{
    "byline": "Terence Tao",
    "charset": "UTF-8",
    "dir": null,
    "excerpt": "I\u2019ve just uploaded to the arXiv the paper \u201cFinite time blowup for an averaged three-dimensional Navier-Stokes equation\u201c, submitted to J. Amer. Math. Soc.. The main purpose of this paper is to formalise the \u201csupercriticality barrier\u201d for the global regularity problem for the Navier-Stokes equation, which roughly speaking asserts that it is not possible to establish global regularity by any \u201cabstract\u201d approach which only uses upper bound function space estimates on the nonlinear part of the equation, combined with the energy identity. This is done by constructing a modification of the Navier-Stokes equations with a nonlinearity that obeys essentially all of the function space estimates that the true Navier-Stokes nonlinearity does, and which also obeys the energy identity, but for which one can construct solutions that blow up in finite time. Results of this type had been previously established by Montgomery-Smith, Gallagher-Paicu, and Li-Sinai for variants of the Navier-Stokes equation without the energy identity, and by Katz-Pavlovic and by Cheskidov for dyadic analogues of the Navier-Stokes equations in five and higher dimensions that obeyed the energy identity (see also the work of Plechac and Sverak and of Hou and Lei that also suggest blowup for other Navier-Stokes type models obeying the energy identity in five and higher dimensions), but to my knowledge this is the first blowup result for a Navier-Stokes type equation in three dimensions that also obeys the energy identity. Intriguingly, the method of proof in fact hints at a possible route to establishing blowup for the true Navier-Stokes equations, which I am now increasingly inclined to believe is the case (albeit for a very small set of initial data).",
    "lang": null,
    "length": 12727,
    "publishedTime": null,
    "siteName": null,
    "title": "Finite time blowup for an averaged three-dimensional Navier-Stokes\u00a0equation"
}