Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off

TL;DR

This paper establishes existence, uniqueness, and exponential stability results for the inhomogeneous Boltzmann equation with hard potentials without angular cut-off, using new regularization estimates in weighted Sobolev spaces.

Contribution

It provides the first well-posedness and stability results in a physically relevant framework with polynomial weights, extending previous work that used equilibrium-prescribed weights.

Findings

Proved existence and uniqueness of solutions near equilibrium.

Established exponential decay rates close to the optimal.

Developed new short-time regularization estimates for the linearized operator.

Abstract

In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a moderate angular singularity). We prove a result of existence and uniqueness of solutions in a close-to-equilibrium regime for this equation in weighted Sobolev spaces with a polynomial weight, contrary to previous works on the subject, all developed with a weight prescribed by the equilibrium. It is the first result in this more physically relevant frameworkfor this equation. Moreover, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation. Let us highlight the fact that a key point of the development of our Cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo-differential tools.