Mathematics

The purpose of this course is to study some dynamical properties of weak solutions to Euler equations in simply-connected domains. A focus will be done on vortex patches which are particular weak solutions where the vorticity is the characteristic functions of a bounded domain. The first part of the course will be devoted to the proof Yudovich's Theorem stating the existence and uniqueness of weak solutions. Then, we will study the emergence of time-periodic vortex patch rigid motion via two approaches: the desingularization of point vortices and the bifurcation from radial solutions. Finally, we will show the existence of periodic non-rigid patch motion within bounded simply-connected domains.