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

Abstract

This paper presents Strichartz estimates for the linearized 1D periodic Dysthe equation on the torus, namely estimate of the $L_{x,t}^6({\mathbb{T}}^2)$ norm of the solution in terms of the initial data, and estimate of the $L_{x,t}^4({\mathbb{T}}^2)$ 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.