main difficulty to be overcome in case of the whole space is the polynomial time decay of solutions which is much slower than the exponential rate in contrast with the torus case., Comment: 33 pages. Wording improved and statements made more precise

The motion of rarefied gases for uniform shear flow at the kinetic level is governed by the spatially homogeneous Boltzmann equation with a deformation force. In the paper we study the corresponding Cauchy problem with initial data of finite mass and energy for the collision kernel in case of hard potentials $0<\gamma\leq 1$ under the cutoff assumption. We prove the global existence and large time behavior of solutions provided that the force strength $\alpha>0$ is small enough. In particular, when the initial perturbation is of order $\alpha^m$ for $m>2$, we make a rigorous justification of the uniform-in-time asymptotic expansion of solutions up to order $\alpha^2$ under a homoenergetic self-similar scaling that can capture the increase of temperature $\theta(t)\sim (1+\gamma \varrho_0\alpha^2 t)^{2/\gamma}$ when time tends to infinity, where $\varrho_0>0$ is a strictly positive constant depending only on the deformation force and the linearized collision operator. Specifically, we establish $$ \theta^{3/2}(t)F(t,\theta^{1/2}(t)v)= \mu+\alpha \sqrt{\mu} G_1(t,v)+\alpha^2 \sqrt{\mu}G_2(t,v)+O(1)\alpha^m(1+\gamma \varrho_0\alpha^2 t)^{-2} $$ as $t\to\infty$, where $\mu$ is a global Maxwellian and $G_1,G_2$ are microscopic bounded functions that can be explicitly determined and decay in time as $G_1\sim (1+\gamma \varrho_0\alpha^2 t)^{-1}$ and $G_2\sim (1+\gamma \varrho_0\alpha^2 t)^{-2}$., Comment: 40 pages. All comments are welcome

In this paper we consider the Boltzmann equation modelling the motion of a polyatomic gas where the integration collision operator in comparison with the classical one involves an additional internal energy variable $I\in\mathbb{R}_+$ and a parameter $\delta\geq 2$ standing for the degree of freedom. In perturbation framework, we establish the global well-posedness for bounded mild solutions near global equilibria on torus. The proof is based on the $L^2\cap L^\infty$ approach. Precisely, we first study the $L^2$ decay property for the linearized equation, then use the iteration technique for the linear integral operator to get the linear weighted $L^\infty$ decay, and in the end obtain $L^\infty$ bounds as well as exponential time decay of solutions for the nonlinear problem with the help of the Duhamel's principle. Throughout the proof, we present a careful analysis for treating the extra effect of internal energy variable $I$ and the parameter $\delta$., Comment: 31 pages. All comments are welcome

In the paper, we develop an $L^1_k\cap L^p_k$ approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in $\mathbb{R}^3$. In particular, only smallness of $\|\mathcal{F}_x{f}_0\|_{L^1\cap L^p (\mathbb{R}^3_k;L^2(\mathbb{R}^3_v))}$ with $3/2

Two fundamental models in plasma physics are given by the Vlasov-Maxwell-Boltzmann system and the compressible Euler-Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions at the kinetic and fluid levels, respectively. It has remained a long-standing open problem to rigorously justify the hydrodynamic limit from the former to the latter as the Knudsen number $\varepsilon$ tends to zero. In this paper we give an affirmative answer to the problem for smooth solutions to both systems near constant equilibrium in the whole space in case when only the dynamics of electrons is taken into account. The explicit rate of convergence in $\varepsilon$ over an almost global time interval is also obtained for well-prepared data. For the proof, one of main difficulties occurs to the cubic growth of large velocities due to the action of the classical transport operator on local Maxwellians and we develop new $\varepsilon$-dependent energy estimates basing on the macro-micro decomposition to characterize the asymptotic limit in the compressible setting., Comment: 68 pages. Any comments are welcome. Modified by the referee report. To appear in "Mathematical Models and Methods in Applied Sciences". arXiv admin note: text overlap with arXiv:2207.01184

