Spatial behavior of the solution to the linearized Boltzmann equation with hard potentials

Abstract

The main goal is to understand the quantitative spatial decay of the solution to the linearized Boltzmann equation for hard potentials with cutoff. We obtain the quantitative space-time behavior of the linearized Boltzmann equation under some slow velocity decay assumption but without any regularity assumption on the initial data. This result extends the classical results of Liu and Yu [Commun. Pure Appl. Math. 57, 1543–1608 (2004); Bull. Inst. Math. Acad. Sin. 1, 1–78 (2006); and ibid. 6, 151–243 (2011)] from hard sphere to hard potential under the same assumption on the initial condition [compactly support in the space variable and polynomial decay in the velocity variable with power β = (3/2)+].