Search

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 ε >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 ε > 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(ε1/2) in L∞x,v and O(ε3/2) in L2x,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-ε estimates for regular solutions to the full compressible Navier-Stokes equations in bounded domains when the parameter ε > 0 is involved in the analysis. [ABSTRACT FROM AUTHOR]

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 [45] and [3, 43] 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 Lx,v∞ framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption 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 LT∞Lv2 norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) 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. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

We are concerned with the Cauchy problem for the relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space. For perturbative initial data with suitable regularity and integrability, we obtain the global classical solutions and the large time decay rates. For the proof, a new structure of the linearized relativistic Fokker–Planck operator is observed so that the coercivity estimates are established, crucial for the optimal large time decay rates is the pivotal Sobolev inequalities and interpolation inequalities as well as the structure of the coupled system. Moreover, it is shown that the energy without any weights decays in time with the polynomial rates varying directly with the derivatives of the solution, and the lower order derivatives of the solution except the magnetic field are further proved to decay faster than the optimal ones which coincide with the linearized equations without electro and magnetic fields, such a new phenomenon occurs only in the case of the relativistic Fokker–Planck model. [ABSTRACT FROM AUTHOR]

We establish the global existence of small-amplitude solutions near a global Maxwellian to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system for non-cutoff soft potentials with weak angular singularity. This extends the work of Duan et al. (2013), in which the case of strong angular singularity is considered, to the case of weak angular singularity. [ABSTRACT FROM AUTHOR]

The incompressible Navier-Stokes-Fourier (INSF) system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos-Levermore-Ukai-Yang [Kinetic equations: Fluid dynamical limits and viscous heating, Bull. Inst. Math. Acad. Sin. 3 (2008) 1-49]. The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in setting in a periodic box. Based on an odd-even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the INSF system with viscous heating and the super-Burnett functions. More importantly, the remainder of the expansion is proven to decay exponentially in time via an approach on the condition that the initial data satisfies the mass, momentum and energy conversation laws. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the nonlinear stability of contact wave for the Vlasov-Poisson-Boltzmann system with two-species. Based on the quasi-neutral Euler system in Lagrangian coordinates, we first construct a nontrivial profile: a local bi-Maxwellian whose parameters are also called 'viscous contact wave' in the sense of hydrodynamics, then we prove that such a nontrivial profile is time-asymptotically stable by a refined energy method. For the proof, the good structure of the macroscopic equations is fruitfully exploited to attain the dissipation of the specific volumes and the electric potential, moreover, both of Lagrangian and Eulerian coordinates transformation are alternatively utilized in order to capture the higher-order energy of the electric potential. [ABSTRACT FROM AUTHOR]

Abstract: Based on a unified energy method, the uniform-in-time stability of solutions in the space is constructed for the both soft potential with angular cutoff and Landau collision kernels in the Boltzmann theory in the whole space. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem of the Cahn–Hilliard equation First, we construct a local smooth solution to the above Cauchy problem, then by combining some a priori estimates, Sobolev''s embedding theorem and the continuity argument, the local smooth solution is extended step by step to all provided that the smooth nonlinear function satisfies a certain local growth condition at some fixed point and that is suitably small. Secondly, we show that the global smooth solution satisfies the following temporal decay estimates: Here , is a constant depending on τ and is any positive constant which can be chosen sufficiently small. At last, we show that, under a strong assumption on the growth of the nonlinear function at , the asymptotics of solutions of the above Cauchy problem is described by . Here , . [Copyright &y& Elsevier]