Bounded solutions to the axially symmetric Navier Stokes equation in a cusp region

TL;DR

This paper proves that in a specific 3D domain touching the axis, axially symmetric Navier-Stokes solutions with slip boundary conditions remain bounded over time, addressing a key regularity problem beyond two dimensions.

Contribution

It establishes the first known case where the Navier-Stokes regularity problem is solved in a three-dimensional setting with a cusp domain.

Findings

Solutions remain bounded for all time in the specified domain.

No finite-time blow-up occurs for the solutions.

The result applies to initial data in a $C^2$ class.

Abstract

A domain in $\mathbb{R}^3$ that touches the $x_3$ axis at one point is found with the following property. For any initial value in a $C^2$ class, the axially symmetric Navier Stokes equations with Navier slip boundary condition has a finite energy solution that stays bounded for any given time, i.e. no finite time blow up of the fluid velocity occurs. The result seems to be the first case where the Navier-Stokes regularity problem is solved beyond dimension 2.