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The Collatz Conjecture is a famous conjecture that has never been proven; nevertheless, there exists a simple heuristic probabilistic argument which supports its truth - in Wikipedia's words, "If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one. (More precisely, the geometric mean of the ratios of outcomes is 3/4.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence."

Is there a similar heuristic probabilistic argument for the Navier-Stokes existence and smoothness conjecture, another problem about a dynamical system? In other words, is there a good reason to believe that this conjecture is true, even if we don't have a proof?

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  • $\begingroup$ A Google search of the title gives a number of results. $\endgroup$ – Alex R. Oct 5 '14 at 3:05
  • $\begingroup$ I did a Google search before I asked the question and couldn't find anything. $\endgroup$ – Craig Feinstein Oct 5 '14 at 3:18
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    $\begingroup$ yes, 100+ years of experiments agree with NS predictions. $\endgroup$ – Piyush Grover Jan 21 at 5:44
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    $\begingroup$ From Tao's blog: The enemy [of probabilistic arguments] is that the Navier-Stokes flow itself might have some perverse entropy-reducing property which somehow makes the average case drift towards (or at least recur near) the worst case over long periods of time. This is incredibly unlikely to be the truth, but we have no tools to prevent it from happening at present. $\endgroup$ – Carlo Beenakker Jan 21 at 7:19
  • $\begingroup$ Does the fact that the viscosity takes away energy from the system increase the probability that the system is stable? $\endgroup$ – Craig Feinstein Jan 22 at 14:54
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The following is the best probabilistic heuristic argument I know, which convinces me that either there is no blow up of the 3D Navier-Stokes equations for any initial data, or a blow up is very unlikely, or at least if there is a blow up it would be difficult to prove that it is a blow up:

See the paper "An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations" by Mattingly and Sinai. In that paper, they put the 3D equations in vorticity formulation and show what it looks like in Fourier space on page 12:

Let $\alpha=2$, $g_k=0$, $\mathcal Z = \mathbb Z^3$, and $k=(k_1,k_2,k_3) \in \mathbb Z^3$. Then they split the system into the equations for the "real and imaginary parts of $\{u_k\}_k$ and $\{\omega_k\}_k$. Letting $$ \begin{split} u_k(t)&=u^{(1)}_k(t)+ iu^{(2)}_k(t),\\ \omega_k(t)&=\omega_k^{(1)}(t)+i \omega_k^{(2)}(t),\:\text{ and}\\ g_k(t)&=g^{(1)}_k(t)+ i g^{(2)}_k(t), \end{split} $$ we obtain \begin{split} \label{3DVorticityRealImag} \frac{d \omega_k^{(1)}(t)}{dt} &= 2 \pi \sum_{\substack{l_1+l_2=k\\ l_1 , l_2 \in \mathcal{Z}}} \Bigl[ \bigl( u^{(1)}_{l_1}(t),k\bigr)\omega^{(2)}_{l_2}(t)+ \bigl(u^{(2)}_{l_1}(t),k\bigr)\omega^{(1)}_{l_2}(t) \\ &\qquad- \bigl(\omega^{(2)}_{l_1}(t),k\bigr)u^{(1)}_{l_2}(t) -\bigl( \omega^{(1)}_{l_1}(t),k\bigr)u^{(2)}_{l_2}(t) \Bigr]\\\\ &\qquad -4 \pi^2 \nu |k|^\alpha \omega_k^{(1)}(t) + g_k^{(1)}(t) \\\\ \frac{d \omega_k^{(2)}(t)}{dt} & = -2 \pi \sum_{\substack{l_1+l_2=k\\ l_1 , l_2 \in \mathcal{Z}}} \Bigl[ \bigl( u^{(1)}_{l_1}(t),k\bigr)\omega^{(1)}_{l_2}(t) - \bigl(u^{(2)}_{l_1}(t),k\bigr)\omega^{(2)}_{l_2}(t)\\ &\qquad - \bigl( \omega^{(1)}_{l_1}(t),k\bigr)u^{(1)}_{l_2}(t) + \bigl( \omega^{(2)}_{l_1}(t),k\bigr)u^{(2)}_{l_2}(t) \Bigr] \\\\ & \qquad + g_k^{(2)}(t) -4 \pi^2 \nu |k|^\alpha \omega_k^{(2)}(t) \end{split}

Notice that the convection part of the right hand side of the two real and imaginary part equations has an equal number of positive and negative terms. (Also note that each $u_k$ is bounded above by the square root of the total initial energy, which is a finite constant.) Because of this, it seems very difficult to find initial data which will prevent the terms of the convection part from approximately cancelling each other out at each time $t$. (The approximate cancellations would stop the enstrophy from going to infinity in finite time.) Also, the diffusion part of the right hand side of the equations can only help to prevent a blow up. Therefore, a blow up of the 3D Navier-Stokes equations seems unlikely, if not impossible.

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