TY - JOUR
T1 - The Dirichlet Problem for Stochastic Degenerate Parabolic-Hyperbolic Equations
AU - Frid , Hermano
AU - Li , Yachun
AU - Marroquin , Daniel
AU - Nariyoshi , João F. C.
AU - Zeng , Zirong
JO - Communications in Mathematical Analysis and Applications
VL - 1
SP - 1
EP - 71
PY - 2022
DA - 2022/01
SN - 1
DO - http://doi.org/10.4208/cmaa.2021-0001
UR - https://global-sci.org/intro/article_detail/cmaa/20157.html
KW - Stochastic partial differential equations, quasilinear degenerate parabolic
equations, divergence measure fields, normal traces.
AB - <p style="text-align: justify;">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.</p>