Although the compressible fluid limit of the Boltzmann equation with cutoff has been well investigated in [6] and [13], it still remains largely open to obtain analogous results in case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, essentially due to the velocity diffusion effect of collision operator such that $L^\infty$ estimates are hard to obtain without using Sobolev embeddings. In the paper, we are concerned with the compressible Euler and acoustic limits of the Landau equation for Coulomb potentials in the whole space. Specifically, over any finite time interval where the full compressible Euler system admits a smooth solution around constant states, we construct a unique solution in a high-order weighted Sobolev space for the Landau equation with suitable initial data and also show the uniform estimates independent of the small Knudsen number $\varepsilon>0$, yielding the $O(\varepsilon)$ convergence of the Landau solution to the local Maxwellian whose fluid quantities are the given Euler solution. Moreover, the acoustic limit for smooth solutions to the Landau equation in an optimal scaling is also established. For the proof, by using the macro-micro decomposition around local Maxwellians together with techniques for viscous compressible fluid and properties of Burnett functions, we design an $\varepsilon$-dependent energy functional to capture the dissipation in the compressible fluid limit with feature that only the highest order derivatives are most singular., Comment: 72 pages. Submitted to a journal on March 29, 2022. Any comments are welcome

In this paper, we study the 1D steady Boltzmann flow in a channel. The walls of the channel are assumed to have vanishing velocity and given temperatures $\theta_0$ and $\theta_1$. This problem was studied by Esposito et al [13,14] where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition. However, a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary. In the regime where the Knudsen number is reasonably small, the slip phenomenon is significant near the boundary. Thus, we revisit this problem by taking into account the slip boundary conditions. Following the lines of [9], we will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points. Then we will establish a uniform $L^\infty$ estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.

This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potentials with angular cutoff. We construct the smooth steady solutions via a perturbation approach when the deformation strength is sufficiently small. The steady solution is a spatially homogeneous non Maxwellian state and may have the polynomial tail at large velocities. Moreover, we also establish the long time asymptotics toward steady states for the Cauchy problem on the corresponding spatially inhomogeneous equation in torus, which in turn gives the non-negativity of steady solutions., Comment: 39 pages. Typos are corrected and the estimates for c in the proof of Lemma 3.1 are simplified with some corrections. All comments are welcome

In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov-Poisson-Landau/Boltzmann system around global Maxwellians in torus or finite channel. The main goal is to establish the global existence and large time behavior of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan-Liu-Sakamoto-Strain [{\it Comm. Pure Appl. Math.} 74 (2021), no.~5, 932--1020] for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems.

In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at $y=\pm L$ moving relative to each other with opposite velocities $(\pm \alpha L,0,0)$ along the $x$-direction. Assuming that the stationary state takes the specific form of $F(y,v_x-\alpha y,v_y,v_z)$ with the $x$-component of the molecular velocity sheared linearly along the $y$-direction, such steady flow is governed by a boundary value problem on a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous non-moving diffuse reflection boundary condition. In case of the Maxwell molecule collisions, we establish the existence of spatially inhomogeneous non-equilibrium stationary solutions to the steady problem for any small enough shear rate $\alpha>0$ via an elaborate perturbation approach using Caflisch's decomposition together with Guo's $L^\infty\cap L^2$ theory. The result indicates the polynomial tail at large velocities for the stationary distribution. Moreover, the large time asymptotic stability of the stationary solution with an exponential convergence is also obtained and as a consequence the nonnegativity of the steady profile is justified., Comment: 55 pages, 1 figure. All comments are welcome

It has been unknown in kinetic theory whether the linearized Boltzmann or Landau equation with soft potentials admits a spectral gap in the spatially inhomogeneous setting. Most of existing works indicate a negative answer because the spectrum of two linearized self-adjoint collision operators is accumulated to the origin in case of soft interactions. In the paper we rather prove it in an affirmative way when the space domain is bounded with an inflow boundary condition. The key strategy is to introduce a new Hilbert space with an exponential weight function that involves the inner product of space and velocity variables and also has the strictly positive upper and lower bounds. The action of the transport operator on such space-velocity dependent weight function induces an extra non-degenerate relaxation dissipation in large velocity that can be employed to compensate the degenerate spectral gap and hence give the exponential decay for solutions in contrast with the sub-exponential decay in either the spatially homogeneous case or the case of torus domain. The result reveals a new insight of hypocoercivity for kinetic equations with soft potentials in the specified situation., Comment: 43 pages. The paper has been rewritten to treat the problem in bounded domain with the inflow boundary condition. All comments are welcome

