Existence time for the Camassa-Holm equation and the critical Sobolev index

2006/01/01 by Peter Byers · 1 citation
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models #Advanced Mathematical Physics Problems #Sobolev space #Mathematics #Camassa–Holm equation #Index (typography) #Mathematical analysis #Integrable system

paper · doi:10.1512/iumj.2006.55.2710

openalex publication_date 2006/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

In this paper, we examine the break-down phenomenon for a particular type of solution to the Camassa-Holm equation-namely, a peakon-antipeakon interaction with equal magnitude. We will show, in particular, that for any s ∈ (1, 3 2) and any e > 0, there exists initial data v 0 ∈ H s with ∥v 0 ∥ H s 3/2, we conclude that So = 3/2 is the critical Sobolev index for the Camassa-Holm equation.

Citations

• A New Integrable Shallow Water Equation
• A note on well-posedness for Camassa–Holm equation
• Classical solutions of the periodic Camassa—Holm equation
• On the ill-posedness of some canonical dispersive equations
• Global Weak Solutions for a Shallow Water Equation
• On the Cauchy problem for the Camassa–Holm equation
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• A shallow water equation as a geodesic flow on the Bott-Virasoro group
• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Wave breaking for nonlinear nonlocal shallow water equations
• Geodesic flow on the diffeomorphism group of the circle
• A Nonlinear Analysis of the Averaged Euler Equations

Cited by

• Camassa–Holm equation [wikipedia]