Absence of simple hyperbolic blow-up for the quasi-geostrophic and Euler equations.

摘要

The quasi-geostrophic thermal active scalar equation is a two-dimensional model for the incompressible 3D Euler equation. Numerical simulation showed a strong front formation when the geometry of the level sets of the thermal active scalar contains a hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. We study solutions of the quasi-geostrophic equation involving simple hyperbolic saddles. It is proved that such breakdown cannot occur in finite time and at large time, the angle of the saddle can not close faster than a double exponential in time. Therefore the gradient of the active scalar may grow at most as a quadruple exponential. Analogous results holds for the incompressible two and three-dimensional Euler equations.