Well-posedness and nonlinear smoothing for the "good" Boussinesq equation on the half-line

TL;DR

This paper establishes local well-posedness and nonlinear smoothing effects for the 'good' Boussinesq equation on the half-line, extending results to low-regularity spaces below L^2 and improving previous work.

Contribution

It provides the first solutions for the initial-boundary value problem of the 'good' Boussinesq equation below L^2, with sharp results within the restricted norm framework.

Findings

Proves local existence, uniqueness, and continuous dependence in low-regularity spaces.

Demonstrates half derivative smoothing of the nonlinear term.

Extends well-posedness results to spaces below L^2.

Abstract

In this paper we study the regularity properties of the "good" Boussinesq equation on the half line. We obtain local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. Moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data, obtaining half derivative smoothing of the nonlinear term in some cases. Our paper improves the result in [Himonas-Mantzavinos 2015], being the first result that constructs solutions for the initial and boundary value problem of the "good" Boussinesq equation below the $L^2$ space. Our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line in [Kenig-Ponce-Vega 1996] and [Farah 2009].