Optimal continuous dependence estimates for fractional degenerate parabolic equations

TL;DR

This paper establishes optimal continuous dependence estimates for solutions of degenerate parabolic equations with fractional diffusion, including the Lipschitz dependence on the fractional order, and connects these results to classical local equations.

Contribution

It provides the first quantitative Lipschitz dependence estimates on the fractional order in the BV-framework for degenerate parabolic equations with fractional diffusion.

Findings

Dependence estimates are stable as fractional order approaches 0 and 2.

The results recover classical local equations in the limit as fractional order approaches 2.

The estimates are shown to be optimal through explicit examples.

Abstract

We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, $\Delta^{\alpha/2}$ for $\alpha \in (0,2)$. Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on $\alpha$ in the $BV$-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits $\alpha \downarrow 0$ and $\alpha \uparrow 2$. In the limit $\alpha \uparrow 2$, $\Delta^{\alpha/2}$ converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231-251, 1999) for local degenerate parabolic equations (thus providing an alternative proof).