Uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion

Tadahisa Funaki (1) , Yueyuan Gao (2) , Danielle Hilhorst (3)

1 Waseda University [Tokyo, Japan]
2 MathAM-OIL
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

In this paper, we prove the 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 in turn coincides with its unique entropy solution.

Mots clés: Uniqueness of entropy solution; Kato inequality; Q-Brownian motion; Stochastic first order conservation law; Associated parabolic problem; -Brownian motion; Existence and uniqueness of the entropy solution

Domaines: Equations aux dérivées partielles [math.AP]

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion. 2020. ⟨hal-02159743v2⟩