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

TL;DR

This paper investigates the behavior of compressible Forchheimer-type fluids in rotating heterogeneous porous media, deriving new estimates for solutions of the governing nonlinear PDEs under complex boundary conditions.

Contribution

It introduces novel $L^eta$-estimates for solutions of nonlinear PDEs modeling Forchheimer flows in rotating media, utilizing advanced weighted Sobolev inequalities.

Findings

Established $L^eta$-estimates for solutions

Derived $L^ abla$-estimates without restrictions on data norms

Developed new weighted Sobolev inequalities

Abstract

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 $L^\alpha$-estimate for the solutions, for any positive number $\alpha$, 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.