| Abstract: |
| The vorticity form of the planar Euler equation says that vorticity is constant along particle trajectories. A vortex patch is a weak solution of the vorticity equation with initial condition the characteristic function of a domain $D_0$. Thus at time $t$ vorticity is the characteristic function of a domain $D_t$. Simulations show that the evolution of $D_t$ is extremely complicated. In spite of this general fact there are some special domains, called $V$-states, whose evolution is just rotation around the center of mass with constant angular velocity. Ellipses are examples of $V$-states. I will discuss Burbea`s proof of existence of other $V$-states and then I will discuss the smoothness of their boundary (joint work with Hmidi and Mateu). For general vortex patches, if the initial condition is the characteristic function of a domain with boundary of class $C^{1+\gamma}$, then the boundary of $D_t$ conserves the regularity for all times (Chemin`s theorem). I will mention a similar result for the aggregation equation in higher dimensions (joint work with Bertozzi, Garnett and Laurent). |
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