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I think I may have just solved a Millennium Problem. I found an exact 3D solution to Navier-Stokes equations that has a finite time singularity. The velocity, pressure, and force are all spatially periodic. The solution has a time singularity at t=T, where T is greater than zero and less than infinity.

I think it's correct, but I may have overlooked something. I don't have a Phd, but I'm not a complete noob. I have a Bachelor of Science in Mechanical Engineering. I posted it on vixra because I'm not endorsed on arxiv. Is the counterexample correct? thanks.

http://vixra.org/abs/1202.0014

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    $\begingroup$ I haven't read your paper in detail, but (assuming your computations are correct) it seems as though what you've shown is "there exist initial conditions $u_0(x,t)$ and forcing $f(x,t)$ such that there is a solution to the N-S equations with a finite-time singularity" whereas what the problem asks for is "There exist initial conditions $u_0(x,t)$ and forcing $f(x,t)$ for which there exist no solutions satisfying appropriate smoothness and finite energy conditions." That is, you've demonstrated the existence of one bad solution, instead of the non-existence of good solutions. $\endgroup$
    – Chris Taylor
    Commented Feb 10, 2012 at 8:34
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    $\begingroup$ As I've understood it it's already been proven by Jean Leray that solutions are unique and smooth for small time according to: ams.org/journals/tran/1971-157-00/S0002-9947-1971-0277929-7/… So doesn't that mean that the solution is unique and hence, there is no "good solution" for those initial conditions and forcing function? $\endgroup$
    – A.G.
    Commented Feb 10, 2012 at 12:36
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    $\begingroup$ Just send this paper to a reputable journal. $\endgroup$
    – Jon
    Commented Feb 10, 2012 at 14:06

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Nice try! But I think it likely there is no singularity at $t=T$ as you claim. The function $$g(x)=\arctan\left(\frac{\sqrt2}2 \tan x\right)$$ has derivative $$g'(x) = \frac{2\sqrt 2}{3+\cos^2x -\sin^2 x},$$ with no blowup at $x=\pi/2$ as you might think.

A result of Constantin and Fefferman from 1993 shows that if the direction of the vorticity remains smooth (Lipschitz is enough), then a Navier-Stokes solution cannot blow up. This also appears to rule out your solution as a counterexample.

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    $\begingroup$ Also, good sensible-looking write-up. Most submissions here that say: please check my arxiv/vixra paper, are clearly from cranks. This one is not. $\endgroup$
    – GEdgar
    Commented Feb 10, 2012 at 15:42

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