# Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction

## TL;DR

This paper proves the global existence of strong solutions for the 3-D Navier-Stokes equations with anisotropic dissipation, under conditions where vertical or horizontal viscosity is sufficiently large relative to initial data.

## Contribution

It establishes global well-posedness for 3-D Navier-Stokes with anisotropic viscosity, considering different viscous coefficients in vertical and horizontal directions.

## Key findings

- Global strong solutions exist when vertical or horizontal viscosity is large enough.
- Anisotropic smallness condition on initial data ensures well-posedness.
- Results extend understanding of Navier-Stokes with directional dissipation.

## Abstract

We consider three dimensional incompressible Navier-Stokes equation $(NS)$ with different viscous coefficient in the vertical and horizontal variables. In particular, when one of these viscous coefficients is large enough compared to the initial data, we prove the global well-posedness of this system. In fact, we obtain the existence of a global strong solution to $(NS)$ when the initial data verify an anisotropic smallness condition which takes into account the different roles of the horizontal and vertical viscosity.