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Conservation laws with a random source. (English) Zbl 0885.35069
The main result of this paper is following: Let \(f(u)\) and \(g(u)\) be real Lipschitz functions, \(g(u)\) with bounded support, and let \(h(x,t,u)\) be a Lipschitz function which has bounded support in \(u\). Let \(u_0(x)\) be a (deterministic) essentially bounded function with bounded support which has finitely many extrema. Then, almost surely, there exists a weak solution to the stochastic conservation law: \[ u_t+ f(x)_x= h(x, t,u)+ g(u)W(t),\;u(x,0;\omega)= u_0(x), \] where \(W(t)\) is the white noise process, and the right-hand side is to be interpreted in the Itô sense. Some numerical results motivated by two-phase flow in porous media are presented.

MSC:
35L65 Hyperbolic conservation laws
35R60 PDEs with randomness, stochastic partial differential equations
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[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. · Zbl 0745.65089 · doi:10.1007/BF01184157
[2] L. Bertini, N. Cancrini, G. Jona-Lasinio, The stochastic Burgers equation, Comm. Math. Phys. 165 (1994), 211–232. · Zbl 0807.60062 · doi:10.1007/BF02099769
[3] Z. Brzezniak, M. Capinski, F. Flandoli, Stochastic partial differential equations and turbulence, Math. Mod. Methods Appl. Sci. 1 (1991), 41–59. · Zbl 0741.60058 · doi:10.1142/S0218202591000046
[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. · Zbl 04508650 · doi:10.1137/1136027
[5] J. M. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974. · Zbl 0302.60048
[6] M. Crandall, A. Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), 285–314. · Zbl 0438.65076 · doi:10.1007/BF01396704
[7] J.-D. Fournier, U. Frisch, L’équation de Burgers déterministe et statistique, J. Méc. Théor. Appl. 2 (1983), 699–750.
[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. · Zbl 0171.46204
[9] S. Gurbatov, A. Malakov, A. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles, Manchester University Press, Manchester, 1991. · Zbl 0860.76002
[10] S. N. Gurbatov, A. I. Saichev, Degeneracy of one-dimensional acoustic turbulence at large Reynolds numbers, Soviet Phys. JETP 53 (1981), 347–354. · Zbl 0483.76062
[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. · Zbl 0851.65064
[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. · Zbl 0658.65085 · doi:10.1016/0898-1221(88)90282-9
[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. · Zbl 0804.35158 · doi:10.1080/03605309408821011
[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. · Zbl 0823.60050
[15] H. Holden, N. H. Risebro, Stochastic properties of the scalar Buckley-Leverett equation. SIAM J. Math. Anal. 51 (1991), 1472–1488. · Zbl 0745.60064 · doi:10.1137/0151073
[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. · Zbl 0789.35103
[17] H. Holden, N. H. Risebro, A fractional steps method for scalar conservation laws without the CFL condition, Math. Comp. 60 (1993), 221–232. · Zbl 0799.35150 · doi:10.1090/S0025-5718-1993-1153165-5
[18] M. Kardar, G. Parisi, Y.-C. Zhang, Dynamic scaling of growing surfaces, Phys. Rev. Lett. 56 (1986), 889–892. · Zbl 1101.82329 · doi:10.1103/PhysRevLett.56.889
[19] S. Kida, Asymptotic properties of Burgers turbulence, J. Fluid. Mech. 93 (1979), 337–377. · Zbl 0436.76031 · doi:10.1017/S0022112079001932
[20] P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. · Zbl 0752.60043
[21] H. Konno, On stochastic Burgers equation, J. Phys. Soc. Japan 54 (1985), 4475–4478. · doi:10.1143/JPSJ.54.4475
[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.
[23] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, Berlin, 1984. · Zbl 0558.76051
[24] R. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, 1991. · Zbl 0847.65053
[25] B. J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), 59–69. · Zbl 0592.65062 · doi:10.1090/S0025-5718-1986-0815831-4
[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. · doi:10.1103/PhysRevA.39.3053
[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. · doi:10.1109/TBME.1981.324751
[28] H. Nakazawa, Stochastic Burgers’ equation in the inviscid limit, Adv. in Appl. Math. 3 (1982), 18–42. · Zbl 0488.60072 · doi:10.1016/S0196-8858(82)80003-1
[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. · doi:10.1103/RevModPhys.61.185
[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. · Zbl 0755.60104 · doi:10.1007/BF02096551
[31] Ya. G. Sinai, Two results concerning asymptotic behavior of solutions of the Burgers equation with force, J. Stat. Phys. 64 (1991), 1–12. · Zbl 0978.35500 · doi:10.1007/BF01057866
[32] Ya. G. Sinai, Statistics of shocks in solutions of inviscid Burgers equation, Comm. Math. Phys. 148 (1992), 601–621. · Zbl 0755.60105 · doi:10.1007/BF02096550
[33] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. · Zbl 0508.35002
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