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A weak solution of a stochastic nonlinear problem. (English) Zbl 1470.35446

Summary: We consider a problem modeling a porous medium with a random perturbation. This model occurs in many applications such as biology, medical sciences, oil exploitation, and chemical engineering. Many authors focused their study mostly on the deterministic case. The more classical one was due to Biot in the 50s, where he suggested to ignore everything that happens at the microscopic level, to apply the principles of the continuum mechanics at the macroscopic level. Here we consider a stochastic problem, that is, a problem with a random perturbation. First we prove a result on the existence and uniqueness of the solution, by making use of the weak formulation. Furthermore, we use a numerical scheme based on finite differences to present numerical results.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35D30 Weak solutions to PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76S05 Flows in porous media; filtration; seepage
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