A time-splitting approach to quasilinear degenerate parabolic stochastic partial differential equations

Abstract

In this paper, we discuss the Cauchy problem for a degenerate parabolic-hyperbolic equation with a multiplicative noise. We focus on the existence of a solution. Using nondegenerate smooth approximations, Debussche, Hofmanová and Vovelle [8] proved the existence of a kinetic solution. On the other hand, we propose to construct a sequence of approximations by applying a time splitting method and prove that this converges strongly in $L^1$ to a kinetic solution. This method will somewhat give us not only a simpler and more direct argument but an improvement over the existence result.