Blowup vs Illposedness of Smooth Solutions of the Incompressible Euler/Navier-Stokes Equations

We present evidence that the problem of breakdown/blowup of smooth solutions of the Euler and Navier-Stokes equations, is closely related to Hadamard’s concepts of wellposedness and illposedness. We present a combined criterion for blowup, based on detecting increasing L2-residuals and stability factors, which can be tested computationally on meshes of finite mesh size. 1 The Clay Navier-Stokes Millennium Problem The Clay Mathematics Institute Millennium Problem on the incompressible Navier-Stokes equations [5, 8] 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. The analogous problem for the inviscid incompressible Euler equations is mentioned briefly in [8] and in [7] described as “a major open problem in PDE theory, of far greater physical importance than the blowup problem for NavierStokes equations, which of course is known to the nonspecialists because it is a Clay Millenium Problem”. In the recent survey [3] the problem is described as “one of the most important and challenging open problems in mathematical fluid mechanics”. Since the viscosity the Millennium Problem is allowed to be arbitarily small and solutions of the Euler equations are defined as viscosity solutions of the Navier-Stokes equations under vanishing viscosity, the Euler equations effectively are included in the Millenium Problem as a limit case. In [16] we presented evidence that a specific initially smooth solution of the Euler equations, potential flow around a circular cylinder, in finite time exhibits blowup into a turbulent non-smooth solution, that is we presented evidence of (II). More generally, we presented evidence that all (non-trivial) initially smooth Euler solutions exhibit blowup into turbulent solutions. In particular, we argued that blowup can be detected computationally on computational meshes of finite mesh size. This work closely connects to the new resolution of d’Alembert’s paradox presented in [15].