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

Abstract

The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = Δu + B(u,u)$, 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 $\partial_t u = Δu + \tilde B(u,u)$ 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.

Figures (9)

• Figure 1: The frequencies $\xi^0_1,\xi^0_2,\xi^0_3$. The frequency variables $\xi_j$ (and later on, the normalised frequencies $\tilde{\xi}_j$) will be localised to within $O(\epsilon_0^3)$ of $\xi_j^0$; this localisation is represented schematically in this figure by the circles around the reference frequencies $\xi_j^0$.
• Figure 2: The pump gate from the $x$ mode to the $y$ mode with coupling constant $\alpha$.
• Figure 3: The amplifier gate from the $x$ mode to the $y$ mode with coupling constant $\alpha$.
• Figure 4: The rotor gate that uses the $z$ mode to exchange energy between the $x$ and $y$ modes using the coupling constant $\alpha$.
• Figure 5: A circuit that creates a delayed, but abrupt, transition of energy from $a$ to $\tilde{a}$.

Theorems & Definitions (44)

• Conjecture 1.1: Navier-Stokes global regularity
• Conjecture 1.2: Navier-Stokes global regularity, again
• Lemma 1.3
• proof
• Remark 1.4
• Theorem 1.5: Finite time blowup for an averaged Navier-Stokes equation
• Remark 1.6
• Definition 3.1: Local cascade operators
• Theorem 3.2: Local cascade operators are averaged Euler operators
• Theorem 3.3: Blowup for a local cascade equation