Seminar

The Boltzmann equation models a high-energy gas with elastic collisions. From the mathematical point of view, it presents a nonlocal degenerate-parabolic PDE with very few coercive quantities. The existence of global smooth solutions remains an open problem, and the state of the art is summarized by the conditional regularity program: as long as the hydrodynamic quantities (mass, energy, and entropy densities) remain “under control” (satisfying four time-independent inequalities), the solution is in fact smooth.

We eliminate two of the four inequalities from the conditional regularity result by showing that solutions to the Boltzmann equation dynamically (and instantly) fill any vacuum regions, even when the initial data contains such regions; the estimates only depend on an initial (possibly small) core of mass. We then examine how this mass spreading effect enhances known results on the construction, regularity estimates, uniqueness, and continuation for solutions starting from very rough initial data.

Regularity and Continuation for the Boltzmann Equation 406 KB application/pdf

Regularity And Continuation For The Boltzmann Equation