# Analysis Seminar

#### Global Existence Solutions and Geometric Properties of the SQG Sharp Front

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 $$C^{\infty}$$ 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.