Abstract
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 are imposed on the degree of the Forchheimer polynomial. We derive, for all time, the interior L ∞ {L^{\infty}} -estimates for the pressure, its gradient and time derivative, and the interior L 2 {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.
Highlights
We study the generalized Forchheimer flows for slightly compressible fluids in porous media
The generalized Forchheimer equations studied in [1, 11,12,13,14] are of the the form: g(|v|)v = −∇p, (2.1)
On the right-hand side of (2.7), the constant κ is very large for most slightly compressible fluids in porous media [16], we neglect its second term and by scaling the time variable, we study the reduced equation
Summary
We study the generalized Forchheimer flows for slightly compressible fluids in porous media. Among a small number of papers on these flows, recent work [14] is focused on studying the pressure and its time derivative in space L∞, the pressure gradient in Ls for s ∈ [1, ∞) and the pressure’s Hessian in L2−δ for δ ∈ (0, 1). It requires the so-called Strict Degree Condition (SDC), that is, the degree of the Forchheimer polynomial is less than 4/(n − 2), where n is the spatial dimension. The issues of the solutions’ estimates for on the entire domain as well as their continuous dependence on the data will be addressed in our future works
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