Continuous dependence estimates for nonlinear fractional convection-diffusion equations

TL;DR

This paper establishes a general framework for deriving error estimates for nonlinear fractional convection-diffusion equations involving nonlocal operators, extending existing results and providing convergence rates for viscosity approximations.

Contribution

It introduces a unified approach to error estimation for nonlocal, nonlinear, and possibly degenerate diffusion equations involving fractional operators, extending prior work.

Findings

Derived continuous dependence estimates on nonlinearities and Levy measures.

Provided convergence rate estimates for nonlocal viscosity approximations.

Extended known error estimates to new classes of fractional equations.

Abstract

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover, and extend to new equations, a large part of the known error estimates in the literature.