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Session S18 - Recent progress in non-linear PDEs and their applications

Tuesday, July 20, 19:00 ~ 19:50 UTC-3

Regularity estimates for the Boltzmann equation without cutoff

Luis Silvestre

University of Chicago, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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 its hydrodynamic quantities: mass density, energy density and entropy density. As a consequence, a classical solution to the equation may fail to exist after a 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 and parabolic equations: a weak Harnack inequality in the style of De Giorgi, and a Schauder-type estimate, for integro-differential equations.

Joint work with Cyril Imbert (CNRS), Clement Mouhot (Cambridge) and Jamil Chaker (Bielefeld).

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