A geometric condition implying energy equality for solutions of 3D Navier-Stokes equation

TL;DR

This paper establishes a geometric condition involving local regularity near a Hölder continuous curve that guarantees energy equality for certain weak solutions of the 3D Navier-Stokes equations, advancing understanding of energy conservation.

Contribution

It introduces a new geometric criterion based on local Hölder continuity that ensures energy equality for weak solutions of the 3D Navier-Stokes equations.

Findings

Weak solutions in specified function spaces satisfy energy equality near a Hölder continuous curve.

Solutions with the geometric condition are shown to be suitable.

The result links local regularity to global energy conservation in fluid dynamics.

Abstract

We prove that every weak solution $u$ to the 3D Navier-Stokes equation that belongs to the class $L^3L^{9/2}$ and $\n u$ belongs to $L^3L^{9/5}$ localy away from a 1/2-H\"{o}lder continuous curve in time satisfies the generalized energy equality. In particular every such solution is suitable.