Interior Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids

TL;DR

This paper develops interior estimates for pressure and its derivatives in generalized Forchheimer flows of slightly compressible fluids within porous media, using advanced iteration techniques and asymptotic analysis.

Contribution

It introduces new interior $L^ abla$-estimates for pressure and derivatives without restrictions on the Forchheimer polynomial degree, applying De Giorgi and Ladyzhenskaya-Uraltseva methods.

Findings

Established interior $L^ abla$-estimates for pressure and derivatives.

Derived asymptotic bounds as time approaches infinity.

Extended analysis to all degrees of Forchheimer polynomials.

Abstract

The generalized Forchheimer flows are studied for slightly compressible fluids 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 $L^\infty$-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 infinity.