Long time dynamics close to large amplitude quasi-periodic traveling waves in two dimensional forced rotating fluids

Abstract

In this paper we consider the $\beta$-plane equation with a smooth external force which is a quasi-periodic traveling wave of large amplitude $O(\lambda^{\alpha - 1})$, $1 < \alpha < 2$, and with large speed of propagation of size $O(\lambda)$. In a previous paper, the second and the third author proved the existence of quasi-periodic traveling wave solutions of large amplitude of order $O(\lambda^{\theta})$, for some $\theta > 0$. The purpose of this paper is to analyze the long time dynamics for smooth initial data close to these traveling wave solutions. In particular, we shall prove that, for initial data sufficiently close to a fixed traveling wave solution (in the $H^s$ topology), the corresponding solution remains close to the traveling wave solution for arbitrary long time (independent of the size of the traveling wave solution). As a consequence, we prove that there are open sets of large initial for which one has almost global existence, namely such that the corresponding solution remains of the same size of the initial datum for arbitrary long time (independent of the size of the initial data). The proof combines several ingredients: an analysis of the linearized PDE at any traveling wave solution via normal form methods, a sharp analysis of the transformed nonlinear problem under the change of coordinates that diagonalizes the linearized equation and energy estimates.