Global solutions in $W_k^{\zeta,p}L^\infty_TL^2_v$ for the Boltzmann equation without cutoff

TL;DR

This paper proves the global existence and long-time behavior of solutions to the Boltzmann equation without cutoff in a specific function space, advancing understanding of kinetic equations in mathematical physics.

Contribution

It establishes the first global solutions in the space $W_k^{zeta,p}L^z_TL^2_v$ for the non-cutoff Boltzmann equation, including both hard and soft potentials.

Findings

Global existence of solutions in the specified function space

Long-time behavior analyzed for different potentials

Norm estimates underpin the proof

Abstract

The Boltzmann equation without an angular cutoff in a three-dimensional periodic domain is considered. The global-in-time existence of solutions in a function space $ W_k^{\zeta,p}L^\infty_TL^2_v $ with $p>1$ and $\zeta>3(1-\frac{1}{p})$ is established in the perturbation framework and the long-time behavior of solutions is also obtained for both hard and soft potentials. The proof is based on several norm estimates.