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

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