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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.02852 |
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Table of Contents:
- 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.