Large, global solutions to the three-dimensional the navier-stokes equations without vertical viscosity

TL;DR

This paper proves the existence of large initial data leading to unique, global solutions of the 3D Navier-Stokes equations without vertical viscosity, leveraging the structure of the nonlinear term and divergence-free condition.

Contribution

It demonstrates the existence of arbitrarily large initial data solutions for the 3D Navier-Stokes equations without vertical viscosity, a case lacking regularization in one direction.

Findings

Existence of global solutions for large initial data

Solutions are slowly varying in the viscosity-lacking direction

The nonlinear structure and divergence-free condition are key to the results

Abstract

The three-dimensional, homogeneous, incompressible Navier-Stokes equations are studied in the absence of viscosity in one direction. It is shown that there are arbitrarily large initial data generating a unique global solution, the main feature of which is that they are slowly varying in the direction where viscosity is missing. The difficulty arises from the complete absence of a regularising effect in this direction. The special structure of the nonlinear term, joint with the divergence-free condition on the velocity field, is crucial in obtaining the result.