Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? [closed]

Solution 1:

http://francis.naukas.com/2014/01/18/la-demostracion-de-otelbaev-del-problema-del-milenio-de-navier-stokes/

This is the proposed counterexample to Theorem 6.1 given at http://dxdy.ru/topic80156-60.html. I used google translate, and then cleaned it up. I also added details here and there.

Let $\hat H = \ell_2$.

Let the operator $A$ be defined by $ Ae_i = e_i $ for $ i < 50$, $ Ae_i = ie_i $ for $ i \ge $ 50

Define the bilinear operator $ L $ to be nonzero only on two-dimensional subspaces $ L (e_ {2n}, e_ {2n +1}) = \frac1n (e_ {2n} + e_ {2n +1}) $, with $ n \ge 25 $.

Check conditions:

U3. Even with a margin of 50 .

U2. $ (e_i, L (e_i, e_i)) = 0 $ for $ i \ge $ 50. This is also true for eigenvectors $ u $ with $ \lambda = 1 $, because for them $ L (u, u) = 0 $.

U4. $ L (e, u) = 0 $ for the eigenvectors $ e $ with $ \lambda = 1 $ also trivial. (Stephen's note: he also needs to check $L_e^*u = L_u^*e = 0$, but that looks correct to me.)

U1. $ (Ax, x) \ge (x, x) $. Also $$ \| L (u, v) \| ^ 2 = \sum_{n \ge 25} u ^ 2_ {2n} v ^ 2_ {2n +1} / n ^ 2 \le C\|(u_n/\sqrt n)\|_4^2 \|(v_n/\sqrt n)\|_4^2 \le C\|(u_n/\sqrt n)\|_2^2 \|(v_n/\sqrt n)\|_2^2 = C \left (\sum u ^ 2_ {n} / n \right) \left (\sum v ^ 2_ {n} / n \right) $$ so we can take $\beta = -1/2$.

And now consider the elements $ u_n =-n (e_ {2n} + e_ {2n +1}) $. Their norms are obviously rising. Let $ \theta = -1 $. Then the $A^\theta $-norms of all these elements are constant. But, $ f_n = u_n + L (u_n, u_n) = 0 $.

Update: Later on in http://dxdy.ru/topic80156-90.html there is a response relayed from Otelbaev in which he asserts he can fix the counterexample by adding another hypothesis to Theorem 6.1, namely the existence of operators $P_N$ converging strongly to the identity such that one has good solvability properties for $u + P_N L(P_N u,P_N u) = f$, in that if $\| f \|$ is small enough then $\| u \|$ is also small.

Terry Tao communicated to me that he thinks a small modification of the counterexample also defeats this additional hypothesis.

Update 2: Terry Tao modified his example to correct for that fact that the statement of Theorem 6.1 is that a bound on $u \equiv \overset0u$ implies a lower bound on $f \equiv \overset0f$ rather than the other way around (i.e. we had a translation error for Theorem 6.1 that I point out above).

Let $\hat H$ be $N$-dimensional Euclidean space, with $N \ge 50$. Let $\theta = -1$ and $\beta = -1/100$. Take $$ A e_n = \begin{cases} e_n & \text{for $n<50$} \\ 50\ 2^{n-50} e_n & \text{for $50 \le n \le N$.}\end{cases}$$ and $$L(e_n, e_n) = - 2^{-(n-1)/2} e_{n+1} \quad\text{for $50 \le n < N$,}$$ and all other $L(e_i,e_j)$ zero.

Axioms (Y.2) and (Y.4) are easily verified. For (Y.1), observe that if $u = \sum_n c_n e_n$ and $v = \sum_n d_n e_n$, then for a universal constant $C$, we have $$ \| L(u,v) \|^2 \le C \sum_n 2^{-n} c_n^2 d_n^2, \\ |c_n| \le C 2^{n/100} \| A^\beta u \| ,\\ |d_n| \le C 2^{n/100} \| A^\beta v \| ,$$ and the claim (Y.1) follows from summing geometric series.

