Small Perturbation Solutions for Parabolic Equations

TL;DR

This paper proves that solutions close to a smooth solution of a parabolic equation remain smooth, even when the ellipticity condition is only locally satisfied, extending regularity results under weaker assumptions.

Contribution

It establishes the stability of smooth solutions under small L1-perturbations for parabolic equations with localized uniform ellipticity.

Findings

Solutions near a smooth solution stay smooth under small perturbations.

Local uniform ellipticity suffices for regularity preservation.

The result extends classical regularity theory to weaker ellipticity conditions.

Abstract

Let $\phi$ be a smooth solution of the parabolic equation $F(D^2u, Du, u, x, t)- u_{t} = 0$: Assume $F$ is uniform elliptic only in a neighborhood of $(D^2\phi, D\phi, \phi, x, t)$, we prove that any solution obtained from small L1-perturbation of $\phi$ remains smooth.