Fractal First-Order Partial Differential Equations

Archive for Rational Mechanics and Analysis volume 182, pages 299–331 (2006)Cite this article

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The present paper is concerned with semi-linear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton–Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one hand, the key a priori estimates for the scalar conservation law and the Hamilton–Jacobi equation and, on the other hand, the smoothing effect of the operator. As far as Hamilton–Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows us to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.

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

Access options

Buy single article

Instant access to the full article PDF.

43,34 €

Tax calculation will be finalised during checkout.

References

  1. 1.

    Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 293–317 (1996)

    MATH  MathSciNet  Google Scholar 

  2. 2.

    Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques et applications. Springer-Verlag, Berlin, 1994

  3. 3.

    Biler, P., Funaki, T., Woyczynski, W.: Fractal Burgers Equations. J. Differential Equations 148, 9–46 (1998)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Biler, P., Karch, G., Woyczynski, W.: Asymptotics for multifractal conservation laws. Studia Math. 135, 231–252 (1999)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Benth, F.E., Karlsen, K.H., Reikvam, K.: Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach. Finance Stoch. 5, 275–303 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    Benth, F.E., Karlsen, K.H., Reikvam, K.: Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution. Finance Stoch. 5, 447–467 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Benth, F.E., Karlsen, K.H., Reikvam, K.: Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs. Stoch. Stoch. Rep. 74, 517–569 (2002)

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    Clavin, P.: Instabilities and nonlinear patterns of overdriven detonations in gases. Nonlinear PDE's in Condensed Matter and Reactive Flows (Eds. Berestycki, H., Pomeau, Y.), Kluwer, pp. 49–97, 2002

  9. 9.

    Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth analysis and control theory. Graduate Texts in Mathematics, 178, Springer, Berlin, 1997

  10. 10.

    Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249, 511–528 (2004)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  11. 11.

    Crandall, M.G., Ishii, H., Lions, P.-L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27, 1–67 (1992)

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Droniou, J.: Vanishing non-local regularization of a scalar conservation law. Electron. J. Differential Equations 2003, 1–20 (2003)

    MATH  Google Scholar 

  13. 13.

    Droniou, J.: Etude théorique et numérique d'équations aux dérivées partielles elliptiques, paraboliques et non-locales. Mémoire d'Habilitation à Diriger les Recherches, Université Montpellier II, France. Available at http:// www-gm3.univ-mrs.fr/~droniou/travaux-en.html

  14. 14.

    Droniou, J., Gallouët, T., Vovelle, J.: Global solution and smoothing effect for a non-local regularization of an hyperbolic equation. J. Evol. Equ. 3, 499–521 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964

  16. 16.

    Garroni, M.G., Menaldi, J.L.: Green functions for second order parabolic integro-differential problems. Longman Scientific and Technical, Burnt Mill, Harlow, 1992

  17. 17.

    Imbert, C.: A non-local reguralization of first order Hamilton-Jacobi equations. J. Differential Equations, 211, 214–246 (2005)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Jakobsen, E.R., Karlsen, K.H.: Continuous dependence estimates for viscosity solutions of integro-fs. Preprint

  19. 19.

    Jakobsen, E.R., Karlsen, K.H.: A maximum principle for semicontinuous functions applicable to integro-partial differential equations. Preprint

  20. 20.

    Krushkov, S.N.: First Order quasilinear equations with several space variables. Math. USSR. Sb. 10, 217–243 (1970)

    Article  Google Scholar 

  21. 21.

    Kuznecov, N.N.: The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation. Ž. Vyčisl. Mat. i Mat. Fiz. 16, 1489–1502 (1976)

    MATH  MathSciNet  Google Scholar 

  22. 22.

    Sayah, A.: Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité, II. Existence de solutions de viscosité. Comm. Partial Differential Equations 16, 1057–1093 (1991)

    MATH  Google Scholar 

  23. 23.

    Soner, H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24, 1110–1122 (1986)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  24. 24.

    Soner, H.M.: Optimal control of jump-Markov processes and viscosity solutions. Stochastic Differential Systems, Stochastic Control Theory and Applications. The IMA Volumes in Mathematics and its Applications, 10, Springer, New York, pp. 501–511, 1988

  25. 25.

    Taylor, M.E.: Partial Differential Equations III (nonlinear equations). Applied Mathematical Sciences 117, Springer-Verlag, New York, 1997

  26. 26.

    Woyczyński, W.A.: Lévy processes in the physical sciences. Lévy Processes, pp. 241–266, Birkhäuser Boston, Boston, MA, 2001

Download references

Author information

Affiliations

  1. Département de Mathématiques, UMR CNRS 5149, CC 051, Université Montpellier II, Place Eugène Bataillon, 34095, Montpellier cedex 5, France

    Jérôme Droniou

  2. Département de Mathématiques, UMR CNRS 5149, CC 051,  ,  

    Cyril Imbert

  3. Polytech'Montpellier, Université Montpellier II, Place Eugène Bataillon, 34095, Montpellier cedex 5, France

    Cyril Imbert

Authors
  1. Jérôme Droniou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cyril Imbert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jérôme Droniou.

Additional information

Communicated by Y. Brenier

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Droniou, J., Imbert, C. Fractal First-Order Partial Differential Equations. Arch Rational Mech Anal 182, 299–331 (2006). https://doi.org/10.1007/s00205-006-0429-2

Download citation

Keywords

  • Weak Solution
  • Viscosity Solution
  • Regular Solution
  • Jacobi Equation
  • Entropy Solution