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 collision operator involving rough coefficients, and this enables us to combine the classical H\"ormander's hypoelliptic techniques together with the global symbolic calculus established for the linearized Boltzmann operator so as to improve the regularity of solutions at positive time., Comment: 57 pages. All comments are welcome

In the paper, we are concerned with the large time asymptotics toward the viscous contact waves for solutions of the Landau equation with physically realistic Coulomb interactions. Precisely, for the corresponding Cauchy problem in the spatially one-dimensional setting, we construct the unique global-in-time solution near a local Maxwellian whose fluid quantities are the viscous contact waves in the sense of hydrodynamics and also prove that the solution tends toward such local Maxwellian in large time. The result is proved by elaborate energy estimates and seems the first one on the dynamical stability of contact waves for the Landau equation. One key point of the proof is to introduce a new time-velocity weight function that includes an exponential factor of the form $\exp (q(t)\langle \xi\rangle^2)$ with $$ q(t):=q_1-q_2\int_0^tq_3(s)\,ds, $$ where $q_1$ and $q_2$ are given positive constants and $q_3(\cdot)$ is defined by the energy dissipation rate of the solution itself. The time derivative of such weight function is able to induce an extra quartic dissipation term for treating the large-velocity growth in the nonlinear estimates due to degeneration of the linearized Landau operator in the Coulomb case. Note that in our problem the explicit time-decay of solutions around contact waves is unavailable but no longer needed under the crucial use of the above weight function, which is different from the situation in [11, 14]., Comment: 60 pages. Comments are most welcome. Slight modifications have been made to simplify some estimates using the conservation of mass

In this paper, we are concerned with the hydrodynamic limit to rarefaction waves of the compressible Euler system for the Landau equation with Coulomb potentials as the Knudsen number $\epsilon>0$ is vanishing. Precisely, whenever $\epsilon>0$ is small, for the Cauchy problem on the Landau equation with suitable initial data involving a scaling parameter $a\in [\frac{2}{3},1]$, we construct the unique global-in-time uniform-in-$\epsilon$ solution around a local Maxwellian whose fluid quantities are the rarefaction wave of the corresponding Euler system. In the meantime, we establish the convergence of solutions to the Riemann rarefaction wave uniformly away from $t=0$ at a rate $\epsilon^{\frac{3}{5}-\frac{2}{5}a}|\ln \epsilon|$ as $\epsilon\to 0$. The proof is based on the refined energy approach combining [19] and [32] under the scaling transformation $(t,x)\to (\epsilon^{-a}t,\epsilon^{-a}x)$., Comment: 48 pages. Submitted. Comments are most welcome

This paper is concerned with the initial-boundary value problem on the full Euler-Poisson system for ions over a half line. We establish the existence of stationary solutions under the Bohm criterion similar to the isentropic case and further obtain the large time asymptotic stability of small-amplitude stationary solutions provided that the initial perturbation is sufficiently small in some weighted Sobolev spaces. Moreover, the convergence rate of the solution toward the stationary solution is obtained. The proof is based on the energy method. A key point is to capture the positivity of the temporal energy dissipation functional and boundary terms with suitable space weight functions either algebraic or exponential depending on whether or not the incoming far-field velocity is critical.

