An \(L^1\)-theory for scalar conservation laws with multiplicative noise on a periodic domain. (English) Zbl 1394.35262

Summary: We study the Cauchy problem for a multi-dimensional scalar conservation law with a multiplicative noise. Our aim is to give the well-posedness of an \(L^1\)-solution characterized by a kinetic formulation under appropriate assumptions. In particular, we focus on the existence of such a solution.


35L04 Initial-boundary value problems for first-order hyperbolic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35L65 Hyperbolic conservation laws
35R60 PDEs with randomness, stochastic partial differential equations
Full Text: Euclid


[1] G. Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal. 204 (2012), 707-743. · Zbl 1261.60062
[2] G. Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 20(2003), 645-668. · Zbl 1031.35077
[3] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., Vol. 44, Cambridge University Press, Cambridge, 1992. · Zbl 0761.60052
[4] A. Debussche, M. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: quasilinear case, arXiv: 1309.5817 [math. A8]. · Zbl 1346.60094
[5] A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259 (2010), 1040-1042. · Zbl 1200.60050
[6] A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, revised version, (2014), http://math.univ-lyon1.fr/ vovelle/DebusscheVovelleRevised.pdf. · Zbl 1200.60050
[7] A. Debussche and J. Vovelle, Invariant measure of scalar first-order conservation laws with stochastic forcing, arXiv: 1310.3779 [math.AP]. · Zbl 1331.60117
[8] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, 1965. · Zbl 0182.16101
[9] J. Feng and D. Nualart, Stochastic scalar conservation laws, J. Funct. Anal. 255 (2008), 313-373. · Zbl 1154.60052
[10] M. Hofmanová, Degenerate parabolic stochastic partial differential equations, Stoch. Pr. Appl. 123 (2013), 4294-4336. · Zbl 1291.60130 · doi:10.1016/j.spa.2013.06.015
[11] J. U. Kim, On a stochastic scalar conservation law, Indiana Univ. Math. J. 52 (2003), 227-256. · Zbl 1037.60064
[12] K. Kobayasi, A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions, J. Differential Equations 230 (2006), 682-701. · Zbl 1105.35004 · doi:10.1016/j.jde.2006.07.008
[13] K. Kobayasi and H. Ohwa, Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. Differential Equations 252 (2012), 137-167. · Zbl 1237.35099
[14] K. Kobayasi and D. Noboriguchi, A stochastic conservation law with nonhomogeneous Dirichlet boundary conditions, to appear in Acta Math. Vietnamica, arXiv: 1506.05758v1 [math-ph]. · Zbl 1364.35451
[15] S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81 (123) (1970) 228-255. · Zbl 0202.11203
[16] P. L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), 169-191. · Zbl 0820.35094
[17] J. Málek, J. Nečas, M. Rokyta and M. R, Weak and measure-valued solutions to evolutionary PDEs, Chapman and Hall, London, Weinheim, New York, 1996. · Zbl 0851.35002
[18] D. Noboriguchi, The equivalence Theorem of Kinetic Solutions and Entropy Solutions for Stochastic Scalar Conservation Laws, Tokyo J. Math. 38 (2015), 575-587. · Zbl 1383.35252
[19] B. Perthame, Kinetic Formulation of Conservation Laws, Oxford Lecture Ser. Math. Appl., Vol. 21, Oxford University Press, Oxford, 2002. · Zbl 1030.35002
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