An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blow-up in finite time?

Abstract

The solutions of time-dependent PDEs may show a bewildering variety of behaviors. The example of the Kuramoto–Sivashinsky (KS) equation illustrates how simple an equation can be with extremely complex solutions. The equations for incompressible fluids are also notorious for the extremely complex behavior of their solutions. But in the latter case, the situation is in a sense far less satisfactory than for the KS equations, since one still ignores if their solution remains smooth as time goes on, with smooth initial data in 3D. This leaves in a quite uncertain state all theories of 3D turbulence, which may be either a way of coping with our ignorance of the true dynamics of real fluids, or a way of analyzing the behavior of the smooth solutions of the Navier–Stokes equations. Below, in this homage to Professor Kuramoto, we present some remarks on this question, limited to the 3D Euler equations without viscosity. The existence of solutions of the 3D incompressible Euler equations for fluids blowing-up in finite time remains an open question if the initial energy is finite. The consensus seems to be that the singularity, if there is any, is local in space and time, although it is difficult to find a straight statement supporting such a claim in the literature. Based on the properties of self-similar equations, we claim that this is not possible at least if one assumes such a self-similar blow-up, including some refinements in the time dependence that extend the class of self-similar solutions. Actually, the possible blow-up solutions are very much constrained by the conservation of energy and of circulation. Based on scaling transformations, we argue that the singularity cannot be a simple self-similar blow-up, but requires some sort of oscillations on a logarithmic time-scale. The space dependence cannot be simple either. The various constraints lead to a self-similar collapse towards a line (at least for an axisymmetric flow with swirl), with a wavelength along the axis decreasing by steps. Our final result is a set of explicit equations such that, if they have a smooth solution satisfying certain conditions, the original Euler equations have a finite-time singularity.