分野別セミナー

In this presentation, we study the well-posedness of a first order conservation law with a multiplicative source term involving a $Q$-Brownian motion. After having presented the definition of a measure-valued entropy solution of the stochastic conservation law, we briefly recall that the existence is proved by the convergence along a subsequence in the sense of Young measures of the discrete solution obtained by a finite volume method as the volume size and time step size tend to zero. The uniqueness of the measure-valued entropy solution is proved as a corollary of the Kato inequality. 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 parabolic problem; we also prove that the strong solution converges to a measure-valued entropy solution of the conservation law in a suitable sense. As a third part, we present numerical simulations for the first order Burgers equation on a one-dimensional torus forced by a stochastic source term. It is joint work with Tadahisa Funaki and Danielle Hilhorst.