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 Rx with n ≥ 3. We use the existence theory of global in time nearby Maxwellian solutions from [13,14]. 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 [3, 29]. 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 Lv(L r x)-norm for any 2 ≤ r ≤ ∞.