Bounded Solutions of the Boltzmann Equation in the Whole Space

TL;DR

This paper constructs bounded classical solutions to the Boltzmann equation in the entire space without smallness assumptions or specific boundary conditions at infinity, broadening the understanding of solution existence.

Contribution

It establishes existence and uniqueness of bounded solutions in the whole space for the Boltzmann equation without limit behavior assumptions at infinity or small initial data.

Findings

Solutions exist for both cutoff and non-cutoff collision kernels.

Solutions include periodic, vacuum, and equilibrium-at-infinity cases.

No smallness condition on initial data is required.

Abstract

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.