Nonlinear stability of boundary layers of the Boltzmann equation for cutoff hard potentials

Abstract

Many physical models have boundaries. For the Boltzmann equation, the study on the boundary layer in the region of the width in the order of the Kundsen number along the boundary is important both in mathematics and physics. In this paper, we consider the nonlinear stability of boundary layer solutions to the Boltzmann equation for hard potentials with angular cut-off. The boundary condition is imposed on incoming particles of the Dirichlet type and the solution tends to a global Maxwellian in the far field. For the existence of the boundary layer solutions, it is proved by Chen et al. [Anal. Appl. 2, 337–363 (2004)] by introducing a weight function which is a function of both position and velocity to overcome the difficulty from the sublinear growth in the collision frequency. Unlike the hard sphere model, even for stability in the case when the Mach number of the far field is less than −1, exponential decay in time cannot be expected for the cutoff hard potentials. Instead, an algebraic decay in time to the boundary layer solution is proved in this paper by using some recursive weighted energy estimates and the bootstrap argument.