Structure of Helicity and Global Solutions of Incompressible Navier-Stokes Equation

TL;DR

This paper introduces a new energy identity based on helicity for 3D incompressible Navier-Stokes equations, enabling the construction of smooth solutions with arbitrarily large critical norms.

Contribution

It derives a novel energy functional related to helicity that is critical and conditionally coercive, advancing understanding of Navier-Stokes solutions.

Findings

New energy identity for Navier-Stokes equations

Construction of smooth solutions with large critical norms

Insights into the structure of helicity in fluid dynamics

Abstract

In this paper we derive a new energy identity for the three-dimensional incompressible Navier-Stokes equations by a special structure of helicity. The new energy functional is critical with respect to the natural scalings of the Navier-Stokes equations. Moreover, it is conditionally coercive. As an application we construct a family of finite energy smooth solutions to the Navier-Stokes equations whose critical norms can be arbitrarily large.