# New estimates for the Landau equation with soft potentials

One of the most interesting physical cases in gas and plasma physics is collision of particles under the influence of Coulomb potential. For this potential the Boltzmann equation is not a valid model anymore. The reason is that the momentum exchanged among particles during a collision is divergent. The physical explanation of such mathematical divergence is that grazing collisions cannot be neglected when the potential is of Coulomb type. This problem was known by Landau, who in 1936 derived from the Boltzmann equation a kinetic equation that describes collisions of particles in plasmas where grazing collisions are predominant. Many properties of this equation are known; however the issue of existence and uniqueness of smooth solution is a long standing open problem. In particular it is not yet clear if solutions become unbounded in a finite time. In this talk we present two new results on smoothness and regularity: any weak radial solution to the Landau equation with Coulomb potential that belongs to $L^{3/2}$ it is automatically bounded and smooth. The second result concerns local bounds which do not deteriorate as time grows for any solution to the Landau equation with moderately soft-potentials. This is a joint work with N. Guillen.