Growth of solutions for QG and 2D Euler equations

The work of Constantin-Majda-Tabak [1] developed an analogy between the Quasi-geostrophic and 3D Euler equations. Constantin, Majda and Tabak proposed a candidate for a singularity for the Quasi-geostrophic equation. Their numerics showed evidence of a blow-up for a particular initial data, where the level sets of the temperature contain a hyperbolic saddle. The arms of the saddle tend to close in finite time, producing a sharp front. Numerics studies done later by Ohkitani-Yamada [8] and Constantin-Nie-Schorghofer [2], with the same initial data, suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time. The study of collapse on a curve was first studied in [1] for the Quasi-geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology. At the time of collapse, the scalar θ is discontinuous across the curve x2 = f(x1) with different limiting values for the temperature on each side of the front. They show that under this topology the directional field remains smooth up to the collapse, which contradicts the following theorem proven in [1]: