Abstract
We are concerned with multidimensional stochastic balance laws. We identify a class of nonlinear balance laws for which uniform spatial BV bound for vanishing viscosity approximations can be achieved. Moreover, we establish temporal equicontinuity in L 1 of the approximations, uniformly in the viscosity coefficient. Using these estimates, we supply a multidimensional existence theory of stochastic entropy solutions. In addition, we establish an error estimate for the stochastic viscosity method, as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions. Various further generalizations of the results are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bouchut F., Perthame B.: Kružkov’s estimates for scalar conservation laws revisited. Trans. Am. Math. Soc 350, 2847–2870 (1998)
Chen G.-Q., Karlsen K.H.: Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Commun. Pure Appl. Anal 4, 241–266 (2005)
Cockburn B., Gripenberg G.: Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differ. Equ 151, 231–251 (1999)
Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3nd edn. Springer, Berlin (2010)
Debussche A., Vovelle J.: Scalar conservation laws with stochastic forcing. J. Funct. Anal 259, 1014–1042 (2010)
W E., Khanin K., Mazel A., Sinai Ya.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. 151, 877–960 (2000)
Feng J., Nualart D.: Stochastic scalar conservation laws. J. Funct. Anal 255, 313–373 (2008)
Holden H., Risebro N.H.: Conservation laws with a random source. Appl. Math. Optim 36, 229–241 (1997)
Karlsen K.H., Risebro N.H.: On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discret. Contin. Dyn. Syst 9, 1081–1104 (2003)
Kim J.U.: On a stochastic scalar conservation law. Indiana Univ. Math. J 52(1), 227–255 (2003)
Kruzkov S.: First order quasilinear equations in several independent variables. Math. USSR Sb 10, 217–243 (1972)
Kurtz, T.G., Protter, E.P.: Weak convergence of stochastic integrals and differential equations II: Infinite-dimensional case. In: CIME School in Probability, Lecture Notes in Math., vol. 1627, pp. 197–285, Springer, Berlin (1996)
Lions P.-L., Souganidis T.: Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math 326(9), 1085–1092 (1998)
Lions P.-L., Souganidis T.: Fully nonlinear stochastic partial differential equations: Nonsmooth equations and applications. C. R. Acad. Sci. Paris Sér. I Math 327(8), 735–741 (1998)
Lions P.-L., Souganidis T.: Fully nonlinear stochastic PDE with semilinear stochastic dependence. C. R. Acad. Sci. Paris Sér. I Math 331(8), 617–624 (2000)
Lions P.-L., Souganidis T.: Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 331(10), 783–790 (2000)
Lucier B.J.: A moving mesh numerical method for hyperbolic conservation laws. Math. Comp 46, 59–69 (1986)
Simon J.: Sobolev, Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl 157(4), 117–148 (1990)
Sinai, Ya. G.: Statistics of shocks in solutions of inviscid Burgers equations. Commun. Math. Phys. 148, 601–621 (1992)
Szepessy A.: An existence result for scalar conservation laws using measure valued solutions. Comm. Partial Differ. Equ. 14, 1329–1350 (1989)
Vallet G., Wittbold P.: On a stochastic first-order hyperbolic equation in a bounded domain. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12, 613–651 (2009)
Walsh, J.B.: An introduction to stochastic partial differential equations. École Dété de Probabilités de Saint-Flour, XIV–1984, Lecture Notes in Math., vol. 1180, pp. 265–439, Berlin, Springer (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Defermos
Rights and permissions
About this article
Cite this article
Chen, GQ., Ding, Q. & Karlsen, K.H. On Nonlinear Stochastic Balance Laws. Arch Rational Mech Anal 204, 707–743 (2012). https://doi.org/10.1007/s00205-011-0489-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-011-0489-9