The Symmetric Regularized-Long-Wave Equation: Ill-posedness and Long Period Limit

TL;DR

This paper investigates the ill-posedness of the Symmetric Regularized-Long-Wave equation for low regularity data and establishes the convergence of periodic solutions to the continuous case as the period increases.

Contribution

It proves ill-posedness for initial data in low regularity Sobolev spaces and characterizes the convergence of periodic solutions to the continuous solution as the period tends to infinity.

Findings

Ill-posedness for data in $H^s$ with $s<0$

Discontinuous flow-map at the origin in low regularity spaces

Convergence of periodic solutions to the continuous solution as period increases

Abstract

In the present work we obtain two important results for the Symmetric Regulraized-Long-Wave equation. First we prove that the initial value problem for this equation is ill-posed for data in $H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}),$ if $s< 0,$ in the sense that the flow-map cannot be continuous at the origin from $H^s(\mathbb{R})\times H^{s-1}(\mathbb{R})$ to even $(\mathcal{D}'(\mathbb{R}))^2.$ We also establish an exact theory of convergence of the periodic solutions to the continuous one, in Sobolev spaces, as the period goes to infinity.