带入射边界条件的 Knudsen 边界层方程：截止碰撞核与远场 Mach 数的完整范围
Knudsen boundary layer equations with incoming boundary condition: full range of cutoff collision kernels and Mach numbers of the far field
AI总结：
本文在 $L^\infty_{x,v}$ 框架下证明带入射边界条件的非线性 Knudsen 层方程在完整截止碰撞核范围和所有远场 Mach 数下的存在唯一性，并通过混合权重与人工阻尼项获得指数衰减。
摘要：
本文建立了带入射边界条件的非线性 Knudsen 层方程的存在性与唯一性。众所周知，该问题的可解性条件随远场 Maxwell 分布 $\mathcal{M}^\infty$ 的 Mach 数而变化。我们在 $L^\infty_{x,v}$ 框架下考虑截止碰撞核的完整范围（即 $-3<\gamma\leq 1$）以及远场的所有 Mach 数。此外，对某个 $c>0$，解表现出指数衰减 $\exp\{-cx^{\frac{2}{3-\gamma}}-c|v|^2\}$。为处理一般角截止碰撞核，我们引入了一个 $(x,v)$ 混合权重 $\sigma$。证明本质上基于添加人工阻尼项。
英文摘要：
This paper establishes tahe existence and uniqueness of the nonlinear Knudsen layer equation with incoming boundary conditions. It is well-known that the solvability conditions of the problem vary with the Mach number of the far Maxwellian $\mathcal{M}^\infty$. We consider full ranges of cutoff collision kernels (i.e., $- 3 < γ\leq 1$) and all the Mach numbers of the far field in the $L^\infty_{x,v}$ framework. Additionally, the solution exhibits exponential decay $\exp \{- c x^\frac{2}{3 - γ} - c |v|^2 \}$ for some $c > 0$. To address the general angular cutoff collision kernel, we introduce a $(x,v)$-mixed weight $σ$. The proof is essentially bsed on adding an artificial damping term.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
