Weak Solutions and Optimal Control of Hemivariational Evolutionary Navier-Stokes Equations under Rauch Condition

Abstract EN

In this paper, we consider the 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 the Rauch condition, we use the Galerkin approximation method and a weak precompactness criterion 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.

At 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.