Skip to main content

Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations

Archive for Rational Mechanics and Analysis volume 213, pages 705–762 (2014)Cite this article

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).

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, (1975)

  2. Alibaud N.: Entropy formulation for fractal conservation laws. J. Evol. Equ. 7(1), 145–175 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alibaud, N., Andreianov, B.: Non-uniqueness of weak solutions for the fractal Burgers equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(4), 997–1016 (2010)

    Google Scholar 

  4. Alibaud N., Cifani S., Jakobsen E.R.: Continuous dependence estimates for nonlinear fractional convection-diffusion equations. SIAM J. Math. Anal. 44(2), 603–632 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alibaud, N., Droniou, J., Vovelle, J.: Occurence and non-appearance of shocks in fractal Burgers equation. J. Hyperb. Differ. Equ. 4(3), 479–499 (2007)

    Google Scholar 

  6. Andreianov N., Bendahmane M., Karlsen K., Ouaro S.: Well-posedness results for triply nonlinear degenerate parabolic equations. J. Differ. Equ. 247(1), 277–302 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Bénilan, P., Crandall, M.G.: The continuous dependence on φ of solutions of \({u_t - \triangle \varphi(u)=0}\). Indiana Univ. Math. J. 30(2), 161–177 (1981)

  8. Biler P., Funaki T., Woyczyński W.: Fractal Burgers Equations. J. Differ. Equ. 148, 9–46 (1998)

    Article  ADS  MATH  Google Scholar 

  9. Biler, P., Karch, G., Imbert, C.: Fractal porous medium equation. C. R. Acad. Sci. Paris, Ser. I 349, 641–645 (2011)

    Google Scholar 

  10. Biler, P., Karch, G., Monneau, R.: A nonlinear diffusion of dislocation density and self-similar solutions. Commun. Math. Physics 294, 145–168 (2010)

    Google Scholar 

  11. Bouchut, F., Perthame, B.: Kružkov’s estimates for scalar conservation laws revisited. Trans. Am. Math. Soc. 7, 2847–2870 (1998)

    Google Scholar 

  12. Brézis, H., Crandall, M.G.: Uniqueness of solutions of the initial–value problem for \({u_t-\triangle \varphi(u)=0}\). J. Math. Pures Appl. (9) 58(2), 153–163 (1979)

    Google Scholar 

  13. Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro–differential equations. Comm. Pure Appl. Math. 62(5), 597–638 (2009)

    Google Scholar 

  14. Caffarelli, L., Silvestre, L.: The Evans-Krylov theorem for nonlocal fully nonlinear equations. Ann. Math. (2) 174(2), 1163–1187 (2011)

    Google Scholar 

  15. Caffarelli, L., Vázquez, J.-L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202(2), 537–565 (2011)

    Google Scholar 

  16. Carrillo, J.: Entropy Solutions for nonlinear Degenerate Problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)

    Google Scholar 

  17. Chan, C.H., Czubak, M.: Regularity of solutions for the critical N-dimensional Burgers equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2), 471–501 (2010)

    Google Scholar 

  18. Chan, C.H., Czubak, M., Silvestre, L.: Eventual regularization of the slightly supercritical fractional Burgers equation. Discret. Contin. Dyn. Syst. 27(2):847–861 (2010)

    Google Scholar 

  19. Chen, G., Karlsen, K.H.: Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Commun. Pure Appl. Anal. 4(2), 241–266 (2005)

    Google Scholar 

  20. Chen, G., Karlsen, K.H.: L 1-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations. Trans. Am. Math. Soc. 358(3), 937–963 (2006)

    Google Scholar 

  21. Cifani, S., Jakobsen, E.R.: Entropy formulation for degenerate fractional order convection-diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(3), 413–441 (2011)

    Google Scholar 

  22. Cifani, S., Jakobsen, E.R.: On numerical methods and error estimates for degenerate fractional convection-diffusion equations. Numer. Math. online first November (2013). doi:10.1007/s00211-013-0590-0

  23. Clavin, P.: Instabilities and Nonlinear Patterns of Overdriven Detonations in Gases. Nonlinear PDE’s in Condensed Matter and Reactive Flows. Kluwer, Dordrecht, 49–97, (2002)

  24. Cockburn, B., Gripenberg, G.: Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differ. Equ. 151(2), 231–251 (1999)

    Google Scholar 

  25. Cockburn, B., Gripenberg, G., Londen, S.-O.: On convergence of entropy solutions to a single conservation law. J. Differ. Equ. 128(1), 206–251 (1996)

    Google Scholar 

  26. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, (2004)

  27. Dafermos, C.M.: Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38, 33–41 (1972)

    Google Scholar 

  28. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin, (2005)

