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

2012/06/20 by Carlos Banquet Brango, Brango, Carlos Banquet
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1206.4726

openalex publication_date 2012/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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(ℝ)× H^s-1(ℝ), if s< 0, in the sense that the flow-map cannot be continuous at the origin from H^s(ℝ)× H^s-1(ℝ) to even (D'(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.

Related

• The lifespan of small data solutions for Intermediate Long Wave equation (ILW)
• Intermediate long wave equation in negative Sobolev spaces
• On the long time behavior of solutions to the Intermediate Long Wave\n equation
• Global well-posedness for the ILW equation in H^s(\mathbbT) for s>-\frac12
• Long-term regularity of 3D gravity water waves