A time splitting approach to quasilinear Degenerate Parabolic Partial Differential Equations

TL;DR

This paper introduces a time splitting method for degenerate parabolic PDEs with noise, providing a simpler and more direct proof of the existence of kinetic solutions, improving upon previous approaches.

Contribution

The paper proposes a novel time splitting approximation scheme that converges strongly to a kinetic solution, offering an improved and more straightforward existence proof for degenerate parabolic equations.

Findings

The time splitting method converges strongly in L^1 to a kinetic solution.

The approach simplifies the proof of existence compared to previous methods.

The method provides an improvement over existing existence results.

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, Hofmanov\'a 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.