Finite time blowup for an averaged three-dimensional Navier-Stokes equation

TL;DR

This paper constructs a smooth solution to an averaged Navier-Stokes equation that blows up in finite time, indicating that resolving the 3D Navier-Stokes regularity problem requires more refined analysis of the nonlinear structure.

Contribution

It introduces an averaged version of the Navier-Stokes equation and demonstrates finite-time blowup, highlighting the need for more detailed nonlinear analysis in the regularity problem.

Findings

Constructed a finite-time blowup solution for an averaged Navier-Stokes equation.

Showed that energy identities alone are insufficient to prevent blowup.

Proposed a program to extend these results to the actual Navier-Stokes equations.

Abstract

The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = \Delta u + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation property $\langle B(u,u), u\rangle=0$ (which is equivalent to the energy identity for the Navier-Stokes equation). In this paper, we consider a modification $\partial_t u = \Delta u + \tilde B(u,u)$ of this equation, where $\tilde B$ is an averaged version of the bilinear operator $B$ (where the average involves rotations and Fourier multipliers of order zero), and which also obeys the cancellation condition $\langle \tilde B(u,u), u \rangle = 0$ (so that it obeys the usual energy identity). By analysing a system of ODE related to (but more complicated than) a dyadic Navier-Stokes model of Katz and Pavlovic, we construct an example of a smooth solution to such a averaged Navier-Stokes equation which blows up in finite time. This demonstrates that any attempt to positively resolve the Navier-Stokes global regularity problem in three dimensions has to use finer structure on the nonlinear portion $B(u,u)$ of the equation than is provided by harmonic analysis estimates and the energy identity. We also propose a program for adapting these blowup results to the true Navier-Stokes equations.