Title: Large amplitude quasi-periodic traveling waves in rotating
fluids: existence and nonlinear stability
Abstract:
The beta-plane equation is a 2D approximation of the 3D
Euler-Coriolis equations for rotating fluids. After a brief overview of
these equations and related problems, I will discuss the dynamics of the
beta-plane equation under a highly oscillating external forcing of
substantially large size — hence outside the perturbative regime —
assumed to be a quasi-periodic traveling wave.
I will present two recent results: the existence of large amplitude
quasi-periodic traveling wave solutions, and their nonlinear stability
for initial data that are small perturbations of a fixed such solution.
As a consequence, one obtains large amplitude solutions that remain
close to a quasi-periodic traveling wave over arbitrarily long times,
independently of its amplitude — and, in particular, open sets of large
initial data for which almost global existence holds. The traveling
condition plays a crucial role in resolving a degeneracy of the
linearized spectrum induced by the rotation of the fluid — a degeneracy
with infinite-dimensional kernel, typical of quasilinear PDEs in space
dimension greater than one, which had so far prevented this kind of
construction outside the one-dimensional setting. After stating the main
theorems, I will outline the main ideas of the proofs, which combine an
approximate solution scheme, normal form methods, and energy estimates.
This is a joint work with Roberta Bianchini, Roberto Feola, Riccardo
Montalto and Shulamit Terracina.