Vol. 9, No. 3, 2016

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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

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.

nonlocal equations, nonsymmetric kernels, covering lemma, crawling ink spots, regularity
Mathematical Subject Classification 2010
Primary: 47G20, 35R09
Received: 3 October 2015
Accepted: 16 December 2015
Published: 17 June 2016
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