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Slightly Compressible Forchheimer Flows in Rotating Porous Media

Emine Celik, Luan Hoang, Thinh Kieu

Abstract

We formulate the the generalized Forchheimer equations for the three-dimensional fluid flows in rotating porous media. By implicitly solving the momentum in terms of the pressure's gradient, we derive a degenerate parabolic equation for the density in the case of slightly compressible fluids and study its corresponding initial, boundary value problem. We investigate the nonlinear structure of the parabolic equation. The maximum principle is proved and used to obtain the maximum estimates for the solution. Various estimates are established for the solution's gradient, in the Lebesgue norms of any order, in terms of the initial and boundary data. All estimates contain explicit dependence on key physical parameters including the angular speed.

Slightly Compressible Forchheimer Flows in Rotating Porous Media

Abstract

We formulate the the generalized Forchheimer equations for the three-dimensional fluid flows in rotating porous media. By implicitly solving the momentum in terms of the pressure's gradient, we derive a degenerate parabolic equation for the density in the case of slightly compressible fluids and study its corresponding initial, boundary value problem. We investigate the nonlinear structure of the parabolic equation. The maximum principle is proved and used to obtain the maximum estimates for the solution. Various estimates are established for the solution's gradient, in the Lebesgue norms of any order, in terms of the initial and boundary data. All estimates contain explicit dependence on key physical parameters including the angular speed.

Paper Structure

This paper contains 11 sections, 22 theorems, 337 equations.

Table of Contents

  1. Introduction and formulation of the problem
  2. Preliminaries
  3. Notation and elementary inequalities
  4. Properties of the function $X$
  5. Maximum estimates
  6. Preparations for the gradient estimates
  7. Gradient estimates (I)
  8. Gradient estimates (II)
  9. Estimates for the $L_{x,t}^{4-a}$-norm
  10. Estimates for higher $L_{x,t}^s$-norms
  11. Gradient estimates (III)

Key Result

Lemma \oldthetheorem

Given any vector $k\in\mathbb R^3$, the function $F_0(v)\stackrel{\rm def}{=} g(|v|)v + k\times v$ is a bijection from $\mathbb R^3$ to $\mathbb R^3$.

Theorems & Definitions (51)

  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 41 more