Entropy dissipation estimates for the Landau equation in the Coulomb case and applications

We present in this paper an estimate which bounds from below the entropy dissipation D(f)D(f) of the Landau operator with Coulomb interaction by a weighted H1H1 norm of the square root of f  . As a consequence, we get a weighted Lt1(Lv3) estimate for the solutions of the spatially homogeneous Landau equation with Coulomb interaction, and the propagation of L1L1 moments of any order for this equation. We also present an application of our estimate to the Landau equation with (moderately) soft potentials, providing thus a new proof of some recent results of [30].