The Schauder estimate for kinetic integral equations

Imbert, Cyril; Silvestre, Luis

doi:10.2140/apde.2021.14.171

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The Schauder estimate for kinetic integral equations

Cyril Imbert and Luis Silvestre

Vol. 14 (2021), No. 1, 171–204

DOI: 10.2140/apde.2021.14.171

Abstract

We establish interior Schauder estimates for kinetic equations with integrodifferential diffusion. We study equations of the form $f_{t} + v \cdot \nabla_{x} f = ℒ_{v} f + c$ , where $ℒ_{v}$ is an integrodifferential diffusion operator of order $2 s$ acting in the $v$ -variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of $ℒ_{v}$ , we obtain an a priori estimate for $f$ in a properly scaled Hölder space.

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Keywords

kinetic integrodifferential equations, Schauder estimates

Mathematical Subject Classification

Primary: 35K70, 35R09

Milestones

Received: 15 January 2019

Accepted: 7 October 2019

Published: 19 February 2021

Authors

Cyril Imbert
CNRS and Instituto de Matemática Pura e Aplicada Rio de Janeiro Brazil

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

Vol. 14, No. 1, 2021