Given a solution $S$ of the Navier-Stokes equations, is there a way to make mathematically precise a statement like: "$S$ is [turbulent](https://en.wikipedia.org/wiki/Turbulence) in the spacetime region $U$"?


And if such a definition exists, are there any known exact solutions of Navier-Stokes exhibiting turbulence?


Reading Wikipedia makes me also want to ask the following related questions:


1. Does a single [vortex](https://en.wikipedia.org/wiki/Vortex) count as turbulence?
2. Is the appearance of vortices a necessary feature of turbulence?
3. What's the difference between a vortex and an [eddy](https://en.wikipedia.org/wiki/Eddy_(fluid_dynamics))?


Pointers to (mathematically rigorous) literature are much appreciated.



Is there a mathematically precise definition of turbulence for solutions of Navier-Stokes?
There is probably no universally accepted mathematical definition of turbulence. (By the way, is there a physical one?) Moreover, the prevailing definitions seem to be highly volatile and time-dependent themselves.


A few notable examples.


* In the Ptolemaic [Landau–Hopf theory](https://en.wikipedia.org/wiki/Landau%E2%80%93Hopf_theory_of_turbulence) turbulence is understood as a cascade of bifurcations from unstable equilibriums via periodic solutions ([the Hopf bifurcation](https://en.wikipedia.org/wiki/Hopf_bifurcation)) to quasiperiodic solutions with arbitrarily large frequency basis.
* According to [Arnold and Khesin](https://books.google.co.uk/books?id=LJapQwAACAAJ&dq=arnold%20topological%20methods&hl=en&ei=dvEhTO_SDY_9_AaNrogq&sa=X&oi=book_result&ct=result), in the 1960's most specialists in PDEs regarded the lack of global existence and uniqueness theorems for solutions of the 3D Navier–Stokes equation as the explanation of turbulence.
* Kolmogorov suggested to study *minimal attractors* of the Navier-Stokes equations and formulated several conjectures as plausible explanations of turbulence. The weakest one says that the maximum of the dimensions of minimal attractors of the Navier–Stokes equations grows along with the Reynolds number Re.
* In 1970 Ruelle and Takens formulated the conjecture that turbulence is the appearance
of *global attractors* with sensitive dependence of motion on the initial conditions in the phase space of the Navier–Stokes equations ([link](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-20/issue-3/On-the-nature-of-turbulence/cmp/1103857186.full)). In spite of the vast popularity of their paper, even the existence of such attractors is still unknown.


**Edit 1.** Concerning explicit solutions to the Navier-Stokes equations, I don't think any of them really exhibit turbulence features. The thing is that the nonlinear term $v\cdot \nabla v$ is equal to $0$ for most known classical explicit solutions. In other words, these solutions
actually solve the linear Stokes equation and don't "see" the nonlinearity of the full Navier-Stokes system. This is probably not what one would expect from a truly turbulent flow.


**Edit 2.** As for the quick reference, you may find helpful the short [survey on turbulence theories](http://www.labma.ufrj.br/%7Errosa/dvifiles/encyturb.pdf) by Ricardo Rosa. ([Wayback Machine](https://web.archive.org/web/20170329151350/http://www.labma.ufrj.br/%7Errosa/dvifiles/encyturb.pdf)) It appears as an article in the [Encyclopedia of Mathematical Physics](http://www.sciencedirect.com/science?_ob=RefWorkIndexURL&_idxType=AU&_cdi=27008&_refWorkId=327&_acct=C000015798&_version=1&_userid=273788&md5=b04306d338adb2412b2379b81f51f3d0).


