Half-space problems in the kinetic theory of gases are of great importance in the study of the asymptotic behavior of solutions of boundary value problems for the Boltzmann equation for small Knudsen numbers. In this work a generally formulated half-space problem, based on generalizations of stationary half-space problems in one spatial variable for the Boltzmann equation - for hard-sphere models of monatomic single species and multicomponent mixtures - is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. Exponential fast convergence is obtained "far away" from the interface. In particular, the exponential decay at regime transitions - where the number of conditions on the indata needed to obtain well-posedness changes - for linearized kinetic half-space problems related to the half-space problem of evaporation and condensation in kinetic theory are considered. The regime transitions correspond to the transition between subsonic and supersonic evaporation/condensation, or the transition between evaporation and condensation. Near the regime transitions, slowly varying modes might occur, preventing uniform exponential speed of convergence there. By imposing extra conditions on the indata at the interface, the slowly varying modes can be eliminated near a regime transition, giving rise to uniform exponential speed of convergence near the regime transition.
Values of the velocity of the flow at the far end, for which regime transitions take place are presented for some particular variants of the Boltzmann equation: for monatomic and polyatomic single species and mixtures, and the quantum variant for bosons and fermions.
Incompressible Navier-Stokes limit of non-bilinear kinetic equations and application to the BGK, nonlinear Fokker-Planck and Boltzmann-Fermi-Dirac equations
Liouville theorems for stable Lane-Emden systems and biharmonic problems
本文研究了由{equation} \label{system_abstract}-Δ u = v ^p，\qquad-Δ v = u ^θ，\qquad\{in} \IR^N，{equation}为$ 1 < p \le θ$和四阶标量方程{equation} \label{fourth_abstract} Δ ^2 u = u ^θ，\qquad\{in $ \IR^N$，}{equation}为$ 1 < θ$的椭圆方程组.证明了正稳定解的各种Liouville型定理。例如，我们证明了（\ref{system_abstract}）（分别是）不存在正稳定解.（\ref{fourth_abstract}））提供了$ N \le 10$和$ 2 \le p \le θ$（分别$ N \le 10$和$1 < θ$）。更高的维度的结果也得到了。这些关于全空间上稳定解的结果暗示了{方程} \label{eq_half_abstract}-Δ u = v ^p，\qquad-Δ v = u ^θ，\qquad\{in} \IR^{N-1}，{方程}在$ \partial \IR^N_+$上有$ u=v=0$的正（可能不稳定）有界解的各种Liouville定理.特别地，对于任何$ 2 \le p \le θ$，如果$ N \le 11$，则不存在（\ref{eq_half_abstract}…
On the Boundary Layer Equations with Phase Transition in the Kinetic Theory of Gases
考虑半空间中单原子硬球气体的具有板对称性的定常玻尔兹曼方程。在半空间的边界处，假定气体与其冷凝相接触。本文讨论了在这种情况下，在体积速度为零的麦克斯韦平衡点附近，温度与凝聚相的温度相同，压力为界面温度下的饱和蒸汽压时，Boltzmann方程的一致衰减边界层型解的存在性和唯一性.这个问题首先由Y. Sone，K.青木和他们的合作者，通过仔细的数值模拟。见[C]第2节。巴尔多斯湾Golse，Y. J. Stat. 124（2006），275-300]的非常详细的介绍这些作品。最近，T. P. Liu和S.- H.羽【大.理.械.肛。209（2013），869-997]已经提出了一种广泛的数学策略来处理Y. Sone，K.青木和他们的团队。本文对[T. P. Liu，S.-]中讨论的一个结果提供了另一种可能更简单的证明。H. Yu，loc. [见前引书]
Constructive Krein-Rutman result for Kinetic Fokker-Planck equations in a domain
Centre de Mathématiques Laurent Schwartz, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院）；Institut Denis Poisson, Université de Tours, Université d’Orléans, CNRS（图尔大学）；Centre de Recherche en Mathématiques de la Décision (CEREMADE, CNRS UMR 7534), Universités PSL & Paris-Dauphine（巴黎第九大学）；Institut Universitaire de France (IUF)（法国高等研究院）
