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Existence and uniqueness of the entropy solution of a stochastic conservation law with a \(Q\)-Brownian motion. (English) Zbl 1451.60065

Summary: 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 \(Q\)-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.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35L60 First-order nonlinear hyperbolic equations
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
60G65 Nonlinear processes (e.g., \(G\)-Brownian motion, \(G\)-Lévy processes)
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