Optimal Control of the Navier-Stokes equations via Pressure Boundary Conditions

TL;DR

This paper investigates an optimal control problem for the Navier-Stokes equations with pressure boundary conditions, establishing well-posedness and regularity results despite challenges posed by boundary conditions.

Contribution

It introduces a novel approach to ensure well-posedness of the control problem using a specific tracking term and provides new regularity results for related Stokes problems.

Findings

Achieved well-posedness of the control problem with pressure boundary conditions.

Established $L^2(I;H^2(; ))$ regularity for solutions to a Stokes problem with mixed boundary conditions.

Demonstrated the effectiveness of the tracking term in handling boundary condition issues.

Abstract

In this work we study an optimal control problem subject to the instationary Navier-Stokes equations, where the control enters via an inhomogeneous Neumann/Do-Nothing boundary condition. Despite the Navier-Stokes equations with these boundary conditions not being well-posed for large times and/or data, we obtain wellposedness of the optimal control problem by choosing a proper tracking type term. In order to discuss the regularity of the optimal control, state and adjoint state, we present new results on $L^2(I;H^2(\Omega))$ regularity of solutions to a Stokes problem with mixed inhomogeneous boundary conditions.