On the BBM-Burgers Equation: Well-posedness, Ill-posedness and Long Period Limit

TL;DR

This paper analyzes the Benjamin-Bona-Mahony-Burgers equation, establishing well-posedness for non-negative Sobolev spaces, ill-posedness below that, and the convergence of periodic solutions to the continuous case as the period increases.

Contribution

It provides a comprehensive well-posedness and ill-posedness analysis for the BBM-Burgers equation and characterizes the long-period limit behavior of solutions.

Findings

Well-posedness in $H^s(\

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Abstract

In this work we study a dispersive equation with a dissipative term, the Benjamin-Bona-Mahony-Burgers equation. First we prove that the initial value problem for this equation is well-posed in $H^s(\mathbb{R}),$ for $s\geq 0$ and ill-posed if $s< 0.$ The ill-posedness is in the sense that the flow-map cannot be continuous at the origin from $H^s(\mathbb{R})$ to even $\mathcal{D}'(\mathbb{R}).$ Additionally, we establish an exact theory of convergence of the periodic solutions to the continuous one, in Sobolev spaces, as the period goes to infinity.