Global existence and full regularity of the Boltzmann equation without angular cutoff

Abstract

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and $C^{\infty }$ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.

References in corpus (2)

• Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials
• Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production

Cited by in corpus (3)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Global Strong Solutions of the Boltzmann Equation without Angular Cut-off
• Optimal Large-Time Behavior of the Vlasov-Maxwell-Boltzmann System in the Whole Space