Interior Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids

The generalized Forchheimer flows are studied for slightly compressible fluids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions are imposed on the degree of the Forchheimer polynomial. We derive, for all time, the interior {L^{\infty}}-estimates for the pressure, its gradient and time derivative, and the interior {L^{2}}-estimates for its Hessian. The De Giorgi and Ladyzhenskaya–Uraltseva iteration techniques are used taking into account the special structures of the equations for both pressure and its gradient. These are combined with the uniform Gronwall-type bounds in establishing the asymptotic estimates when time tends to infinity.