Local well-posedness of the Boltzmann equation with polynomially decaying initial data

TL;DR

This paper proves short-time well-posedness for the inhomogeneous non-cutoff Boltzmann equation with polynomial decay initial data, using new estimates and a mixed Sobolev space approach, extending previous results.

Contribution

It introduces a novel framework for establishing well-posedness with polynomial decay initial data, relaxing regularity restrictions and broadening parameter ranges.

Findings

Established short-time existence in a mixed Sobolev space.

Improved parameter range for well-posedness with polynomial decay.

Combined with regularity estimates to enable solution continuation.

Abstract

We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed $L^2$ and $L^\infty$ space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as relax the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities remain under control. This continuation criterion was previously only available in the restricted range of parameters of previous well-posedness results for polynomially decaying initial data.