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

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].