The Dirichlet Problem for Stochastic Degenerate Parabolic-Hyperbolic Equations

The Dirichlet Problem for Stochastic Degenerate Parabolic-Hyperbolic Equations

Year:    2022

Author:    Hermano Frid, Yachun Li, Daniel Marroquin, João F. C. Nariyoshi, Zirong Zeng

Communications in Mathematical Analysis and Applications, Vol. 1 (2022), Iss. 1 : pp. 1–71

Abstract

We consider the Dirichlet problem for a quasilinear degenerate parabolic stochastic partial differential equation with multiplicative noise and nonhomogeneous Dirichlet boundary condition. We introduce the definition of kinetic solution for this problem and prove existence and uniqueness of solutions. For the uniqueness of kinetic solutions we prove a new version of the doubling of variables method and use it to deduce a comparison principle between solutions. The proof requires a delicate analysis of the boundary values of the solutions for which we develop some techniques that enable the usage of the existence of normal weak traces for divergence measure fields in this stochastic setting. The existence of solutions, as usual, is obtained through a two-level approximation scheme consisting of nondegenerate regularizations of the equations which we show to be consistent with our definition of solutions. In particular, the regularity conditions that give meaning to the boundary values of the solutions are shown to be inherited by limits of nondegenerate parabolic approximations provided by the vanishing viscosity method.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cmaa.2021-0001

Communications in Mathematical Analysis and Applications, Vol. 1 (2022), Iss. 1 : pp. 1–71

Published online:    2022-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    71

Keywords:    Stochastic partial differential equations quasilinear degenerate parabolic equations divergence measure fields normal traces.

Author Details

Hermano Frid

Yachun Li

Daniel Marroquin

João F. C. Nariyoshi

Zirong Zeng