Nihonkai Mathematical Journal

An $L^1$-theory for scalar conservation laws with multiplicative noise on a periodic domain

Dai Noboriguchi

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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.


The author has been supported by Waseda University Grant for Special Research Projects (No. 2015S-042).

Article information

Nihonkai Math. J., Volume 28, Number 1 (2017), 43-53.

Received: 29 December 2015
Revised: 2 July 2016
First available in Project Euclid: 7 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L04: Initial-boundary value problems for first-order hyperbolic equations
Secondary: 60H15: Stochastic partial differential equations [See also 35R60]

stochastic partial differential equations conservation laws kinetic formulation initial value problem


Noboriguchi, Dai. An $L^1$-theory for scalar conservation laws with multiplicative noise on a periodic domain. Nihonkai Math. J. 28 (2017), no. 1, 43--53.

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  • G. Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal. 204 (2012), 707–743.
  • 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.
  • G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., Vol. 44, Cambridge University Press, Cambridge, 1992.
  • A. Debussche, M. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: quasilinear case, arXiv: 1309.5817 [math. A8].
  • A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259 (2010), 1040–1042.
  • A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, revised version, (2014),
  • A. Debussche and J. Vovelle, Invariant measure of scalar first-order conservation laws with stochastic forcing, arXiv: 1310.3779 [math.AP].
  • R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, 1965.
  • J. Feng and D. Nualart, Stochastic scalar conservation laws, J. Funct. Anal. 255 (2008), 313–373.
  • M. Hofmanová, Degenerate parabolic stochastic partial differential equations, Stoch. Pr. Appl. 123 (2013), 4294–4336.
  • J. U. Kim, On a stochastic scalar conservation law, Indiana Univ. Math. J. 52 (2003), 227–256.
  • K. Kobayasi, A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions, J. Differential Equations 230 (2006), 682–701.
  • 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.
  • 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].
  • S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81 (123) (1970) 228-255.
  • 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.
  • J. Málek, J. Nečas, M. Rokyta and M. R\ružička, Weak and measure-valued solutions to evolutionary PDEs, Chapman and Hall, London, Weinheim, New York, 1996.
  • D. Noboriguchi, The equivalence Theorem of Kinetic Solutions and Entropy Solutions for Stochastic Scalar Conservation Laws, Tokyo J. Math. 38 (2015), 575–587.
  • B. Perthame, Kinetic Formulation of Conservation Laws, Oxford Lecture Ser. Math. Appl., Vol. 21, Oxford University Press, Oxford, 2002.