We study the Cauchy problem for a multi-dimensional scalar conservation law with a multiplicative noise. Our aim is to give the well-posedness of an $L^1$-solution characterized by a kinetic formulation under appropriate assumptions. In particular, we focus on the existence of such a solution.
The author has been supported by Waseda University Grant for Special Research Projects (No. 2015S-042).
"An $L^1$-theory for scalar conservation laws with multiplicative noise on a periodic domain." Nihonkai Math. J. 28 (1) 43 - 53, 2017.