Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst

Research output: Contribution to journalArticle

Abstract

In this paper, we prove the existence and uniqueness of the entropy solution for a first-order stochastic conservation law with a multiplicative source term involving a (Formula presented.) -Brownian motion. After having defined a measure-valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure-valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure-valued entropy solution of the stochastic conservation law, which proves the existence of the measure-valued entropy solution.

Original languageEnglish
Pages (from-to)5860-5886
Number of pages27
JournalMathematical Methods in the Applied Sciences
Volume43
Issue number9
DOIs
Publication statusPublished - 2020 Jun 1

Keywords

  • Kato inequality
  • Q-Brownian motion
  • associated parabolic problem
  • existence and uniqueness of the entropy solution
  • stochastic first-order conservation law

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

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