Well-posedness for a two-dimensional dispersive model arising from capillary-gravity flows

Abstract

This paper is aimed to establish well-posedness in several settings for the Cauchy problem associated to a model arising in the study of capillary-gravity flows. More precisely, we determinate local wellposedness conclusions in classical Sobolev spaces and some spaces adapted to the energy of the equation.A key ingredient is a commutator estimate involving the Hilbert transform and fractional derivatives. We also study local well-posedness for the associated periodic initial value problem. Additionally, by determining well-posedness in anisotropic weighted Sobolev spaces as well as some unique continuation principles, we characterize the spatial behavior of solutions of this model. As a further consequence of our results, we derive new conclusions for the Shrira equation which appears in the context of waves in shear flows.

On some regularity properties for the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov Equation

“…This equation was deduced as a simplified model to describe a two-dimensional weakly nonlinear long-wave perturbation on the background of a boundary-layer type plane-parallel shear flow (see [31]). For some references dealing with LWP issues see [5,33].…”