DCDS
Persistence of Hölder continuity for non-local integro-differential equations
Kyudong Choi
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 1741-1771 doi: 10.3934/dcds.2013.33.1741
In this paper, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in $C^\beta$ for all time if its initial data lies in $C^\beta$. This result has an application for a fully non-linear problem, which is used in the field of image processing. In addition, we show Hölder regularity for solutions of drift diffusion equations with supercritical fractional diffusion under the assumption $b\in L^\infty C^{1-\alpha}$ on the divergent-free drift velocity. The proof is in the spirit of [23] where Kiselev and Nazarov established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation.
keywords: nonlinear partial differential equations super-critical fractional diffusion. Image processing nonlocal operators drift diffusion equations integro-differential equations

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