Knudsen boundary layer equations for full ranges of cutoff collision kernels: Maxwell reflection boundary with all accommodation coefficients in [0,1]

TL;DR

This paper proves existence, uniqueness, and exponential decay of solutions to the Knudsen layer equations with Maxwell boundary conditions across all cutoff kernels, addressing nondissipative boundary challenges.

Contribution

It introduces a novel framework for analyzing the Knudsen layer equations with full range of cutoff kernels and accommodates nondissipative boundary conditions.

Findings

Proved existence and uniqueness of solutions.

Established exponential decay rates.

Developed a new iterative approach for boundary energy control.

Abstract

In this paper, we prove the existence and uniqueness of the Knudsen layer equation imposed on Maxwell reflection boundary condition with full ranges of cutoff collision kernels and accommodation coefficients (i.e., $- 3 < \gamma \leq 1$ and $0 \leq \alpha_* \leq 1$, respectively) in the $L^\infty_{x,v}$ framework. Moreover, the solution enjoys the exponential decay $\exp \{- c x^\frac{2}{3 - \gamma} - c |v|^2 \}$ for some $c > 0$. In order to study the general angular cutoff collision kernel $-3 < \gamma \leq 1$, we should introduce a $(x,v)$-mixed weight $\sigma$. The biggest difficulty in this paper is the nondissipative boundary condition, hence, the boundary temperature and velocity $(T_w, u_w)$ on $\{ x = 0 \}$ and $(T, \mathfrak{u})$ on $\{ x = + \infty \}$ do not guarantee the nonnegativity of the $L^2$ boundary energy. We also do not assume that $(T_w, u_w)$ and $(T, \mathfrak{u})$ are very closed to each other. We first derive the Nondissipative boundary lemma to pull the boundary energy to the interior weighted $L^2$ norms with higher power of $x$-polynomial weights. Then a so-called spatial-velocity indices iteration approach is developed to shift the higher power $x$-polynomial weights to $|v|$-polynomial weights. Finally, we construct an interleaved iteration process such that the boundary energy is successfully dominated.