# The Boltzmann equation, Besov spaces, and optimal decay rates in R^n

# The Boltzmann equation, Besov spaces, and optimal decay rates in R^n

In this talk, we will study the large-time convergence to the global Maxwellian of perturbative classical solutions to the Boltzmann equation on R^n, for n geq 3, without the angular cut-off assumption. We prove convergence of the k-th order derivatives in the norm L^r_x(L^2_v), for any 2 leq r leq infinity, with optimal decay rates, in the sense that they are equal to the rates which one obtains for the corresponding linear equation. The initial data are assumed to lie in a mixed norm space involving the negative homogeneous Besov space of order geq -n/2 in the space variable, without a smallness assumption on the appropriate norm. The space for the initial data is physically relevant since it contains spaces of the type L^p_x (L^2_v), by the Besov-Lipschitz space embeddings. Due to the nature of the vector valued spaces, we need to use a vector analogue of the Calderon-Zygmund theory to prove the necessary nonlinear energy estimates. These results hold both in the hard and soft potential case. Furthermore, in the soft potential case, in the case of special initial data, we prove additional optimal decay results if the initial data belongs to [-(n+2)/2,-n/2). The latter result requires a closer study of the spectrum of the linearized Boltzmann operator for small frequencies dual to the spatial variable. This is a joint work with Robert Strain.