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DCDS

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.

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