Strichartz and Multi-linear Estimates for the One-dimensional Periodic Dysthe equation

TL;DR

This paper establishes Strichartz and multilinear estimates for the 1D periodic Dysthe equation, advancing understanding of its well-posedness and ill-posedness in various function spaces.

Contribution

It provides new Strichartz and multilinear estimates for the linearized Dysthe equation on the torus, and analyzes well-posedness and ill-posedness in Bourgain and Sobolev spaces.

Findings

Derived $L^6_{x,t}$ and $L^4_{x,t}$ estimates for solutions

Established bilinear and trilinear estimates for local well-posedness

Proved ill-posedness results in Sobolev spaces

Abstract

This paper presents Strichartz estimates for the linearized 1D periodic Dysthe equation on the torus, namely estimate of the $L^6_{x,t}(\mathbb{T}^2)$ norm of the solution in terms of the initial data, and estimate of the $L^4_{x,t}(\mathbb{T}^2)$ norm in terms of the Bourgain space norm. The paper also presents other results such as bilinear and trilinear estimates pertaining to local well-posedness of the 1-dimensional periodic Dysthe equation in a suitable Bourgain space, and ill-posedness results in Sobolev spaces.