Regularity for parabolic integro-differential equations with very irregular kernels

Schwab, Russell W.; Silvestre, Luis

doi:10.2140/apde.2016.9.727

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Regularity for parabolic integro-differential equations with very irregular kernels

Russell W. Schwab and Luis Silvestre

Vol. 9 (2016), No. 3, 727–772

DOI: 10.2140/apde.2016.9.727

Abstract

We prove Hölder regularity for a general class of parabolic integro-differential equations, which (strictly) includes many previous results. We present a proof that avoids the use of a convex envelope as well as give a new covering argument that is better suited to the fractional order setting. Our main result involves a class of kernels that may contain a singular measure, may vanish at some points, and are not required to be symmetric. This new generality of integro-differential operators opens the door to further applications of the theory, including some regularization estimates for the Boltzmann equation.

Keywords

nonlocal equations, nonsymmetric kernels, covering lemma, crawling ink spots, regularity

Mathematical Subject Classification 2010

Primary: 47G20, 35R09

Milestones

Received: 3 October 2015

Accepted: 16 December 2015

Published: 17 June 2016

Authors

Russell W. Schwab
Department of Mathematics Michigan State University East Lansing, MI 48824 United States

Luis Silvestre
Department of Mathematics The University of Chicago Chicago, IL 60637 United States

Vol. 9, No. 3, 2016