Propagation of regularity and decay of solutions to nonlinear dispersive equations

Abstract

This article is mainly concentrated in the recent discovering of an intrinsic regularity property for solutions of nonlinear dispersive models that we refer as propagation of regularity. Roughly, we will describe how regularity on the initial data is transferred to the corresponding solutions with infinite speed. To illustrate the propagation of regularity property we will use solutions of the initial value problem associated to the generalized Korteweg-de Vries (KdV) equation. We then will discuss extensions of this special property to other nonlinear dispersive equations as quasilinear KdV type equations and nonlocal nonlinear equations in one and higher dimensions. We will also briefly comment on additional regularity properties for solutions to nonlinear dispersive models. Finally, we will introduce some relevant open problems regarding the propagation of regularity, decay and smoothing of solutions to nonlinear dispersive equations.