Seminar

Analysis and PDEs Seminar

Time quasi-periodic vortex patch solutions of the 2D-Euler equations

Speaker:  Zineb Hassainia (New York University Abu Dhabi)
Date:  Wednesday, 01 June 2022 - 15:00
Online:  s06web.zoom.us/j/81491079139?pwd=OWo1TWc0NEExNDJZUFljWVRIdkZRQT09 (ID: 814 9107 9139; Access code: 937701)

Abstract:

In this talk I will discuss recent results concerning the emergence of time quasi-periodic vortex patch solutions of the 2D-Euler equations set either in the whole plan or in the unit disc. In the first case, the search of such solutions near  Rankine vortices is not clear due to the resonances of the linear frequencies and the absence of an exterior parameter. However, we are able to prove the existence of these  structures  close to Kirchhoff elliptical vortices  when aspect ratios belongs to a Borel set of asymptotically full Lebesgue measure. In the case of Euler equations in the unit disc, we  highlight the importance of the  boundary effects on the construction of quasi periodic vortex patches solutions close to Rankine vortices with radius belonging to a suitable massive Cantor-like set. Both proofs are based on Nash-Moser implicit function theorem and  KAM theory in infinite dimensional spaces.  The first result is joint work with Massimiliano Berti and Nader Masmoudi, whereas   the second is  joint work with Emeric Roulley.