Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations

Abstract

We establish a new a priori estimate on solutions to the space-inhomogeneous Landau and Boltzmann equations. As a consequence, we prove a new continuation criterion, based on a weighted $L^{\infty }$-norm, without requiring bounds on the hydrodynamic quantities. This complements existing conditional regularity results from a rather different perspective. Consequently, we show that the singularities present in the fluid equations are largely incompatible with the Boltzmann and Landau equations. More precisely, we largely rule out ``lifting a singularity'' from the 3D Euler equations to the physical range of kinetic equations, a widely expected mechanism for singularity formation. Under general considerations, this mechanism is essentially excluded for soft potentials, whereas for hard potentials the situation is more nuanced: one cannot produce blowup through the standard hydrodynamic ansatz using known imploding solutions to the Euler equations.

23 pages, 1 figure