Nonexistence of Simple Hyperbolic Blow-Up for the Quasi-Geostrophic Equation

The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations. In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak 7 suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: If the level set topology in the temperature field for the 2D quasi-geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle, then no finite time singularity is possible. Numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. The main theorem we present in this paper shows that such breakdown cannot occur in finite time. We also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time. Using the same techniques, we see that analogous results hold for incompressible 2D and 3D Euler. These results were announced in 9, but with a slight difference in the definition of a simple hyperbolic saddle. The definition given in Section 4 generalizes the one given in the announcement. See also Constantin 4, discussed in Section 7, Remark 5 below. The whole work described in this paper is basically part of the author's thesis. I am particularly grateful to my thesis advisor Charles Fefferman for his attention, support, guidance and advice. I am indebted to D. Christodoulou and P. Constantin for helpful corrections and suggestions. I wish to thank A. Majda for suggesting the subject and E. Tabak for discussions and com-