A counterexample to the smoothness of the solution to an equation arising in fluid mechanics

TL;DR

This paper presents a counterexample demonstrating that the solution to a fluid mechanics equation derived from the Eulerian-Lagrangian framework may lose smoothness over time, challenging assumptions about long-term regularity.

Contribution

It provides the first counterexample showing the breakdown of smoothness in solutions to an Eulerian-Lagrangian fluid equation, and discusses implications for Navier-Stokes regularity.

Findings

Short-time smoothness of Constantin's fluid description is not guaranteed.

The inverse of the 'back to coordinates map' can become non-smooth over time.

A plausibility argument for Navier-Stokes global regularity is proposed.

Abstract

We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for the Navier-Stokes equation.