Seminar

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract:

The Boltzmann equation models the space and velocity distribution of the particles in a diffuse gas. Microscopic interactions lead to a quadratic, nonlocal (in velocity), collision operator that behaves somewhat like a fractional Laplacian. In recent years there has been substantial progress on the regularity and continuation program for the Cauchy problem. Notably, a smooth and unique solution exists for as long as the so-called hydrodynamic quantities remain "under control'': the mass, energy, and entropy densities must stay bounded above uniformly in space and the mass density must stay bounded below uniformly in space. The last condition is crucial for smoothing since it gives the collision operator elliptic properties in certain velocity directions. We show that the solution to the Boltzmann equation (even starting from initial data that contains large regions of vacuum) instantaneously fills space. That is, the gas diffuses and spreads positive mass to every space and velocity coordinate at any positive times. We obtain this result dynamically through barrier arguments for moving mass through space and a De Giorgi type iteration for spreading mass to arbitrary velocities. A consequence is that the above continuation criterion can now be weakened; it is no longer necessary to assume that the mass density is bounded from below for continuation of smooth solutions. Joint work with Christopher Henderson and Stanley Snelson.

Self-Generating Lower Bounds for the Boltzmann Equation