For the Boltzmann equation with cutoff hard potentials, we construct the unique global solution converging with an exponential rate in large time to global Maxwellians not only for the specular reflection boundary condition with the bounded convex C^3 domain but also for a class of large amplitude initial data where the L^infty norm with a suitable velocity weight can be arbitrarily large but the relative entropy need to be small. A key point in the proof is to introduce a delicate nonlinear iterative process of estimating the gain term basing on the triple Duhamel iteration along the linearized dynamics., Comment: 36 pages, 1 figure

The uniform shear flow for the rarefied gas is governed by the time-dependent spatially homogeneous Boltzmann equation with a linear shear force. The main feature of such flow is that the temperature may increase in time due to the shearing motion that induces viscous heat and the system becomes far from equilibrium. For Maxwell molecules, we establish the unique existence, regularity, shear-rate-dependent structure and non-negativity of self-similar profiles for any small shear rate. The non-negativity is justified through the large time asymptotic stability even in spatially inhomogeneous perturbation framework, and the exponential rates of convergence are also obtained with the size proportional to the second order shear rate. The analysis supports the numerical result that the self-similar profile admits an algebraic high-velocity tail that is the key difficulty to overcome in the proof., Comment: 51 pages

This paper proves the existence of small-amplitude global-in-time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well-known works (Guo, 2002) and (Gressman-Strain-2011, AMUXY-2012) on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an $L^\infty_{x,v}$ framework, similar to that in (Guo-2010), for the Boltzmann equation with cutoff in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity-diffusion-type collision operator in the non-cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus, or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the $L^\infty_T L^2_v$ norm, in velocity and time, of the distribution function is in the Wiener algebra $A(\Omega)$ in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large-time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables., Comment: 65 pages

Let the motion of a rarefied gas between two parallel infinite plates of the same temperature be governed by the Boltzmann equation with diffuse reflection boundaries, where the left plate is at rest and the right one oscillates in its normal direction periodically in time. For such boundary-value problem, we establish the existence of a time-periodic solution with the same period, provided that the amplitude of the right boundary is suitably small. The positivity of the solution is also proved basing on the study of its large-time asymptotic stability for the corresponding initial-boundary value problem. For the proof of existence, we develop uniform estimates on the approximate solutions in the time-periodic setting and make a bootstrap argument by reducing the coefficient of the extra penalty term from a large enough constant to zero.

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 the proof, we develop new estimates in the time-periodic setting., Comment: 36 pages. This paper is dedicated to Professor Philippe G. Ciarlet on the occasion of his 80th birthday

In the paper, assuming that the motion of rarefied gases in a bounded domain is governed by the angular cutoff Boltzmann equation with diffuse reflection boundary, we study the effects of both soft intermolecular interaction and non-isothermal wall temperature upon the long-time dynamics of solutions to the corresponding initial boundary value problem. Specifically, we are devoted to proving the existence and dynamical stability of stationary solutions whenever the boundary temperature has suitably small variations around a positive constant. For the proof of existence, we introduce a new mild formulation of solutions to the steady boundary-value problem along the speeded backward bicharacteristic, and develop the uniform estimates on approximate solutions in both $L^2$ and $L^\infty$. Such mild formulation proves to be useful for treating the steady problem with soft potentials even over unbounded domains. In showing the dynamical stability, a new point is that we can obtain the sub-exponential time-decay rate in $L^\infty$ without losing any velocity weight, which is actually quite different from the classical results as in [11,43] for the torus domain and essentially due to the diffuse reflection boundary and the boundedness of the domain., Comment: 65 pages

It is well known that the full compressible Navier-Stokes equations can be deduced via the Chapman-Enskog expansion from the Boltzmann equation as the first-order correction to the Euler equations with viscosity and heat-conductivity coefficients of order of the Knudsen number $\epsilon>0$. In the paper, we carry out the rigorous mathematical analysis of the compressible Navier-Stokes approximation for the Boltzmann equation regarding the initial-boundary value problems in general bounded domains. The main goal is to measure the uniform-in-time deviation of the Boltzmann solution with diffusive reflection boundary condition from a local Maxwellian with its fluid quantities given by the solutions to the corresponding compressible Navier-Stokes equations with consistent non-slip boundary conditions whenever $\epsilon>0$ is small enough. Specifically, it is shown that for well chosen initial data around constant equilibrium states, the deviation weighted by a velocity function is $O(\epsilon^{1/2})$ in $L^\infty_{x,v}$ and $O(\epsilon^{3/2})$ in $L^2_{x,v}$ globally in time. The proof is based on the uniform estimates for the remainder in different functional spaces without any spatial regularity. One key step is to obtain the global-in-time existence as well as uniform-in-$\epsilon$ estimates for regular solutions to the full compressible Navier-Stokes equations in bounded domains when the parameter $\epsilon>0$ is involved in the analysis.

