Concentrations in kinetic transport equations and hypoellipticity

TL;DR

This paper develops new hypoelliptic estimates for kinetic transport equations, showing that compactness in velocity variables implies compactness in all variables, which is useful for low-regularity problems like the Boltzmann equation with rough forces.

Contribution

It introduces a novel method to establish hypoelliptic estimates based on phase space decomposition, addressing the lack of averaging lemmas in low regularity settings.

Findings

Established improved hypoelliptic estimates for kinetic equations.

Demonstrated that velocity compactness implies full variable compactness.

Provided a new approach applicable to equations with low regularity right-hand sides.

Abstract

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family $f_\lambda(x,v)\in L^p$ satisfying some appropriate transport relation $$v\cdot\nabla_x f_\lambda = (1-\Delta_x)^\frac{\beta}{2}(1-\Delta_v)^\frac{\alpha}{2}g_\lambda$$ may be inferred solely from its compactness in $v$. This method is introduced as an alternative to the lack of known suitable averaging lemmas in $L^1$ when the right-hand side of the transport equation has very low regularity, due to an external force field for instance. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field.