We consider the problem of the evolution of sharp fronts for the surface quasigeostrophic (QG) equation. This problem is,the analogue to the vortex patch problem for the two-dimensional Euler equation.The special interest of the quasi-geostrophic equation lies in its strong similarities with the three-dimensional Euler equation, while being a two-dimensional model. In particular, an analogue of the problem considered here, the evolution of. sharp fronts for QG, is the evolution of a vortex line for the three-dimensional Euler equation. The rigorous derivation of an equation for the evolution of a vortex line is still an open problem. The influence of the singularity appearing in the velocity when using the Biot-Savart law still needs to be understood.We present two derivations for the evolution of a periodic sharp front. The first one, heuristic, shows the presence of a logarithmic singularity in the velocity, while the second, making use of weak solutions, obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve.Finally, using a Nash-Moser argument as the main tool, we obtain local existence and uniqueness of a solution for the derived equation in the C-∞ case. © 2004 Wiley Periodicals, Inc.
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