Is the Clay Navier-Stokes Problem Wellposed? JOHAN HOFFMAN 1 and CLAES JOHNSON 1 April 25, 2008 1 School of Computer Science and Communication Royal Institute of Technology 10044 Stockholm, Sweden. email: jhoffman@csc.kth.se, cgjoh@csc.kth.se Abstract We discuss the formulation of the Clay Mathematics Institute Millen- nium 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 incompress- ible 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 mean- ingful 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 pertur- bations: 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 prob- lem formulation itself illposed in the sense that a resolution is either trivial or impossible [10]. 1