Interior Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids

The generalized Forchheimer ows are studied for slightly compressible uids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions on the degree of the Forchheimer polynomial are imposed. We derive, for all time, the interior L 1 -estimates for the pressure and its partial derivatives, 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 innity.