Time-dependent singularities in the Navier-Stokes system

TL;DR

This paper constructs solutions to the Navier-Stokes equations that are smooth everywhere except along a specified curve where they exhibit singularities, effectively modeling pointwise singularities along a moving curve.

Contribution

It introduces a method to generate Navier-Stokes solutions with prescribed singularities along a continuous curve in space-time.

Findings

Existence of solutions with singularities on a given curve

Solutions are smooth outside the singular curve

Distributional solutions incorporate singular forces on the curve

Abstract

We show that, for a given H\"older continuous curve in $\{(\gamma(t),t)\,:\, t>0\} \subset R^3\times R^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $R^3$ which is smooth outside this curve and singular on it. This is a pointwise solution of the system outside the curve, however, as a distributional solution on $R^3\times R^+$, it solves an analogous Navier-Stokes system with a singular force concentrated on the curve.