On Well-posed Boundary Conditions for the Linear Non-homogeneous Moment Equations in Half-space

TL;DR

This paper establishes a necessary and sufficient condition for the well-posedness of linear non-homogeneous Grad moment equations in half-space, addressing boundary condition stability and providing analytical solutions.

Contribution

It introduces a new criterion for well-posed boundary conditions in Grad moment equations, improving stability analysis and solution methods.

Findings

Grad boundary condition is unstable for non-homogeneous problems

Proposed criteria verify well-posedness of modified boundary conditions

Analytical expressions of solutions are derived for efficient problem solving

Abstract

We propose a necessary and sufficient condition for the well-posedness of the linear non-homogeneous Grad moment equations in half-space. The Grad moment system is based on Hermite expansion and regarded as an efficient reduction model of the Boltzmann equation. At a solid wall, the moment equations are commonly equipped with a Maxwell-type boundary condition named the Grad boundary condition. We point out that the Grad boundary condition is unstable for the non-homogeneous half-space problem. Thanks to the proposed criteria, we verify the well-posedness of a class of modified boundary conditions. The technique to make sure the existence and uniqueness mainly includes a well-designed preliminary simultaneous transformation of the coefficient matrices and Kreiss' procedure about the linear boundary value problem with characteristic boundaries. The stability is established by a weighted estimate. At the same time, we obtain the analytical expressions of the solution, which may help solve the half-space problem efficiently.