Smoothing estimates for weakly nonlinear internal waves in a rotating ocean

Abstract

In this paper we study the effect of rotation on nonlinear wave phenomena in weakly dispersive media modeled by the Korteweg-de Vries equation on the real line. It is well known that smoothing in the case of the KdV equation with periodic boundary conditions is a result of the presence of high frequency waves that weaken the nonlinearity through time averaging, \cite{bit}, \cite{et1}. High frequency interactions through normal form transformations can be thought of as adding a fast rotating term in certain toy PDE models that arrest the development of singularities, \cite{bit}. It is crucial for this phenomena that the zero Fourier mode can be removed due to the conservation of the mean. On the real line this mechanism breaks down as the resonance sets close to zero frequency are sizable and normal form transformations are not useful, \cite{imt}, and hence smoothing fails. The model we study is a perturbation of the KdV equation on a rotating frame of reference. A combination of a suitable normal form transformation along with the restricted norm method of Bourgain \cite{Bou2} implies smoothing on the real line.