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Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations


We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, \({\triangle^{\alpha/2}}\) for \({\alpha \in (0,2)}\). Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on α in the BV-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits \({\alpha \downarrow 0}\) and \({\alpha \uparrow 2}\). In the limit \({\alpha \uparrow 2}\), \({\triangle^{\alpha/2}}\) converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231–251, 1999) for local degenerate parabolic equations (thus providing an alternative proof).

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Correspondence to Espen R. Jakobsen.

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Communicated by A. Bressan

This research was supported by the Research Council of Norway (NFR) projects DIMMA and “Integro-PDEs: Numerical methods, Analysis, and Applications to Finance,” and by the “French ANR project CoToCoLa, no. ANR-11-JS01-006-01.”

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Alibaud, N., Cifani, S. & Jakobsen, E.R. Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations. Arch Rational Mech Anal 213, 705–762 (2014).

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  • Continuous Dependence
  • Entropy Solution
  • Entropy Inequality
  • Degenerate Parabolic Equation
  • Weak Entropy Solution