标题： $H^s_x$静止玻尔兹曼方程在入射边界条件下的解的正则性 显示英文标题

标题： $H^s_x$ regularity of solutions to the stationary Boltzmann equation with the incoming boundary condition

摘要： 我们考虑具有形式为$B(|v - \tilde{v}, \theta|) = B_0 |v - \tilde{v}|^\gamma \cos \theta \sin \theta$的截面的稳态玻尔兹曼方程，对于$-3 < \gamma \leq 1$在有界凸区域中，在入射边界条件下。 在本文中，我们将证明在加权$L^\infty$空间中存在一个解，该解具有分数 Sobolev 正则性，而无需假设边界上的高斯曲率的正性。 对于足够光滑且接近标准麦克斯韦分布的边界数据，解具有$H^{1-}_x$正则性对于$-2 \leq \gamma \leq 1$，而对于$-3 < \gamma < -2$只能得到较差的正则性。 我们首先证明在加权$L^2$空间上线性化问题的适定性，并发展了不考虑随机循环的$L^2-L^\infty$估计。 我们接下来研究线性化问题解的$H^s_x$正则性。 速度平均引理在我们的分析中起着关键作用。 我们最终推导出一个双线性估计，以将线性化问题的结果扩展到弱非线性问题。 显示英文摘要

摘要： We consider the stationary Boltzmann equation with the cross section of the form $B(|v - \tilde{v}, \theta|) = B_0 |v - \tilde{v}|^\gamma \cos \theta \sin \theta$ for $-3 < \gamma \leq 1$ in a bounded convex domain under the incoming boundary condition. In this article, we shall show the existence of a solution in a weighted $L^\infty$ space with fractional Sobolev regularity without assuming the positivity of the Gaussian curvature on the boundary. For boundary data sufficiently smooth and close to the standard Maxwellian, the solution has $H^{1-}_x$ regularity for $-2 \leq \gamma \leq 1$, while only worse regularity is obtained for $-3 < \gamma < -2$. We first show the well-posedness of the linearized problem on a weighted $L^2$ space and develop the $L^2-L^\infty$ estimate without the stochastic cycle. We next investigate $H^s_x$ regularity of the solution to the linearized problem. The velocity averaging lemma plays a key role in our analysis. We finally derive a bilinear estimate to extend results on the linearized problem to the weakly nonlinear problem.