The formulation of the Clay Mathematics Institute millennium problem on the Navier-Stokes equations is discussed and a possible approach towards resolution is presented.
The Clay Mathematics Institute Navier-Stokes Millennium Problem
The Clay Mathematics Institute has, in order to celebrate mathematics in the new millennium, named seven one million dollar prize problems, one of which concerns the Navier-Stokes equations described as follows:
- Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
Clay image of turbulence illustrating the nature of the Clay problem.
Evidently the hidden secret of Navier-Stokes equations is the secret of turbulence, the main unsolved mystery of classical mechanics. The Clay problem thus asks for progress towards a mathematical theory of turbulence. It is known that turbulence typically appears in slightly viscous fluids such as air and water, and thus the Clay problem essentially concerns solutions of the Navier-Stokes equations with small viscosity, or viscosity solutions of the Euler equations.
The official formulation given by Charles Fefferman asks for a mathematical analytical proof of either of the following propositions:
- (I) existence + smoothness of solutions for all smooth data
- (II) the converse: non-(existence + smoothness) or blowup for some smooth data.
Fefferman’s official formulation does not have any reference to turbulence and ends with the following reflection:
- Let me end with a few words about the signiﬁcance of the problems posed here. Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behavior of solutions of the Euler and Navier–Stokes equations. Since we don’t even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas.
Fefferman thus carefully avoids the main issue of turbulence apparently taking for granted that an analytical mathematical theory of turbulence is so far out of reach that it cannot even be mentioned. The two formulations of the problem, one informal focussing on turbulence and one formal seemingly focussing on something without turbulence, do not match.
But there is a connection through (II): Blowup is the same as turbulence. A turbulent solution is a non-smooth solution, and blowup from smooth data initial data is the same as transition from laminar to turbulent flow. And (II) shows to be the true alternative, not (I).
Analytical mathematics is performed by writing symbols on paper and can be very powerful, but also has very severe limitations as concerns solution of non-linear differential equations such as the Navier-Stokes equations. Very few analytical solutions are known, in particular no analytical turbulent solutions are known. What Fefferman describes is a complete dead-lock of analytical mathematical theory of turbulence, which is also witnessed by Terence Tao as why global regularity for Navier-Stokes is hard.
Computational Mathematical Theory of Turbulence
But today the computer offers new tools to mathematics in the form of computational solution of differential equations. In particular, computational solution of the Navier-Stokes/Euler equations is possible, and turbulent solutions can be computed on a modern lap-top as shown in the book Computational Turbulent Incompressible Flow and the article Blow up of incompressible Euler solutions 
It is shown by computation that initially smooth solutions of the Navier-Stokes/Euler equations with small viscosity, develop into turbulent viscosity solutions of the Euler equations, that is, evidence of (II) is given. It is also shown that turbulent solutions are not well defined pointwise or illposed, since turbulent solutions fluctuate in space and time, while certain meanvalues such as drag and lift, are wellposed and can be computed, see in particular the Knol article on d’Alembert’s paradox. Specifically it is shown that the drag and lift of cars and airplanes can be predicted by computing turbulent viscosity solutions to the Euler equations equations. See the related Knols on why it is possible to fly and why a topspin tennis ball curves down.
In Fefferman’ s formulation wellposedness is not an issue, only existence + smoothness. But existence without some form of wellposedness, e.g. in a meanvalue sense, has no mathematical meaning nor physical. Fefferman’s formulation is thus unfortunate from both mathematical and physical point of view, as is the argument that the mathematics of the Navier-Stokes equations is something very different from the fluid mechanics of the same equations.
Altogether a step towards the solution of the Clay Navier-Stokes problem can be made by computing turbulent solutions of the Navier-Stokes/Euler equations, which can be inspected and reveal secrets of turbulence. A key point is that the quality of computed solutions are tested by a posteriori error estimation guaranteeing the that computed solutions are mathematical solutions. Computation thus opens to a mathematical theory of turbulence as requested in the millennium problem.
Terence Tao argues that turbulence cannot be defined mathematically, but this is not so: The flow of a slightly viscous flow is turbulent if the viscous dissipation is comparable to the kinetic energy. And blowup can be identified with turbulence, see .