A domain hemivariational inequality for 2D and 3D convective Brinkman-Forchheimer extended Darcy equations

TL;DR

This paper establishes existence, energy equality, and uniqueness of weak solutions for non-stationary 2D and 3D convective Brinkman-Forchheimer equations with hemivariational inequalities, extending results to Navier-Stokes systems.

Contribution

It introduces a regularized Galerkin scheme for hemivariational inequalities in CBFeD equations, extending to 3D Navier-Stokes and providing conditions for solution uniqueness.

Findings

Existence of weak solutions for all admissible exponents and viscosity.

Solutions satisfy energy equality under certain conditions.

Uniqueness holds for specified absorption exponents and parameters.

Abstract

This paper investigates domain hemivariational inequality problems arising from the non-stationary two- and three-dimensional convective Brinkman-Forchheimer extended Darcy (CBFeD) equations, which describe the flow of viscous incompressible fluids through saturated porous media in bounded domains. These equations may be regarded as generalized Navier-Stokes systems incorporating both damping and pumping mechanisms. For all admissible absorption exponents $r \ge 1 $ and effective viscosity $\mu > 0 $, the existence of weak solutions to the non-stationary 2D and 3D CBFeD equations with hemivariational inequalities is established via a regularized Galerkin approximation scheme, based on a suitable regularization of the Clarke subdifferential. A noteworthy aspect of the analysis is that the existence results extend to the three-dimensional non-stationary Navier-Stokes equations. Moreover, under appropriate conditions on the absorption exponent, specifically, $r \ge 1 $ in two dimensions and $ r \ge 3 $ in three dimensions, it is shown that weak solutions satisfy the energy equality. In addition, uniqueness of solutions is proved for $ r \ge 1$ in 2D and $r \ge 3$ in 3D, with the additional requirement $2\beta \mu > 1 $ in the critical case $r = 3 $.