Optimal continuous dependence estimates for fractional degenerate parabolic equations
Résumé
We obtain optimal continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, $\Delta^{\alp/2}$ for $\alp \in (0,2)$, the generator of a pure jump Lévy process. Our results cover the dependence on the nonlinearities, and for the first time, also the explicit dependence on~$\alp$. The former estimate (dependence on nonlinearity) shows a clear dependence on $\alp$, and it is stable in the limits $\alp\ra0$ and $\alp\ra2$. In the limit $\alp\ra2$, $\Delta^{\alp/2}$ converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of \cite{CoGr99} for local degenerate parabolic equations (thus providing an alternative proof).
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