Well-posedness of a boundary hemivariational inequality for stationary and non-stationary 2D and 3D convective Brinkman-Forchheimer equations

Abstract

This paper investigates boundary hemivariational inequality problems associated with both stationary and non-stationary two and three-dimensional convective Brinkman-Forchheimer equations (or Navier-stokes equations with damping), which model the flow of viscous incompressible fluids through saturated porous media. The governing equations are nonlinear in both velocity and pressure and are subject to nonstandard boundary conditions. Specifically, we impose the no-slip condition along with a Clarke subdifferential relation between pressure and the normal velocity components. For the stationary case, we establish the existence and uniqueness of weak solutions using a surjectivity theorem for pseudomonotone operators. The existence of weak solutions to the non-stationary hemivariational inequality is established via a limiting process applied to a temporally semi-discrete scheme, where the time derivative is approximated using the backward Euler method-commonly referred to as the Rothe method. It is demonstrated that the discrete problem admits solutions, which possess a weakly convergent subsequence as the time step tends to zero, and that any such weak limit satisfies the original hemivariational inequality. A novel outcome of this paper is that the existence results obtained in this work is applicable to 3D non-stationary Navier-Stokes equations also. Moreover, under appropriate conditions on the absorption exponent, we show that Leray-Hopf weak solutions satisfies the energy equality, the solution is shown to be unique and to depend continuously on the given data.