Advances in Differential Equations

Qualitative study of generalized Forchheimer flows with the flux boundary condition

Luan Hoang and Akif Ibragimov

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This article is focused on qualitative properties of solutions to generalized Forchheimer equations for slightly compressible fluids in porous media subject to the flux condition on the boundary. The pressure and pressure gradient are proved to depend continuously on the boundary flux 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 obtained. 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 inequalities. Also, additional flux-related quantities are introduced as controlling parameters of fluid flows for large time in the case of unbounded fluxes.

Article information

Adv. Differential Equations Volume 17, Number 5/6 (2012), 511-556.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B35: Stability 35B40: Asymptotic behavior of solutions 35K20: Initial-boundary value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations 35Q35: PDEs in connection with fluid mechanics 35Q86: PDEs in connection with geophysics


Hoang, Luan; Ibragimov, Akif. Qualitative study of generalized Forchheimer flows with the flux boundary condition. Adv. Differential Equations 17 (2012), no. 5/6, 511--556.

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