Qualitative study of generalized Forchheimer flows with the flux boundary condition

Abstract

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.