| 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. |
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