Finite time blowup for an averaged three-dimensional Navier-Stokes equation
By Terence Tao. 2014
268 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/eafa4006e0906ce901d5ad4adbc17b619ab27944
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In Applied Mathematics
In Control and Systems Engineering
In Mathematical Physics
Citations in the corpus
- Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system (2014, result): “Our results are also relevant in view of the analysis in [Tao14] (see Remark 5.2 therein), since they confirm that the condition (1.7) is optimal, when general non–linear terms with the same scaling are considered.”