Global well-posedness of three-dimensional Navier-Stokes equations with partial viscosity under helical symmetry

TL;DR

This paper proves the global existence and uniqueness of solutions for 3D Navier-Stokes equations with partial viscosity under helical symmetry, highlighting the crucial role of horizontal viscosity in fluid dynamics.

Contribution

It demonstrates the global well-posedness of helical flows with horizontal viscosity, extending previous results to a more general symmetric setting.

Findings

Global existence and uniqueness of weak and strong solutions

Horizontal viscosity is essential for well-posedness

Generalizes previous results on viscous helical fluids

Abstract

In this paper, we investigate the global well-posedness of three-dimensional Navier-Stokes equations with horizontal viscosity under a special symmetric structure: helical symmetry. More precisely, by a revised Ladyzhenskaya-type inequality and utilizing the behavior of helical flow, we prove the global existence and uniqueness of weak and strong solution to the three-dimensional helical flows. Our result reveals that for the issue of global well-posedness of the viscous helical fluids, the horizontal viscosity plays the important role. To some extent, our work can be seen as a generalization of the result by Mahalov-Titi-Leibovich [Arch. Ration. Mech. Anal. 112 (1990), no. 3, 193-222].