OPTIMAL TIME DECAY OF THE NON CUT-OFF BOLTZMANN EQUATION IN THE WHOLE SPACE

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 R-x(n) 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(-n/2+n/2r)) in the L-v(2)(L-x(r))-norm for any 2 <= r <= infinity.