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

TL;DR

This paper establishes well-posedness results for a 2D dispersive model from capillary-gravity flows, analyzing solutions in various Sobolev spaces and exploring implications for related wave equations.

Contribution

It provides new well-posedness results in classical and adapted Sobolev spaces, including periodic and weighted spaces, and characterizes solution behavior with unique continuation principles.

Findings

Local well-posedness in classical Sobolev spaces

Well-posedness in energy-adapted Sobolev spaces

Implications for the Shrira wave equation

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 well-posedness 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.