Existence of boundary layers to the Boltzmann equation with cutoff soft potentials

Abstract

When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of the width in the order of the Knudsen number along the boundary. There have been extensive studies on the existence and stability of boundary layers to the Boltzmann equation in different physical settings. Based on the previous work, in this paper, we consider the existence of boundary layer solutions to the Boltzmann equation for the cutoff soft potentials when its parameter γ satisfying −2<γ⩽0. The boundary condition is imposed on the incoming particles of Dirichlet type, and the solution is assumed to approach to a global Maxwellian at the far field. As for the hard sphere model and the cutoff hard potentials, the existence of solutions is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data which gives the codimension of the boundary data related to the number of the positive characteristic speeds. In the following analysis, even though the weight function of both the position and velocity introduced in [Chen, C.-C., Liu, T. P., and Yang, T., “Existence of boundary layer solutions to the Boltzmann equation,” Anal. Appl. 2, 337–363 (2004)]. plays an important role, the result in this paper extends the previous works on the existence and the classification of the boundary data for the hard sphere model and cutoff hard potentials to the cutoff soft potentials with weaker collisional interactions.