University of Chicago
Regularity estimates for the Boltzmann equation without cutoff
We study the regularization effect of the inhomogeneous Boltzmann equation without cutoff. We obtain a priori estimates for all derivatives of the solution depending only on bounds of the hydrodynamic quantities: mass density, energy density and entropy density. As a consequence, a classical solution to the equation may fail to exists after certain time \(T\) only if at least one of these hydrodynamic quantities blows up. Our analysis applies to the case of moderately soft and hard potentials. We use methods that originated in the study of nonlocal elliptic equations: a weak Harnack inequality in the style of De Giorgi, and a Schauder-type estimate.