One of the 7 Clay Mathematics Institute $1 million Millennium Problems concerns existence and regularity of solutions to the Navier-Stokes equations. The problem was formulated in 2000 and I discussed certain very problematic aspects of the formulation in the following articles from 2008:
- On the Uniqueness of Weak Solutions of the Navier-Stokes Equations
- Is the Clay Navier-Stokes Problem Wellposed?
also addressed in a blog post in 2009. I suggested a reformulation of the Clay Problem into 1 million €1 problems, each one solved by computing turbulent solutions to the Navier-Stokes equations. Clay did not respond to my suggestion.
There is no progress towards a solution of the problem as formulated by Clay, but the technique of computing turbulent solutions to the Navier-Stokes has advanced substantially: It is now possible to compute virtually any Navier-Stokes problem in the most difficult case of small viscosity with solutions always partially turbulent. It is further possible to do this in an fully automated way using the free software Unicorn as a FEniCS App. One of the problems solved is presented in Computation Can Now Replace Wind Tunnel.
It is thus now possible to solve the reformulated Clay Navier-Stokes problem and harvest $1 million.
As concerns the original formulation all solutions to Navier-Stokes equations with small viscosity are turbulent, and a turbulent solution is non-smooth. The Clay NS problem thus has a negative solution: smooth solutions do not exist, in the case of small viscosity.