Continuous dependence estimates for nonlinear fractional convection-diffusion 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.

In this version we have corrected Example 3.4 explaining the link with the results in [51,59]

References in corpus (10)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations
• Nonlinear porous medium flow with fractional potential pressure
• Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations
• Nonlinear diffusion of dislocation density and self-similar solutions
• Global well-posedness of the critical Burgers equation in critical Besov spaces
• On convergence of solutions of fractal Burgers equation toward rarefaction waves
• Non-uniqueness of weak solutions for the fractal Burgers equation
• The discontinuous Galerkin method for fractal conservation laws
• The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
• Stability of Entropy Solutions for Levy Mixed Hyperbolic-Parabolic Equations

Cited by in corpus (4)

• Optimal continuous dependence estimates for fractional degenerate parabolic equations
• A framework for non-local, non-linear initial value problems
• Sensitivity analysis for solutions to heterogeneous nonlocal systems
• Nonlocal degenerate parabolic hyperbolic equations on bounded domains