Conditional \(L^{\infty }\) Estimates for the Non-cutoff Boltzmann Equation in a Bounded Domain

Abstract

We consider weak solutions of the inhomogeneous non-cutoff Boltzmann equation in a bounded domain with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection. When the mass, energy and entropy densities are bounded above, and the mass density is bounded away from a vacuum, we obtain an estimate of the \(L^\infty \) norm of the solution depending on the macroscopic bounds on these hydrodynamic quantities only. This is a regularization effect in the sense that the initial data is not required to be bounded. We present a proof based on variational ideas, which is fundamentally different to the proof that was previously known for the equation in periodic spatial domains.