Continuous dependence estimates for nonlinear fractional convection-diffusion equations

Nathael Alibaud (1, 2) , Simone Cifani (3) , Espen Jakobsen (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Department of Mathematics
3 Department of Mathematical Sciences

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 L'evy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the L'evy 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.

Mots clés: Fractional/fractal conservation laws; Nonlinear parabolic equations; Pure jump Levy processes; Continuous dependence estimates

Domaines: Equations aux dérivées partielles [math.AP]

Nathael Alibaud, Simone Cifani, Espen Jakobsen. Continuous dependence estimates for nonlinear fractional convection-diffusion equations. SIAM Journal on Mathematical Analysis, 2012, 44 (2), pp.603--632. ⟨10.1137/110834342⟩. ⟨hal-00593376v2⟩