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

TL;DR

This paper introduces new a priori estimates and continuation criteria for the Landau and Boltzmann equations, showing that fluid singularities are incompatible with these kinetic models, especially for soft potentials.

Contribution

It provides a novel a priori estimate and continuation criterion based on weighted L-infinity norms, expanding understanding of singularity formation in kinetic equations.

Findings

Singularities in fluid equations are largely incompatible with kinetic equations.

The new continuation criterion does not require bounds on hydrodynamic quantities.

Mechanisms for blowup via Euler singularities are largely excluded for these kinetic models.

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.