On the geometry of solutions of the quasi-geostrophic and Euler equations.
We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.
Cordoba
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Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA.