I am looking for feedback on this attempt to revisit the criticality of the Navier-Stokes equation (NSE) for incompressible fluids in 3 dimensions. It has been said, that the NSE is supercritical (see T. Tao, Why proving global regularity conjecture for Navier-Stokes is hard (terrytao.wordpress.com, March 18, 2007)), because the rescaling that preserves the equation is
$$\vec{u} \rightarrow \frac{1}{k}\vec{u}$$
while the energy is an integral of $|\vec{u}|^2$ and thus the rescaling that preserves the energy is given by
$$\vec{u} \rightarrow \frac{1}{k^{3/2}} \vec{u}$$
thereby showing that for the NSE in 3 dimensions energy is being rescaled more severely; this amplifies the rescaling at small scales $k\ll1$. Hence, the NSE appears to be energy-supercritical.
The attempt here is to show that in this reasoning, the assumption is only true in part; becaise the rescaling that preserves the NSE is not $\vec{u} \rightarrow \frac{1}{k}\vec{u}$, but it is actually
$$\vec{u}^\prime = k^{\alpha_x-\alpha_t}\vec{u},$$
where
$$ \begin{split} (x,y,z)^\prime &= k^{\alpha_x}(x,y,z)\\ t^\prime & = k^{\alpha_t}t \end{split} $$
and these are identical with the above rescaling laws only when $\alpha_x=1$, $\alpha_t=2$ but not in general. This fact was discovered only recently, see A. Ercan and M. L. Kavvas, Chaos 25, 123126 (2015).
But this changes everything, because to keep the NSE invariant under scaling transformation, the velocity and energy transform as $$ \begin{split} \vec{u}^\prime & = k^{\alpha_x-\alpha_t}\vec{u} \\ E^\prime & = k^{5\alpha_x-2\alpha_t} E \end{split} $$ and the energy $E$ at a given moment $t$, would be scale-invariant if $\vec{u}$ transformed according to the law
$$\vec{u}^\prime = k^{-\frac{3}{2}\alpha_x}\vec{u}.$$
Then, appllying T. Tao's definitions of critical, subcritical and supercritical (see T. Tao, Current developments in mathematics 2006 , 255(2008)) we find that the NSE is
- energy-subcritical, when $\frac{\alpha_t}{\alpha_x}>\frac{5}{2}$,
- energy-critical, when $\frac{\alpha_t}{\alpha_x}=\frac{5}{2}$,
- energy-supercritical, when $\frac{\alpha_t}{\alpha_x}<\frac{5}{2}$.
Do you find this reasoning correct/credible? NSE criticality is quite important and relates to the smoothness of solutions
Your kind thoughts and suggestions are welcome. It is better to have them put together as an answer, pointing mistakes or showing an alternative explanations, rather than having extended discussion/comments
Thank you
You are only scaling the magnitude of the velocity field, whereas the scaling that is actually preserved by Navier-Stokes equation is $$ u^{(\lambda)}(x,t) = \frac{1}{\lambda}u\left(\frac{t}{\lambda^2},\frac{x}{\lambda}\right), \quad p^{(\lambda)}(x,t) = \frac{1}{\lambda^2}u\left(\frac{t}{\lambda^2},\frac{x}{\lambda}\right). $$ One can then find the critical Sobolev spaces of the Navier-Stokes equation in $\mathbb{R}^3$ by computing how Sobolev norms of different orders $H^\alpha$ scale with $\lambda$. (Using the Fourier version of the Sobolev norm makes this much easier). Then sub/super/criticality can be established by seeing which norms decrease/increase/stay the same as $\lambda\to 0$. This computation is carried out at the end of these notes and a more qualitative treatment is given in this post.