Analysis of PDEs (Mathematics)
Finite time blowup for an averaged three-dimensional Navier-Stokes equation
Terence Tao constructs a finite-time blowup example for a modified 3D Navier-Stokes equation, revealing fundamental limits of current analytic tools for the Millennium Prize regularity problem.
Imagine water flowing through a pipe, and you want to know: could the water ever start moving infinitely fast at some point, breaking the rules of physics as we model them mathematically? The Navier-Stokes equations are the math rules that govern fluid motion, and nobody knows for certain whether following these rules could ever lead to this kind of 'blowup' in three-dimensional space. Tao created a slightly modified, simplified version of these equations - one that still obeys the same basic energy conservation rules as the real equations - and proved that HIS version can blow up in finite time. It's like building a simplified model airplane that crashes, to learn something about why real airplanes might crash, even though the model isn't exactly the same as the real thing. This tells mathematicians that just using the 'energy conservation' argument alone isn't enough to rule out blowup in the real equations - they need to find something extra, more specific about how fluids behave.
- 01
Constructs an averaged version of the Navier-Stokes bilinear operator that still satisfies the energy identity cancellation property
- 02
Analyzes a complex system of ODEs related to the dyadic Navier-Stokes model of Katz and Pavlovic
- 03
Constructs a smooth solution to the averaged Navier-Stokes equation that blows up in finite time
- 04
Demonstrates that resolving the true Navier-Stokes global regularity problem requires finer structural information beyond harmonic analysis estimates and the energy identity
- 05
Proposes a program for adapting the blowup construction to the true (non-averaged) Navier-Stokes equations
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.