A variational approach to Navier-Stokes

TL;DR

This paper introduces a variational method using stabilized WIDE functionals to solve the incompressible Navier-Stokes equations, providing a new approach to obtain weak solutions through regularization and minimization.

Contribution

It proposes a novel variational framework with WIDE functionals for Navier-Stokes, enabling elliptic-in-time regularization and recovery of weak solutions.

Findings

WIDE functionals effectively regularize Navier-Stokes equations.

Minimization of these functionals yields weak solutions.

The approach bridges variational methods with classical weak solution theory.

Abstract

We present a variational resolution of the incompressible Navier-Stokes system by means of stabilized Weighted-Inertia-Dissipation-Energy (WIDE) functionals. The minimization of these parameter-dependent functionals corresponds to an elliptic-in-time regularization of the system. By passing to the limit in the regularization parameter along subsequences of WIDE minimizers one recovers a classical Leray-Hopf weak solution.