Solving Linearized Landau Equation Pointwisely

Abstract

We study the pointwise (in the space and time variables) behavior of the linearized Landau equation for hard and moderately soft potentials. The solution has very clear description in the $(x,t)-$variables, including large time behavior and asymptotic behavior. More precisely, we obtain the pointwise fluid structure inside finite Mach number region, and exponential or sub-exponential decay, depending on interactions between particles, in the space variable outside finite Mach number region. The spectrum analysis, regularization effect and refined weighted energy estimate play important roles in this paper.