Hemivariational inequality for stationary Navier-Stokes equations: existence, dependence and optimal control

TL;DR

This paper investigates the existence, dependence, and optimal control of solutions to hemivariational inequalities related to stationary Navier-Stokes equations, avoiding pseudo-monotone operator theory and using Rauch's assumption.

Contribution

It introduces a new approach to hemivariational inequalities for Navier-Stokes equations without pseudo-monotone operators, including novel dependence results and an optimal control framework.

Findings

Existence of solutions under Rauch's assumption.

Dependence of solutions on boundary conditions and external forces.

Existence of an optimal control for external force-based control problems.

Abstract

In this paper we study existence, dependence and optimal control results concerning solutions to a class of hemivariational inequalities for stationary Navier-Stokes equations but without making use of the theory of pseudo-monotone operators. To do so, we consider a classical assumption, due to J. Rauch, which constrains us to make a slight change on the definition of a solution. The Rauch assumption,, although insure the existence of a solution, does not allow the conclusion that the non-convex functional is locally Lipschitz. Moreover, two dependence results are proved, one with respect to changes of the boundary condition and the other with respect to the density of external forces. The later one will be used to prove the existence of an optimal control to the distributed parameter optimal control problem where the control is represented by the external forces.