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

Emine Celik, Luan Hoang, Thinh Kieu

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)3617-3650
Number of pages34
JournalNonlinearity
Volume31
Issue number8
DOIs
StatePublished - Jun 27 2018

Keywords

  • Forchheimer flows
  • Moser iteration
  • compressible fluids
  • doubly nonlinear equation
  • nonlinear Robin condition
  • porous media

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