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Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations

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Abstract

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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Authors and Affiliations

  1. UMR CNRS 6623, Université de Franche-Comté, 25030, Besançon, France

    Nathaël Alibaud

  2. Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, 90112, Thailand

    Nathaël Alibaud

  3. Department of Mathematics, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Simone Cifani

  4. Department of Mathematics, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Espen R. Jakobsen

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  1. Nathaël Alibaud
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  2. Simone Cifani
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  3. Espen R. Jakobsen
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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). https://doi.org/10.1007/s00205-014-0737-x

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