Ill-posedness for the Camassa-Holm and related equations in Besov spaces

2021/04/13 by Jinlu Li, Li, Jinlu, Yanghai Yu +3 · 1 citation
Mathematics · Physics and Astronomy · #35Q53 #37K10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2104.05973

openalex publication_date 2021/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we give a construction of u₀∈ B^σ_p,∞ such that corresponding solution to the Camassa-Holm equation starting from u₀ is discontinuous at t = 0 in the metric of B^σ_p,∞, which implies the ill-posedness for this equation in B^σ_p,∞. We also apply our method to the b-equation and Novikov equation.

Citations

• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Wave breaking for nonlinear nonlocal shallow water equations
• Ill-posedness of the Camassa–Holm and related equations in the critical space

Cited by

• The local well-posedness, global existence and ill-posedness for the fifth order Camassa-Holm model

Related

• Sharp ill-posedness for the generalized Camassa-Holm equation in Besov spaces
• Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces
• A new result for the local well-posedness of the Camassa-Holm type equations in critial Besov spaces B^1+(1)/(p)_p,1,1≤ p
• Ill-posedness for the periodic Camassa--Holm type equations in the end-point critical Besov space B¹_∞,1
• Ill-posedness for the Camassa-Holm equation in B_p,1¹∩ C^0,1