Optimal time decay of the non cut-off Boltzmann equation in the whole space

Abstract

In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space $\mathbb{R}^n_x$ with $n≥3$ .We use the existence theory of global in time nearby Maxwellian solutions from [12,11].It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption [26,1]. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of $O(t^{-\frac{N}{2}+\frac{N}{2r}})$ in the $L^2_v$$(L^r_x)$-norm for any $2\leq r\leq \infty$.