F. Hamel - Nonlinear Phenomena: between ODEs and PDEs

Radial symmetry for the Euler equations and semilinear elliptic equations

In this talk, I will discuss radial symmetry properties for the stationary incompressible Euler equations in two-dimensional circular domains and some related semilinear elliptic equations. I will show that, in an annulus, a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured
plane, with some suitable conditions at infinity or at the origin. I will also discuss the case of the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow, which are trajectories of families of first-order ODEs, and on Liouville type results for some elliptic PDEs satisfied by the stream function. The talk is based on some joint works with N. Nadirashvili.