Regularity for the Boltzmann Equation Conditional to Pressure and Moment Bounds

Abstract

We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in \(L^\infty \) in the case of hard potentials. As a consequence, we derive \(C^{\infty }\) estimates and decay estimates for all derivatives, conditional to these macroscopic bounds. Our \(L^\infty \) estimates are uniform in the limit \(s \nearrow 1\) and hence we recover the same results also for the Landau equation.