A Note on the analytic solutions of the Camassa–Holm equation

2005/11/08 by Maria Carmela Lombardo, M. Sammartino, Vincenzo Sciacca · 7 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Camassa–Holm equation #Mathematics #Analytic function #Function (biology) #Pure mathematics #Mathematical analysis #Integrable system

paper · doi:10.1016/j.crma.2005.10.006

published in Comptes Rendus Mathématique 341(11), 659-664 (Elsevier BV)

openalex publication_date 2005/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23

Abstract

In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using the Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic, belongs to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">/</mml:mo> <mml:mn>2</mml:mn> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">‖</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msub> <mml:mo stretchy="false">‖</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mo>∞</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:math> does not change sign, we prove that the solution stays analytic globally in time.

Citations

• A note on well-posedness for Camassa–Holm equation
• Classical solutions of the periodic Camassa—Holm equation
• 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
• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Gevrey class regularity for the solutions of the Navier-Stokes equations
• On the Blow-Up of Solutions of a Periodic Shallow Water Equation

Cited by

• Camassa–Holm equation [wikipedia]