On a nonlinear dispersive equation with time-dependent coefficients

Abstract

As a first step, we consider an evolution linear problem, the symbol of which is a real polynomial of degree three with time-dependent coefficients. We get for this problem smoothing effects known when these coefficients are constant. In particular, by using the theory of Calderón-Zygmund operators and the David and Journé T1 Theorem, we establish a local smoothing effect on the solution of the linear problem. In a second step, we study a nonlinear dispersive equation the linear part of which is the one studied above. We use the previous smoothing properties and a regularization method to establish that the Cauchy problem is locally well-posed in the Sobolev spaces $H^s(\mathbb R)$ for $s>3/4$.