A representation of solutions to a scalar conservation law in several dimensions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:405

Article

Albeverio, Sergio^1,2,3,4,5  Rozanova, Olga⁶ 

[1] Univ Bonn, Inst Angew Math, Abt Stochast, D-53115 Bonn, Germany

[2] HCM, Bonn, Germany

[3] IZKS, Bonn, Germany

[4] BiBoS, Bielefeld, Germany

[5] CERFIM, Locarno, Switzerland

[6] Moscow MV Lomonosov State Univ, Dept Mech & Math, Moscow 119991, Russia

关键词: Scalar conservation law;    The Cauchy problem;    Representation of solution;    Associated conservation laws;    Stochastic perturbation;   

DOI  :  10.1016/j.jmaa.2013.04.039

来源: Elsevier

【 摘 要 】

We find a representation of smooth solutions to the Cauchy problem for a scalar multidimensional conservation law as a small diffusion limit of a stochastic perturbation along characteristics. It helps, in particular, to study the process of singularity formation. Further, we introduce an associated system of balance laws that can be interpreted as describing the motion of a continuum with some specific pressure term. This term arises only after the instant when the solution to the initial Cauchy problem loses its smoothness. Before this instant the system coincides partly with the one known as pressure free gas dynamics. (c) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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