Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media

This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source term and boundary flux are integrated into the model. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressur...

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Bibliographic Details
Published in:Nonlinearity 2018-08, Vol.31 (8), p.3617-3650
Main Authors: Celik, Emine, Hoang, Luan, Kieu, Thinh
Format: Article
Language:English
Subjects:
compressible fluids
doubly nonlinear equation
Forchheimer flows
Moser iteration
nonlinear Robin condition
porous media
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Summary:This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source term and boundary flux are integrated into the model. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressure, and study its initial value problem subject to a general nonlinear Robin boundary condition. The growth rates in the source term and the boundary condition are arbitrarily large. The maximum of the solution, for positive time, is estimated in terms of certain Lebesgue norms of the initial and boundary data. The gradient estimates are obtained under a theoretical condition which, indeed, is relevant to the fluid flows in applications. In dealing with the complexity and generality of the equation and boundary condition, suitable trace theorems and Sobolev's inequalities are utilized, and a well-adapted Moser's iteration is implemented.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/aabf05