2D Constrained Navier-Stokes Equations

TL;DR

This paper investigates 2D Navier-Stokes equations with an energy constraint, establishing existence, uniqueness, and convergence to Euler solutions as viscosity approaches zero.

Contribution

It introduces a novel constrained Navier-Stokes model and proves key properties including global existence, uniqueness, and the zero-viscosity limit.

Findings

Existence and uniqueness of global solutions for the constrained model.

Convergence of constrained Navier-Stokes solutions to Euler solutions as viscosity vanishes.

Applicable on both ^2 and torus domains.

Abstract

We study 2D Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\R^2$ and $\T$, by a fixed point argument. We also show that the solution of constrained Navier-Stokes converges to the solution of Euler equation as viscosity $\nu$ vanishes.