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Global existence solutions and geometric properties of the SQG Sharp front

PDE and Applied Math Seminar

Speaker: Diego Cordoba, ICMAT Spain and Princeton
Location: 2112 MSB
Start time: Wed, Feb 11 2015, 4:10PM

A particular kind of weak solutions for a 2D active scalar are the so called “sharp fronts”, i.e., solutions for which the scalar is a step function. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem to be treated as a non-local one dimensional equation for the contour. In this setting we will present several analytical results for the surface quasi-geostrophic equation (SQG): the existence of convex $C^{\infinity}$ global rotating solutions, elliptical shapes are not rotating solutions (as opposed to 2D Euler equations) and the existence of convex solutions that lose their convexity in finite time.