Did Prof Otelbaev Solve the Clay Mathematics Institute Problem?

analysisfluid dynamicspartial differential equations

Recently, mathematician Mukhtarbay Otelbaev published a paper Existence of a strong solution of the Navier-Stokes equations, in which he claim that he solved one of the Millennium Problems: existence and smoothness of the Navier-Stokes Equation. Unfortunately, the paper is in Russian but I cannot read Russian. There is a summary in English at the end of the paper, in which I found that the problem he proved was

Let $Q \equiv (0,2\pi)^3\subseteq \mathbb{R}^3$ be a 3-dim domain, $\Omega=(0,a)\times Q$, a>0.

$\textbf{Navier-stokes problem}$ is to find unknowns: a speed vector $u(t) = (u_1(t,x), u_2(t,x), u_3(t,x))$ and a scalar pressure function $p(t,x)$ at the points $x\in Q$ and time $t\in (0,a)$ satisfying the system of the equations

…

initial
$$ u(t,x)\vert_{t=0} =0$$

…

But the problem that stated by the Clay Mathematics Institute was, see http://www.claymath.org/sites/default/files/navierstokes.pdf,

$\textbf{(B) Existence and smoothness of Navier–Stokes solutions in $\mathbb{R}^3/\mathbb{Z}^3$.}$ … Let $u^0$ be any smooth, divergence-free vector field satisfying $u^0(x+e_j) = u^0(x)$; we take $f(x, t)$ to be identically zero. Then there exist smooth functions $p(x,t)$, $u_i(x,t)$ on $\mathbb{R}^3 \times[0,\infty)$. that …

So my question is:

1) Can the $a>0$ in his proof be $\infty$? Or is it enough to prove the problem for arbitrary $a>0$?

2) Is there any theory that can turn the problem with arbitrary initial value $u^0$ (of course satisfying some condition) into a problem with initial value being $0$?

3) The problem that formulated by the Clay Institute assumes $f\equiv 0$. Prof Otelbaev proved his result for all $f\in L_2(\Omega)$. Is this result much stronger?

Update: There is an article stating that the $L_2$ estimate is not enough to solve the problem. (in Spanish)

Best Answer

  1. Proving $\|\Delta u\|_{L^2(\Omega)} < \infty$ for any $a>0$ is sufficient to solve the Millennium problem. In fact $\|(-\Delta)^{3/4} u\|_{L^2(\Omega)} < \infty$ is enough. Also there is no problem if $C$ and $l$ depend upon $a$, as long as they are finite for every $a$.
  2. Setting $\nu = 1$ is not a big deal - this is easily overcome by putting the equation into dimensionless units. (Since the units of length are fixed, you can rescale the velocity and time units, but that is enough degrees of freedom to get the Reynold's number equal to 1.)
  3. The assumption $u(x,0) = 0$ - this one I am not sure about. I don't immediately see an argument that you can deduce the $u(x,0) \ne 0$, $f \equiv 0$ case from the $u(x,0) = 0$, $f \in L_2(\Omega)$ case. Even if such an argument isn't available, this paper, if correct, still seems to be an enormous advance.

Update: Actually 3 is also not an issue. See user121805's answer.

So if this paper is correct, it solves the Clay Millennium problem.

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