Qualitative study of generalized forchheimer flows with the flux boundary condition

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Abstract

This article is focused on qualitative properties of solutions to generalized Forchheimer equations for slightly compressible uids in porous media subject to the ux condition on the boundary. The pres- sure and pressure gradient are proved to depend continuously on the boundary ux and coefficients of the Forchheimer polynomial in the momentum equation. In particular, the asymptotic dependence of the shifted solution on the asymptotic behavior of the boundary data is ob- tained. In order to improve various a priori estimates for the pressure, its gradient and time derivative, we prove and utilize suitable Poincaré- Sobolev and nonlinear Gronwall inequalities, as well as obtain uniform Gronwall-type inequalities from a system of coupled differential inequali- ties. Also, additional ux-related quantities are introduced as controlling parameters of uid ows for large time in the case of unbounded uxes.

Original languageEnglish
Pages (from-to)511-556
Number of pages46
JournalAdvances in Differential Equations
Volume17
Issue number5-6
StatePublished - 2012

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