An Optimal Control Problem for the Navier--Stokes Equations with Point Sources

TL;DR

This paper investigates an optimal control problem for 2D Navier--Stokes equations with point source controls, addressing reduced regularity issues using weighted Sobolev spaces and deriving optimality conditions.

Contribution

It introduces a framework using Muckenhoupt weights to handle point source controls in Navier--Stokes equations, establishing existence and optimality conditions.

Findings

Existence of optimal solutions established.

First and second order optimality conditions derived.

Framework effectively manages reduced regularity from point sources.

Abstract

We analyze, in two dimensions, an optimal control problem for the Navier--Stokes equations where the control variable corresponds to the amplitude of forces modeled as point sources; control constraints are also considered. This particular setting leads to solutions to the state equation exhibiting reduced regularity properties. We operate under the framework of Muckenhoupt weights, Muckenhoupt-weighted Sobolev spaces, and the corresponding weighted norm inequalities and derive the existence of optimal solutions and first and, necessary and sufficient, second order optimality conditions.