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In this paper, the authors study the 1D steady Boltzmann flow in a channel. The walls of the channel are assumed to have vanishing velocity and given temperatures θ0 and θ1. This problem was studied by Esposito-Lebowitz-Marra (1994, 1995) 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, they revisit this problem by taking into account the slip boundary conditions. Following the lines of [Coron, F., Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation, J. Stat. Phys., 54(3–4), 1989, 829–857], the authors 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 they will establish a uniform L∞ estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously. [ABSTRACT FROM AUTHOR]

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]

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]

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 = ± ∞ . 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 → + ∞ . 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. [ABSTRACT FROM AUTHOR]

Abstract: Even though the system of the compressible Navier–Stokes equations is not a limiting system of the Boltzmann equation when the Knudsen number tends to zero, it is the second order approximation by applying the Chapman–Enskog expansion. The purpose of this paper is to justify this approximation rigorously in mathematics. That is, if the difference between the initial data for the compressible Navier–Stokes equations and the Boltzmann equation is of the second order of the Knudsen number, so is the difference between two solutions for all time. The analysis is based on a refined energy method for a fluid-type system using the techniques for the system of viscous conservation laws. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the diffusive expansion for solutions of the rescaled Boltzmann equation in the whole space with prescribed initial data Our main purpose is to justify the global validity of the diffusive expansion for a solution of the rescaled Boltzmann equation (0.1) in the whole space for all with initial data satisfying the initial expansion Here is a normalized global Maxwellian. Under the assumption that the fluid components of the coefficients of the initial expansion have divergence-free velocity fields as well as temperature fields , if we assume further that the velocity-temperature fields of have small amplitude in , we can determine these coefficients in the diffusive expansion (0.3) uniquely by an iteration method and energy method. The hydrodynamic component of these coefficients satisfies the incompressible condition, the Boussinesq relations and/or the Navier–Stokes–Fourier system respectively, while the microscopic component of these coefficients is determined by a recursive formula. Compared with the corresponding problem inside a periodic box studied in Y. Guo (2006) , the main difficulty here is due to the fact that Poincaré''s inequality is not valid in the whole space and this difficulty is overcome by using the -estimate on the Riesz potential. Moreover, by exploiting the energy method, we can also deduce certain the space–time energy estimates on these coefficients . Once the coefficients in the diffusive expansion (0.3) are uniquely determined and some delicate estimates have been obtained, the uniform estimates with respect to ϵ on the remainders are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion (0.3) in the large provided that Notice that for , is no longer a divergence-free vector and it is worth to pointing out that, for , it was in deducing certain estimates on by the -estimate on the Riesz potential that we need to require that . [Copyright &y& Elsevier]