  29. de Pablo, A., Quiros, F., Rodriguez, A., Vázquez, J.L.: A fractional porous medium equation. Adv. Math. 226(2), 1378–1409 (2011)

    Google Scholar 

  30. de Pablo, A., Quiros, F., Rodriguez, A., Vázquez, J.L.: A general fractional porous medium equation. Comm. Pure Appl. Math. 65(9), 1242–1284 (2012)

    Google Scholar 

  31. de Pablo, A., Quiros, F., Rodriguez, A., Vázquez, J.L.: Classical solutions for a logarithmic fractional diffusion equation. J. Math. Pures Appl. (9) (to appear)

  32. DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New-York, (1993)

  33. Droniou, J.: Vanishing non-local regularization of a scalar conservation law. Electron. J. Differ. Equ. 2003(117), 1–20 (2003)

    Google Scholar 

  34. Droniou, J., Gallouët, T., Vovelle, J.: Global solution and smoothing effect for a nonlocal regularization of a hyperbolic equation. J. Evol. Equ. 4(3), 479–499 (2003)

    Google Scholar 

  35. Droniou, J., Imbert, C.: Fractal first order partial differential equations. Arch. Ration. Mech. Anal. 182(2), 299–331 (2006)

    Google Scholar 

  36. Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws. Applied Mathematical Sciences, 152. Springer, Berlin, (2007)

  37. Imbert, C.: A non-local regularization of first order Hamilton-Jacobi equations. J. Differ. Equ. 211(1), 214–246 (2005)

    Google Scholar 

  38. Jakobsen, E.R., Karlsen, K.H.: Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differ. Equ. 212(2), 278–318 (2005)

    Google Scholar 

  39. Jakubowski, V.G., Wittbold, P.: On a nonlinear elliptic/parabolic integro-differential equation with L 1-data. J. Differ. Equ. 197(2), 427–445 (2003)

    Google Scholar 

  40. Karlsen, K.H., Risebro, N.H.: On the uniqueness and stability of entropy solutions of non- linear degenerate parabolic equations with rough coefficients. Discret. Contin. Dyn. Syst. 9(5), 1081–1104 (2003)

    Google Scholar 

  41. Karlsen, K.H., Ulusoy, S.: Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations. Electron. J. Differ. Equ. 2011(116), 1–23 (2011)

  42. Kassmann, M., Schwab, R.W.: Regularity results for nonlocal parabolic equations. Riv. Math. Univ. Parma (N.S.) 5(1) (2014)

  43. Kiselev, A., Nazarov, F., Shterenberg, R.: Blow up and regularity for fractal Burgers equation. Dyn. PDE 5(3), 211–240 (2008)

    Google Scholar 

  44. Kruzhkov, S.N.: First order quasilinear equations with several independent variables. Math. Sb. (N.S.) 81(123), 228–255 (1970)

    Google Scholar 

  45. Kuznetsov, N.N.: Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR. Comput. Math. Math. Phys. Dokl. 16(6), 105–119 (1976)

  46. Landkof, N.S.: Foundations of Modern Potential Theory. Die Grundlehren der mathematischen Wissenschaften, 180. Springer, New York, (1972)

  47. Lions, P.-L., Perthame, B., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related local equations. J. Am. Math. Soc. 7, 169–191 (1994)

    Google Scholar 

  48. Lucier, B.J.: A moving mesh numerical method for hyperbolic conservation laws. Math. Comp. 46, 59–69 (1980)

    Google Scholar 

  49. Lukkari, T.: Stability of solutions to nonlinear diffusion equations. arXiv:1206.2492 [math.AP]

  50. Rohde, C., Yong, W.-A.: The nonrelativistic limit in radiation hydrodynamics. I. Weak entropy solutions for a model problem. J. Differ. Equ. 234(1), 91–109 (2007)

    Google Scholar 

  51. Rosenau P.: Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. Phys. Rev. A 40, 7193–7196 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  52. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, (1999)

  53. Sayah, A.: Equations d’Hamilton-Jacobi du premier ordre avec termes intégro-différentiels, Parties I et II. Comm P.D.E. 16(6–7), 1057–1093 (1991)

  54. Schochet, S., Tadmor, E.: Regularized Chapman-Enskog expansion for scalar conservation laws. Arch. Ration. Mech. Anal. 119, 95–107 (1992)

    Google Scholar 

  55. Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, (2007)

  56. Woyczyński, W.: Lévy processes in the physical sciences. Lévy process. Birkhäuser, Boston, 241–266, (2001)

Download references

Author information

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

Authors
  1. Nathaël Alibaud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simone Cifani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Espen R. Jakobsen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Espen R. Jakobsen.

Additional information

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.”

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-014-0737-x

Keywords

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