Streamlines concentration and application to the incompressible Navier-Stokes equations

TL;DR

This paper investigates the behavior of divergence-free vector fields near singularities in smooth domains, focusing on streamline geometry to exclude certain singularities in the context of incompressible Navier-Stokes equations.

Contribution

It introduces a novel approach analyzing streamline geometry to exclude specific isolated singularities for divergence-free vector fields.

Findings

Streamline geometry constrains possible singularities.

Certain singularities are excluded based on streamline analysis.

Results apply to incompressible Navier-Stokes equations.

Abstract

For a smooth domain $D$ containing the origin, we consider a vector field $u \in C^1(D\setminus\{0\},\mathbb{R}^3)$ with $\divg u \equiv 0$ and exclude certain types of possible isolated singularities at the origin, based on the geometry of streamlines that go near that possible singular point.