Finally, set $$ u = \sum_{n=50}^N 2^{n/2} e_n $$ then one calculates that $$ \| A^\theta u \| < C $$ for an absolute constant C, and $$ u + L(u,u) = 2^{50/2} e_{50} $$ so $$ \| u + L(u,u) \| \le C $$ but that $$ \| u \| \ge 2^{N/2}. $$ Since $N$ is arbitrary, this gives a counterexample to Theorem 6.1.

By writing the equation $u+L(u,u)=f$ in coordinates we obtain $f_n = u_n$ for $n \le 50$, and $f_n = u_n + 2^{-n/2} u_{n-1}^2$ if $50<n\le N$. Hence we see that $u$ is uniquely determined by $f$. From the inverse function theorem we see that if $\| f \|$ is sufficiently small then $\| u \| < 1/2$, so the additional axiom Otelbaev gives to try to fix Theorem 6.1 is also obeyed (setting $P_N$ to be the identity).

Update 3: on Feb 14, 2014, Professor Otelbaev sent me this message, which I am posting with his permission:

Dear Prof. Montgomery-Smith,

To my shame, on the page 56 the inequality (6.34) is incorrect therefore the proposition 6.3 (p. 54) isn't proved. I am so sorry.

Thanks for goodwill.

Defects I hope to correct in English version of the article.

Solution 2:

I have started to translate the paper so that English speakers can explore it. I've only had time for the abstract, introduction, and main result statement, but that already gives an important part of the picture. Any further contributions are welcome. https://github.com/myw/navier_stokes_translate

Solution 3:

OK, I spent an afternoon getting help with Russian, and I think I understand a lot more.

So first he actually proves a rather abstract theorem (Theorem 2), and strong solutions of the Navier-Stokes is merely a corollary. He shows the existence of solutions satisfying certain bounds to $$ \dot u + Au + B(u,u) = f , \quad u(0) = 0,$$ where $A$ and $B$ satisfy rather mild hypotheses that, for example, the replacements $A = E-\Delta$, and $B(u,v) = e^t u \cdot \nabla v + \nabla p$ where $p$ is a scalar chosen so that $B(u,v)$ is divergence free.

(I always thought the proof or counterexample would use the special structure of $B(u,v)$ that comes with the Navier-Stokes equation.)

In Chapter 5, he outlines how he will turn it into a different abstract problem, explaining that it is sufficient to find a bound on $\overset{0}{v} = \dot u + Au$. He constructs an equation for a quantity $v(\xi) \equiv v(\xi,t,x)$, so that in effect it is a time dependent velocity field described by a parameter $\xi$. He creates a differential equation in $\xi$, which morphs $v(0) = \overset 0v$ into $v(\xi_1)$, where $\|v(\xi)\| = \|\overset0v\|$, but $v(\xi_1)$ is easier to work with. This equation is given by equations (5.2) and (5.3).

So far, the only part of equation (5.3) that I am beginning to understand is the $-\alpha(\xi) R(v(\xi))$ part. $R(v)$ measures how far $v$ is from being an eigenvector of $A^\theta$. And so the differential equation $$ \frac{dv}{d\xi} = -\alpha(\xi) R(v(\xi)) $$ pushes $v$ into becoming closer to become an eigenvector.

Anyway, it looks like Chapter 6 is the meat of the paper. Theorem 6.1 seems to be the main result. However it has a rather odd condition, namely that the dimension of the eigenspace corresponding to the smallest eigenvalue of $A$ should be at least 20. So I will be interested to see how he converts the Navier-Stokes into an equation with this property.

Solution 4:

A full translation of the main theorem (Theorem 6.1) and conditions (Y.1)-(Y.4), due to Sergei Chernyshenko, can be found at

http://go.warwick.ac.uk/jcrobinson/lf/otelbaev

There is also a brief discussion of the method of proof used by Otelbaev.

[Responses below are to an earlier version of this post in which I thought I had found an error in the Galerkin argument used in the final part of the proof of Theorem 6.1.]

Solution 5:

what do you make of Definition 2? He defines a strong solution so that all terms of NSE are required to live in $L^2$. Global regularity of NSE calls for pressure and velocity fields in $C^\infty$. Is it just me or are we speaking of a very, very, very weak notion of strong solution, which has nothing to do with the millenium problem?