The paper is concerned with the propagation of ion-acoustic shock waves in a collision dominated plasma. We firstly establish the existence and uniqueness of a small-amplitude smooth travelling wave, then justify its approximation to the shock profile of the KdV-Burgers equations in a suitable asymptotic regime where dissipation in terms of viscosity coefficient is much stronger than dispersion by the Debye length, and prove in the end the large time asymptotic stability of travelling waves under suitably small smooth perturbations.

In this paper we study the Boltzmann equation near global Maxwellians in the $d$-dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin-Lerner type space with respect to the phase variable $(x,v)$. Both hard and soft potentials with angular cutoff are considered. The new function space for global well-posedness is introduced to essentially treat the case of soft potentials, and the key point is that the velocity variable is taken in the weighted supremum norm, and the space variable is in the $s$-order Besov space with $s\geq d/2$ including the spatially critical regularity. The proof is based on the time-decay properties of solutions to the linearized equation together with the bootstrap argument. Particularly, the linear analysis in case of hard potentials is due to the semigroup theory, where the extra time-decay plays a role in coping with initial data in $L^2$ with respect to space variable. In case of soft potentials, for the time-decay of linear equations we borrow the results basing on the pure energy method and further extend them to those in $L^\infty$ framework through the technique of $L^2$--$L^\infty$ interplay. In contrast to hard potentials, $L^1$ integrability in $x$ of initial data is necessary for soft potentials in order to obtain global solutions to the nonlinear Cauchy problem., Comment: 25pages

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 large-amplitude solutions to equilibrium in large time. The isothermal diffuse reflection boundary condition plays a vital role in the analysis. The most key ingredients in our strategy of the proof include: (i) $L^2_{x,v}$--$L^\infty_xL^1_v$--$L^\infty_{x,v}$ estimates along a bootstrap argument; (ii) Pointwise estimates on the upper bound of the gain term by the product of $L^\infty$ norm and $L^2$ norm; (iii) An iterative procedure on the nonlinear term., Comment: 48 pages, 1 figure

The ellipsoidal BGK model was introduced in \cite{Ho} to fit the correct Prandtl number in the Navier-Stokes approximation of the classical BGK model. In the paper we establish the global existence of mild solutions to the Cauchy problem on the model for a class of initial data allowed to have large oscillations. The proof is motivated by a recent study of the same topic on the Boltzmann equation in \cite{DHWY}., Comment: 15 pages

The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new $L^\infty_xL^1_{v}\cap L^\infty_{x,v}$ approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted $L^\infty$ norm under some smallness condition on $L^1_xL^\infty_v$ norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in $L^\infty_{x,v}$ norm with explicit rates of convergence is also studied., Comment: 34 pages. Typos corrected

The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in [42]. It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-trivial long-term dynamics of the binary system. In the paper, we consider the problem in the setting of rarefaction waves. The analytical tool is based on the macro-micro decomposition introduced in [59] that we can be able to develop into the case for the two-component Boltzmann equations around local bi-Maxwellians. Our focus is to explore how the disparate masses and charges of particles play a role in the analysis of the approach of the complex coupling system time-asymptotically toward a non-constant equilibrium state whose macroscopic quantities satisfy the quasineutral nonisentropic Euler system., Comment: 64 pages

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 initial perturbation has sufficient regularity and velocity-integrability., Comment: 41 pages

This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima-Shizuta stability condition formulated in \cite{UKS84,SK85}, and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (cf.~the Timoshenko system and the Euler-Maxwell system). Moreover, it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity-loss type (cf.~\cite{D,IHK08,IK08,USK,UK}). Therefore our purpose of this paper is to formulate a new structural condition which include the Kawashima-Shizuta condition, and to analyze the weak dissipative structure for general systems with non-symmetric relaxation., Comment: 25 pages

