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

TL;DR

This work establishes the well-posedness and exponential decay of Knudsen boundary layer solutions for the Boltzmann equation under Maxwell boundary conditions and full-range cutoff kernels with accommodation α_*∈[0,1], in an $L^∞$-weighted framework featuring a novel $(x,v)$-mixed weight $\sigma(x,v)$. The authors develop a nondissipative boundary energy lemma, a spatial-velocity indices iteration, and an interleaved iteration to transfer weights and dominate boundary contributions, while introducing a damping mechanism that is removed via a vanishing-sources argument and a freezing-point method. They solve a connection auxiliary problem as a stepping stone to the main Knudsen layer equations, proving uniform estimates and then extending the results from the linear problem to a nonlinear setting with small data, all while ensuring a codimension-4 vanishing-sources structure. The results push forward the theory of Knudsen layers to cover non-dissipative Maxwell boundaries and non-cutoff angular regimes, with implications for hydrodynamic limits and boundary-layer analysis in kinetic theory.

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., $- 3 < γ\leq 1$ and $0 \leq α_* \leq 1$, respectively) in the $L^\infty_{x,v}$ framework. Moreover, the solution enjoys the exponential decay $\exp \{- c x^\frac{2}{3 - γ} - c |v|^2 \}$ for some $c > 0$. In order to study the general angular cutoff collision kernel $-3 < γ\leq 1$, we should introduce a $(x,v)$-mixed weight $σ$. 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.

Figures (3)

• Figure 1: Interleaved iteration process.
• Figure 2: Derivation of uniform bounds for the connection auxiliary equation \ref{['A1']}. Here we denote by $g_\sigma = e^{\hbar \sigma} g$.
• Figure :

Theorems & Definitions (52)

• Theorem 1.1
• Remark 1.1: Exponential decay
• Remark 1.2: Background temperature $T$ vs boundary temperature $T_w$
• Remark 1.3: Re-characterization of the vanishing sources set $\mathbb{VSS}_{\alpha_*}$
• Theorem 1.2
• Remark 1.4: Re-characterization of the vanishing sources set $\widetilde{\mathbb{VSS}}_{\alpha_*}$
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4