Applied Mathematics and Optimization

, Volume 36, Issue 2, pp 229–241 | Cite as

Conservation laws with a random source

  • H. Holden
  • N. H. Risebro


We study the scalar conservation law with a noisy nonlinear source, namely,u l + f(u)x = h(u, x, t) + g(u)W(t), whereW(t) is the white noise in the time variable, and we analyse the Cauchy problem for this equation where the initial data are assumed to be deterministic. A method is proposed to construct approximate weak solutions, and we then show that this yields a convergent sequence. This sequence converges to a (pathwise) solution of the Cauchy problem. The equation can be considered as a model of deterministic driven phase transitions with a random perturbation in a system of two constituents. Finally we show some numerical results motivated by two-phase flow in porous media.

Key Words

Conservation laws Stochastic partial differential equations Phase transitions 

AMS Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Bensoussan, R. Glowinski, A. Rascanu, Approximation of some stochastic differential equations by the splitting up method, Appl. Math. Optim. 25 (1992), 81–106.zbMATHCrossRefGoogle Scholar
  2. 2.
    L. Bertini, N. Cancrini, G. Jona-Lasinio, The stochastic Burgers equation, Comm. Math. Phys. 165 (1994), 211–232.zbMATHCrossRefGoogle Scholar
  3. 3.
    Z. Brzezniak, M. Capinski, F. Flandoli, Stochastic partial differential equations and turbulence, Math. Mod. Methods Appl. Sci. 1 (1991), 41–59.zbMATHCrossRefGoogle Scholar
  4. 4.
    A. V. Bulinskii, S. A. Molchanov, Asymptotical normality of a solution of Burgers’ equation with random initial data, Theory Probab. Appl. 36 (1992), 217–236.CrossRefGoogle Scholar
  5. 5.
    J. M. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974.zbMATHGoogle Scholar
  6. 6.
    M. Crandall, A. Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), 285–314.zbMATHCrossRefGoogle Scholar
  7. 7.
    J.-D. Fournier, U. Frisch, L’équation de Burgers déterministe et statistique, J. Méc. Théor. Appl. 2 (1983), 699–750.zbMATHGoogle Scholar
  8. 8.
    S. K. Godunov, Finite difference methods for numerical computations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), 271–295.Google Scholar
  9. 9.
    S. Gurbatov, A. Malakov, A. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles, Manchester University Press, Manchester, 1991.zbMATHGoogle Scholar
  10. 10.
    S. N. Gurbatov, A. I. Saichev, Degeneracy of one-dimensional acoustic turbulence at large Reynolds numbers, Soviet Phys. JETP 53 (1981), 347–354.Google Scholar
  11. 11.
    H. Holden, L. Holden, First-order nonlinear scalar hyperbolic conservation laws in one dimension, in Ideas and Methods in Mathematical Analysis, Stochastics, and Applications (S. Albeverio, J. E. Fenstad, H. Holden, T. Lindstrøm, eds.), Cambridge University Press, Cambridge, 1992, pp. 480–510.Google Scholar
  12. 12.
    H. Holden, L. Holden, R. Høegh-Krohn, A numerical method for first-order nonlinear scalar conservation laws in one-dimension, Comput. Math. Appl. 15 (1988), 595–602.zbMATHCrossRefGoogle Scholar
  13. 13.
    H. Holden, T. Lindstrøm, B. Øksendal, J. Ubøe, T.-S. Zhang, The Burgers equation with a noisy force, Comm. Partial Differential Equations 19 (1994), 119–141.zbMATHCrossRefGoogle Scholar
  14. 14.
    H. Holden, T. Lindstrøm, B. Øksendal, J. Ubøe, T.-S. Zhang, The stochastic Wick-type Burgers equation, in Stochastic Partial Differential Equations (A. Etheridge, ed.), London Mathematical Society Lecture Note Series, Vol. 216, Cambridge University Press, Cambridge, 1995, pp. 141–161.Google Scholar
  15. 15.
    H. Holden, N. H. Risebro, Stochastic properties of the scalar Buckley-Leverett equation. SIAM J. Math. Anal. 51 (1991), 1472–1488.zbMATHCrossRefGoogle Scholar
  16. 16.
    H. Holden, N. H. Risebro, A stochastic approach to conservation laws, Proc. Third International Conference on Hyperbolic Problems. Theory. Numerical Methods and Applications, Uppsala, 1990 (B. Engquist, B. Gustafsson, eds.), Studentlitteratur/Chartwell-Bratt, Lund-Bromley, 1991, pp. 575–587.Google Scholar
  17. 17.
    H. Holden, N. H. Risebro, A fractional steps method for scalar conservation laws without the CFL condition, Math. Comp. 60 (1993), 221–232.zbMATHCrossRefGoogle Scholar
  18. 18.
    M. Kardar, G. Parisi, Y.-C. Zhang, Dynamic scaling of growing surfaces, Phys. Rev. Lett. 56 (1986), 889–892.zbMATHCrossRefGoogle Scholar
  19. 19.
    S. Kida, Asymptotic properties of Burgers turbulence, J. Fluid. Mech. 93 (1979), 337–377.zbMATHCrossRefGoogle Scholar
  20. 20.
    P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.zbMATHGoogle Scholar
  21. 21.
    H. Konno, On stochastic Burgers equation, J. Phys. Soc. Japan 54 (1985), 4475–4478.CrossRefGoogle Scholar
  22. 22.
    H. Kunita, First-order stochastic partial differential equations, in Stochastic Analysis, Taniguchi Symposium, Katata and Kyoto, 1982 (K. Ito, ed.), North-Holland, Amsterdam, 1984 pp. 249–269.Google Scholar
  23. 23.
    Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, Berlin, 1984.zbMATHGoogle Scholar
  24. 24.
    R. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, 1991.Google Scholar
  25. 25.
    B. J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), 59–69.zbMATHCrossRefGoogle Scholar
  26. 26.
    E. Medina, T. Hwa, M. Kardar, T.-C. Zhang, Burgers equation with correlated noise: Renormalizationgroup analysis and applications to directed polymers and interface growth, Phys. Rev. 39A (1989), 3053–3075.Google Scholar
  27. 27.
    T. Musha, Y. Kosugi, G. Matsumoto, M. Suzuki, Modulation of the time relation of action potential impulses propagating along the axon, IEEE Trans. Biomed. Engrg. 28 (1981), 616–623.CrossRefGoogle Scholar
  28. 28.
    H. Nakazawa, Stochastic Burgers’ equation in the inviscid limit, Adv. in Appl. Math. 3 (1982), 18–42.zbMATHCrossRefGoogle Scholar
  29. 29.
    S. F. Shandarin, Ya. B. Zeldovich, The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Mod. Phys. 61 (1989), 185–220.CrossRefGoogle Scholar
  30. 30.
    Z.-S. She, E. Aurell, U. Frisch, The inviscid Burgers equation with initial data of Brownian motion, Comm. Math. Phys. 148 (1992), 623–641.zbMATHCrossRefGoogle Scholar
  31. 31.
    Ya. G. Sinai, Two results concerning asymptotic behavior of solutions of the Burgers equation with force, J. Stat. Phys. 64 (1991), 1–12.zbMATHCrossRefGoogle Scholar
  32. 32.
    Ya. G. Sinai, Statistics of shocks in solutions of inviscid Burgers equation, Comm. Math. Phys. 148 (1992), 601–621.zbMATHCrossRefGoogle Scholar
  33. 33.
    J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • H. Holden
    • 1
  • N. H. Risebro
    • 2
  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of MathematicsUniversity of OsloOsloNorway

Personalised recommendations