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

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 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. KEYWORDS associated parabolic problem, existence and uniqueness of the entropy solution, Kato inequality, Q-Brownian motion, stochastic first-order conservation law MSC CLASSIFICATION 60H15; 35L60; 35A07 Several articles have been devoted to the study of stochastic perturbations of nonlinear first-order hyperbolic problems. Let us mention the article of Bauzet-Vallet-Wittbold 1 who prove the existence and uniqueness of a stochastic entropy Math Meth Appl

Convergence of a finite volume scheme for a stochastic conservation law involving a <inline-formula><tex-math id="M1">\begin{document}$Q$\end{document}</tex-math></inline-formula>-brownian motion

“…Remark 2. We remark that, if the flux function f is monotone, and F G (a, b) is the Godunov scheme, then the flux term in (15) coincides with the upwind scheme. Indeed, suppose that f is increasing, we use the definition of the Godunov flux (9) to deduce that F G (a, b) = f (a) for all a, b ∈ R. Thus the flux term in the scheme…”

“…In a forthcoming work [15], we will show that the measured-value entropy solution u is unique and coincides with the unique weak stochastic entropy solution; this will ensure that the whole approximate sequence {u T ,k } converges to the entropy solution u.…”