Optimal control of the 2D constrained Navier-Stokes equations

TL;DR

This paper investigates the optimal control of 2D Navier-Stokes equations with energy conservation constraints, establishing existence, uniqueness, and optimality conditions using Galerkin approximation and Lagrange methods.

Contribution

It introduces a framework for constrained 2D Navier-Stokes control, proving existence, uniqueness, and deriving first-order optimality conditions.

Findings

Existence and uniqueness of global solutions for constrained 2D Navier-Stokes equations.

Lipschitz continuity and Fréchet differentiability of the solution map.

Derivation of first-order necessary optimality conditions using Lagrange multipliers.

Abstract

We study the 2D Navier-Stokes equations within the framework of a constraint that ensures energy conservation throughout the solution. By employing the Galerkin approximation method, we demonstrate the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on the torus $\mathbb{T}^2$. Moreover, we investigate the linearized system associated with the 2D-constrained Navier-Stokes equations, exploring its existence and uniqueness. Subsequently, we establish the Lipschitz continuity and Fr$\'{e}$chet differentiability properties of the solution mapping. Finally, employing the formal Lagrange method, we prove the first-order necessary optimality conditions.