We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth and remain smooth as long as the mass, energy, and entropy densities remain under control. For very soft potentials, we obtain the same conclusion with the additional assumption that a sufficiently high moment of the solution in the velocity variable remains bounded. Our proof relies on the iteration of local Schauder-type estimates.
Abstract We consider a parabolic equation in nondivergence form, defined in the full space [ 0 , ∞ ) × R d , with a power nonlinearity as the right-hand side. We obtain an upper bound for the solution in terms of a weighted control in L p . This upper bound is applied to the homogeneous Landau equation with moderately soft potentials. We obtain an estimate in L ∞ ( R d ) for the solution of the Landau equation, for positive time, which depends only on the mass, energy and entropy of the initial data. We establish a simplified form for the Landau equation with Maxwellian-type molecules. We study in details the Cauchy problem associated to this equation, and some qualitative features of the solution. Explicit solutions are given. We study the Cauchy problem for the homogeneous Landau equation of kinetic theory, in the case of hard potentials. We prove that for a large class of initial data, there exists a unique weak solution to this problem, which becomes immediately smooth and rapidly decaying at infinity.