The paper aims at investigating two types of decay structure for linear symmetric hyperbolic systems with non-symmetric relaxation. Precisely, the system is of the type $(p,q)$ if the real part of all eigenvalues admits an upper bound $-c|\xi|^{2p}/(1+|\xi|^2)^{q}$, where $c$ is a generic positive constant and $\xi$ is the frequency variable, and the system enjoys the regularity-loss property if $p

This paper is devoted to the study of the nonlinear stability of the rarefaction waves of the Vlasov-Poisson-Boltzmann system with slab symmetry in the case where the electron background density satisfies an analogue of the Boltzmann relation. We allows that the electric potential may take distinct constant states at both far-fields. The rarefaction wave whose strength is not necessarily small is constructed through the quasineutral Euler equations coming from the zero-order fluid dynamic approximation of the kinetic system. We prove that the local Maxwellian with macroscopic quantities determined by the quasineutral rarefaction wave is time-asymptotically stable under small perturbations for the corresponding Cauchy problem on the Vlasov-Poisson-Boltzmann system. The main analytical tool is the combination of techniques we developed in [10] for the viscous compressible fluid with the self-consistent electric field and the reciprocal energy method based on the macro-micro decomposition of the Boltzmann equation around a local Maxwellian. Both the time decay property of the rarefaction waves and the structure of the Poisson equation play a key role in the analysis., Comment: 53 pages

In the paper we are concerned with the large time behavior of solutions to the one-dimensional Navier-Stokes-Poisson system in the case when the potential function of the self-consistent electric field may take distinct constant states at $x=\pm\infty$. Precisely, it is shown that if initial data are close to a constant state with asymptotic values at far fields chosen such that the Riemann problem on the corresponding quasineutral Euler system admits a rarefaction wave whose strength is not necessarily small, then the solution exists for all time and tends to the rarefaction wave as $t\to+\infty$. The construction of the nontrivial large-time profile of the potential basing on the quasineutral assumption plays a key role in the stability analysis. The proof is based on the energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid., Comment: 26 pages. Accepted for publication in J. Differential Equations

This paper is concerned with the large-time behavior of solutions to the Cauchy problem on the two-fluid Euler-Maxwell system with collisions when initial data are around a constant equilibrium state. The main goal is the rigorous justification of diffusion phenomena in fluid plasma at the linear level. Precisely, motivated by the classical Darcy's law for the nonconductive fluid, we first give a heuristic derivation of the asymptotic equations of the Euler-Maxwell system in large time. It turns out that both the density and the magnetic field tend time-asymptotically to the diffusion equations with diffusive coefficients explicitly determined by given physical parameters. Then, in terms of the Fourier energy method, we analyze the linear dissipative structure of the system, which implies the almost exponential time-decay property of solutions over the high-frequency domain. The key part of the paper is the spectral analysis of the linearized system, exactly capturing the diffusive feature of solutions over the low-frequency domain. Finally, under some conditions on initial data, we show the convergence of the densities and the magnetic field to the corresponding linear diffusion waves with the rate $(1+t)^{-5/4}$ in $L^2$ norm and also the convergence of the velocities and the electric field to the corresponding asymptotic profiles given in the sense of the geneneralized Darcy's law with the faster rate $(1+t)^{-7/4}$ in $L^2$ norm. Thus, this work can be also regarded as the mathematical proof of the Darcy's law in the context of collisional fluid plasma., Comment: 46 pages. The paper was modified according to referees' reports. Main modifications: Title, definition of the asymptotic profile (1.5)-(1.8), the statement of Theorem 1.1 where the assumption that |x| B_0 is L^1_x etc was removed, and some other corrected typos or minor mistakes pointed out in the report. The comments by anonymous reviewers are quite appreciated!

The unique global strong solution in the Chemin-Lerner type space to the Cauchy problem on the Boltzmann equation for hard potentials is constructed in perturbation framework. Such solution space is of critical regularity with respect to spatial variable, and it can capture the intrinsic property of the Botlzmann equation. For the proof of global well-posedness, we develop some new estimates on the nonlinear collision term through the Littlewood-Paley theory., Comment: 32 pages