Local well-posedness for the Boltzmann equation with hard potentials

TL;DR

This paper proves local existence and uniqueness of weak solutions for the inhomogeneous Boltzmann equation with hard potentials, addressing challenges due to velocity moment loss using hypoelliptic estimates.

Contribution

It establishes the first local well-posedness results for the non-cutoff Boltzmann equation with hard potentials in a non-perturbative setting.

Findings

Proves local-in-time existence and uniqueness of weak solutions.

Handles velocity moment loss with hypoelliptic estimates.

Requires polynomial decay in initial data.

Abstract

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with hard potentials in the non-perturbative setting. For initial data with polynomial decay in the velocity variable, we establish the local-in-time existence and uniqueness of weak solutions, conditional to pointwise bounds on the hydrodynamic quantities (mass, energy, and entropy). Compared to the soft potential case, the key challenge for full-range hard potentials lies in the more severe loss of velocity moments. The proof combines a hypoelliptic estimate with interpolation inequalities to handle the moment-loss terms.