SMOOTHNESS OF WEAK SOLUTIONS OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION

On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations

Weak Solutions of the Stochastic Landau-lifshitz-gilbert Equation

Abstract. The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative spacedependent noise is considered for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere S. The regularity of weak solutions is also discussed. Some of the regularity results are new even for the deterministic Landau-Lifshitz-Gilbert e...

Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz’s space S(RN ). The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the sing...

Existence and smoothness of the density for spatially homogeneous SPDEs

In this paper, we extend Walsh’s stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang’s one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a gen...

Gevrey Regularity for the Solution of the Spatially Homogeneous Landau Equation

This is a non-linear diffusion equation, and the coefficients āi j, c̄ depend on the solution f . Here we are mainly concerned with the Gevrey class regularity for the solution of the Landau equation. This equation is obtained as a limit of the Boltzmann equation when the collisions become grazing (see [8] and references therein). Recently, a lot of progress has been made on the study of the Sob...