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Motivated by porous medium equations with randomly perturbed velocity field,this paper considers a class of nonlinear degenerate diffusion equations withnonlinear conservative noise in bounded domains. The existence, uniqueness and$L_{1}$-stability of non-negative entropy solutions under the homogeneousDirichlet boundary condition are proved. The approach combines Kruzhkov'sdoubling variables technique with a revised strong entropy condition that isautomatically satisfied by the solutions of approximate equations.

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