Continuous dependence estimates for nonlinear fractional convection-diffusion equations

Details Continuous dependence estimates for nonlinear fractional convection-diffusion equations Continuous dependence estimates for nonlinear fractional convection-diffusion equations

2011-05-11

arxiv.org/abs/1105.2288v1

SIAM J. Math. Anal. 44 (2012) 603-632

Mathematics

Analysis of PDEs

Scientific paper

10.1137/110834342

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.

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