Ill-posedness for the Camassa-Holm equation in B_p,1¹∩ C^0,1

2022/01/08 by Jinlu Li, Yanghai Yu, Li, Jinlu +5
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2201.02839

openalex publication_date 2022/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the Cauchy problem for the Camassa-Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space B_p,1¹∩ C^0,1 with p∈(2,∞] is discontinuous at origin. More precisely, u₀∈ B_p,1¹∩ C^0,1 can guarantee that the Camassa-Holm equation has a unique local solution in W^1,p∩ C^0,1, however, this solution is instable and can have an inflation in B_p,1¹∩ C^0,1 for certain initial data.

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