Is the Clay Navier-Stokes Problem Wellposed?

We discuss the formulation of the Clay Mathematics Institute Millennium Prize Problem on the Navier-Stokes equations in the perspective of Hadamard’s notion of wellposedness. 1 The Clay Navier-Stokes Millennium Problem The Clay Mathematics Institute Millennium Prize Problem on the incompressible Navier-Stokes equations [3, 7] asks for a proof of (I) global existence of smooth solutions for all smooth data, or a proof of the converse (II) non global existence of a smooth solution for some smooth data, referred to as breakdown or blowup. In [10, 12, 13, 14] we have discussed the formulation of the Millennium Prize Problem and pointed to a possible reformulation and resolution. Central to our discussion is Hadamard’s concept [8] of wellposed solution of a differential equation. Hadamard makes the observation that perturbations of data (forcing and initial/boundary values) have to be taken into account. If a vanishingly small perturbation can have a major effect on a solution, then the solution (or problem) is illposed, and in this case the solution may not carry any meaningful information and thus may be meaningless from both mathematical and applications points of view. According to Hadamard, only a wellposed solution, for which small perturbations have small effects (in some suitable sense), can be meaningful. Hadamard, thus makes a distinction between a wellposed and illposed solution through a quantitative measure of the effects of small perturbations: For a wellposed problem the effects are small and for an illposed large. A wellposed solution is meaningful, an illposed not. In this perspective it is remarkable that the issue of wellposedness does not appear in the formulation of the Millennium Problem [7]. The purpose of this note is to seek an explanation of this fact, which threatens to make the problem formulation itself illposed in the sense that a resolution is either trivial or impossible [10].