Stochastic system for generalized polytropic filtration |
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Authors: | Ali Zakaria |
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Affiliation: | Department of Mathematical Sciences, University of South Africa, Johannesburg, 0003 Florida, South Africa |
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Abstract: | In this paper, we study a certain class of stochastic quasilinear parabolic equations describing a generalized polytropic elastic filtration in the framework of variable exponents Lebesgue and Sobolev spaces. We establish an existence result in the infinite dimensional framework of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions, and the governing equations are subjected to cylindrical Wiener processes. We use a Galerkin method, derive crucial a priori estimates for the approximate solutions, and combine profound analytic and probabilistic compactness results in order to pass to the limit. Several difficulties arise in obtaining these uniform bounds and passing to the limit since the nonlinear elliptic part of the leading operator admits nonstandard growth. Apart from adapting the above essential tools, we extend classical methods of monotonicity to the present situation. |
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Keywords: | generalized weak solution monotone method non-Newtonian polytropic filtration stochastic partial differential equations stochastic systems |
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