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On Nonlinear Stochastic Balance Laws

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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.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford, OX1 3LB, UK

    Gui-Qiang Chen

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Gui-Qiang Chen

  3. Department of Mathematics, Northwestern University, Evanston, IL, 60208-2730, USA

    Gui-Qiang Chen & Qian Ding

  4. Centre of Mathematics for Applications, University of Oslo, P. O. Box 1053, Blindern, 0316, Oslo, Norway

    Kenneth H. Karlsen

Authors
  1. Gui-Qiang Chen
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  2. Qian Ding
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  3. Kenneth H. Karlsen
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Correspondence to Gui-Qiang Chen.

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Communicated by C. Defermos

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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

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  • DOI: https://doi.org/10.1007/s00205-011-0489-9

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