On the evolution of sharp fronts for the quasi-geostrophic equation.

摘要

In this dissertation we consider the problem of the evolution of sharp fronts for the surface quasi-geostrophic (QG) equation. More precisely, we consider the evolution of a solution &thetas; to the QG equation that only takes the values 0 and 1 on two different regions bounded by a smooth curve ϕ, the front. This problem is the analog to the vortex patch problem for the 2-D Euler equation.; The special interest of the quasi-geostrophic equation lies in its strong similarities with the 3-D Euler equation, while being a 2-D model. These analogies, originally noticed by Constantin, Majda and Tabak provide us with a 2-D model with many of the interesting features appearing in the Euler equation. In particular an analog of the problem considered here, the evolution of sharp fronts for QG, is the evolution of a vortex line for 3-D Euler. 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.; In this dissertation 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 of the front, 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.