We consider the spatially inhomogeneous Landau equation with soft potentials. First, we establish the short-time existence of solutions, assuming the initial data has sufficient decay in the velocity variable and regularity (no decay assumptions are made in the spatial variable). Next, we show that the evolution instantaneously spreads mass throughout the domain. The resulting lower bounds are sub-Gaussian, which we show is optimal. The proof of mass-spreading is based on a stochastic process, and makes essential use of nonlocality. By combining this theorem with prior results, we derive two important applications: C∞-smoothing, even for initial data with vacuum regions, and a continuation criterion (the solution can be extended as long as the mass and energy densities stay bounded from above). This is the weakest condition known to prevent blow-up. In particular, it does not require a lower bound on the mass density or an upper bound on the entropy density.
ASJC Scopus subject areas
- Applied Mathematics