Special Session 16: 

Existence and uniqueness results for a first order conservation law involving a Q-Brownian motion

Yueyuan Gao
MathAM-OIL, AIST and Tohoku University
Japan
Co-Author(s):    Tadahisa Funaki, Danielle Hilhorst
Abstract:
We consider a first order stochastic conservation law with a multiplicative source term involving a Q-Brownian motion. We first recall a result stating that the discrete solution obtained by a finite volume method converges along a subsequence in the sense of Young measures to a measure-valued entropy solution as the maximum diameter of the volume elements and the time step tend to zero [1]. This convergence result yields the existence of a measure-valued entropy solution. We then present the Kato's inequality and as a corollary we deduce the uniqueness of the measure-valued entropy solution as well as the uniqueness of the weak entropy solution. The Kato's inequality is proved by a doubling of variables method; in order to apply this method, we prove the existence and the uniqueness of the weak solution of an associated nonlinear parabolic problem [2]. This is joint work with Tadahisa Funaki and Danielle Hilhorst. [1] T. Funaki, Y. Gao and D. Hilhorst, Convergence of a finite volume scheme for a stochastic conservation law involving a Q-Brownian motion, Accepted for publication in DCDS B, AIMS, hal-01404119. [2] T. Funaki, Y. Gao and D. Hilhorst, Uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion, in preparation.