Persistence of H\"{o}lder continuity for non-local integro-differential equations

TL;DR

This paper proves that solutions to certain non-local integro-differential equations maintain their initial H"{o}lder continuity over time, with applications in image processing and connections to the SQG equation.

Contribution

It establishes the persistence of H"{o}lder continuity for solutions of non-local equations under natural kernel assumptions, extending previous results.

Findings

Solutions remain in C^β if initially in C^β

Applicable to fully non-linear problems in image processing

Method inspired by Kiselev and Nazarov's work on SQG

Abstract

In this paper, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of H\"{o}lder 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. The proof is in the spirit of the paper [18] of Kiselev and Nazarov where they established H\"{o}lder continuity of the critical surface quasi-geostrophic (SQG) equation.