The "good" Boussinesq equation on the half-line with Robin boundary conditions

TL;DR

This paper establishes the local well-posedness of the 'good' Boussinesq equation on the half-line with Robin boundary conditions, using explicit linear solutions and Sobolev space analysis.

Contribution

It provides the first well-posedness proof for the Boussinesq equation with Robin boundary conditions on the half-line, employing Fokas's unified transform method.

Findings

Proves local Hadamard well-posedness in Sobolev spaces.

Utilizes explicit solution formulas for the linear problem.

Extends well-posedness to low regularity solutions.

Abstract

We prove the local Hadamard well-posedness of the ``good'' Boussinesq equation formulated on the half-line with nonzero Robin boundary conditions. These boundary data involve the Dirichlet and Neumann boundary values as well as the second spatial derivative of the solution evaluated at the boundary. The nonlinear analysis crucially relies on the linear estimates established through the explicit solution formula obtained for the forced linear counterpart of the problem via Fokas's unified transform. The two pieces of initial data and the two pieces of boundary data belong in appropriate Sobolev spaces. The corresponding solution is established in the natural Hadamard solution space of continuous/continuously differentiable functions from a suitable time interval to the Sobolev spaces associated with the two initial data. Furthermore, in line with the well-posedness theory of the Cauchy problem, in the case of low regularity (namely, below the spatial continuity threshold) the solution space is refined by also including an appropriate spatiotemporal Lebesgue space.