On compressible fluid flows of Forchheimer-type in rotating heterogeneous porous media

We study the dynamics of compressible fluids in rotating heterogeneous porous media. The fluid flow is of {F}orchheimer-type and is subject to a mixed mass and volumetric flux boundary condition. The governing equations are reduced to a nonlinear partial differential equation for the pseudo-pressure. This parabolic-typed equation can be degenerate and/or singular in the spatial variables, the unknown and its gradient. We establish the $L^α$-estimate for the solutions, for any positive number $α$, in terms of the initial and boundary data and the angular speed of rotation. It requires new elliptic and parabolic Sobolev inequalities and trace theorem with multiple weights that are suitable to the nonlinear structure of the equation. The $L^\infty$-estimate is then obtained without imposing any conditions on the $L^\infty$-norms of the weights and the initial and boundary data.

💡 Research Summary

This paper investigates the dynamics of compressible fluids flowing through rotating heterogeneous porous media, where the flow obeys a Forchheimer-type (non‑Darcy) law. The authors begin by formulating the governing equations in a rotating reference frame, incorporating Coriolis and centrifugal forces, and allowing the permeability and porosity to vary spatially. Two classes of compressible fluids are considered: (i) isentropic gases with the equation of state (p=c\rho^\gamma) and (ii) slightly compressible fluids described by ( \frac{1}{\rho}\frac{d\rho}{dp}= \varpi). In both cases the authors introduce a pseudo‑pressure (u) such that (\rho\nabla p = \nabla u) and (\rho = \bar c, u^{\lambda}) with (\lambda\in(0,1]).

By eliminating the velocity from the momentum balance using a nonlinear operator (F_{x,z}) (which combines the Forchheimer resistance and the rotational terms) and invoking the invertibility of (F_{x,z}), the authors derive a single nonlinear parabolic equation for (u): \