Weak solutions and optimal control of hemivariational evolutionary Navier-Stokes equations under Rauch condition

TL;DR

This paper investigates weak solutions and optimal control for evolutionary Navier-Stokes equations with nonslip boundary conditions and Clarke subdifferential relations, using Galerkin approximation and stability analysis under Rauch and directional growth conditions.

Contribution

It introduces a novel approach to analyze weak solutions and control problems for hemivariational Navier-Stokes equations under Rauch and Naniewicz conditions.

Findings

Established convergence of Galerkin approximations to weak solutions.

Proved stability of the control system with respect to external forces.

Reexamined results under more general directional growth conditions.

Abstract

In this paper we consider evolutionary Navier-Stokes equations subject to the nonslip boundary condition together with a Clarke subdifferential relation between the dynamic pressure and the normal component of the velocity. Under Rauch condition, we use the Galerkin approximation method and a weak precompactness criteria to ensure the convergence to a desired solution. Moreover a control problem associated with such system of equations is studied with the help of a stability result with respect to the external forces. In the end of this paper, a more general condition due to Z. Naniewicz, namely the directional growth condition, is considered and all the results are reexamined.