Entropy solutions to the Dirichlet problem for nonlinear diffusion equations with conservative noise

Motivated by porous medium equations with randomly perturbed velocity field, this paper considers a class of nonlinear degenerate diffusion equations with nonlinear conservative noise in bounded domains. The existence, uniqueness and $L_1$-stability of non-negative entropy solutions under the homogeneous Dirichlet boundary condition are proved. The approach combines Kruzhkov’s doubling variables technique with a revised strong entropy condition that is automatically satisfied by the solutions of approximate equations.