Analysis and Optimal Control of a System of Hemivariational Inequalities Arising in Non-Stationary Navier-Stokes Equation with Thermal Effects

Abstract

In this paper, we primarily investigate the existence, dependence and optimal control results related to solutions for a system of hemivariational inequalities pertaining to a non-stationary Navier-Stokes equation coupled with an evolution equation of temperature field. The boundary conditions for both the velocity field and temperature field incorporate the generalized Clarke gradient. The existence and uniqueness of the weak solution are established by utilizing the Banach fixed point theorem in conjunction with certain results pertaining to hemivariational inequalities. The finite element method is used to discretize the system of hemivariational inequalities and error bounds are derived. Ultimately, a result confirming the existence of a solution to an optimal control problem for the system of hemivariational inequalities is elucidated.