Funaki, Tadahisa; Gao, Yueyuan; Hilhorst, Danielle Existence and uniqueness of the entropy solution of a stochastic conservation law with a \(Q\)-Brownian motion. (English) Zbl 1451.60065 Math. Methods Appl. Sci. 43, No. 9, 5860-5886 (2020). 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. Cited in 1 Document 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) Keywords:associated parabolic problem; existence and uniqueness of the entropy solution; Kato inequality; \(Q\)-Brownian motion; stochastic first-order conservation law PDFBibTeX XMLCite \textit{T. Funaki} et al., Math. Methods Appl. Sci. 43, No. 9, 5860--5886 (2020; Zbl 1451.60065) Full Text: DOI