# Slightly Compressible Forchheimer Flows in Rotating Porous Media

## TL;DR

This paper develops a mathematical model for slightly compressible fluid flows in rotating porous media, deriving a degenerate parabolic equation and analyzing its properties, including maximum principles and gradient estimates.

## Contribution

It introduces a generalized Forchheimer model for rotating porous media and provides detailed analysis of the resulting nonlinear parabolic equation, including maximum principles and explicit parameter dependence.

## Key findings

- Maximum principle established for the degenerate parabolic equation
- Explicit gradient estimates in Lebesgue norms derived
- Dependence of estimates on physical parameters like angular speed clarified

## 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.