Vol. 14, No. 1, 2021

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The Schauder estimate for kinetic integral equations

Cyril Imbert and Luis Silvestre

Vol. 14 (2021), No. 1, 171–204
Abstract

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

Keywords
kinetic integrodifferential equations, Schauder estimates
Mathematical Subject Classification
Primary: 35K70, 35R09