Archive for Rational Mechanics and Analysis

, Volume 182, Issue 2, pp 299–331 | Cite as

Fractal First-Order Partial Differential Equations

  • Jérôme Droniou
  • Cyril Imbert


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.


Weak Solution Viscosity Solution Regular Solution Jacobi Equation Entropy Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 293–317 (1996)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques et applications. Springer-Verlag, Berlin, 1994Google Scholar
  3. 3.
    Biler, P., Funaki, T., Woyczynski, W.: Fractal Burgers Equations. J. Differential Equations 148, 9–46 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Biler, P., Karch, G., Woyczynski, W.: Asymptotics for multifractal conservation laws. Studia Math. 135, 231–252 (1999)zbMATHMathSciNetGoogle 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)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle 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, 2002Google Scholar
  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, 1997Google Scholar
  10. 10.
    Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249, 511–528 (2004)zbMATHMathSciNetCrossRefADSGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Droniou, J.: Vanishing non-local regularization of a scalar conservation law. Electron. J. Differential Equations 2003, 1–20 (2003)zbMATHGoogle 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://
  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)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964Google Scholar
  16. 16.
    Garroni, M.G., Menaldi, J.L.: Green functions for second order parabolic integro-differential problems. Longman Scientific and Technical, Burnt Mill, Harlow, 1992Google Scholar
  17. 17.
    Imbert, C.: A non-local reguralization of first order Hamilton-Jacobi equations. J. Differential Equations, 211, 214–246 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jakobsen, E.R., Karlsen, K.H.: Continuous dependence estimates for viscosity solutions of integro-fs. PreprintGoogle Scholar
  19. 19.
    Jakobsen, E.R., Karlsen, K.H.: A maximum principle for semicontinuous functions applicable to integro-partial differential equations. PreprintGoogle Scholar
  20. 20.
    Krushkov, S.N.: First Order quasilinear equations with several space variables. Math. USSR. Sb. 10, 217–243 (1970)CrossRefGoogle 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)zbMATHMathSciNetGoogle 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)zbMATHGoogle Scholar
  23. 23.
    Soner, H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24, 1110–1122 (1986)zbMATHMathSciNetCrossRefADSGoogle 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, 1988Google Scholar
  25. 25.
    Taylor, M.E.: Partial Differential Equations III (nonlinear equations). Applied Mathematical Sciences 117, Springer-Verlag, New York, 1997Google Scholar
  26. 26.
    Woyczyński, W.A.: Lévy processes in the physical sciences. Lévy Processes, pp. 241–266, Birkhäuser Boston, Boston, MA, 2001Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Département de Mathématiques, UMR CNRS 5149, CC 051Université Montpellier IIMontpellier cedex 5France
  2. 2.Département de Mathématiques, UMR CNRS 5149, CC 051  
  3. 3.Polytech'MontpellierUniversité Montpellier IIMontpellier cedex 5France

Personalised recommendations