Global Lagrangian solutions of the Camassa-Holm equation

Abstract

In this paper we construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We first express the equation in Lagrangian flow variable $\eta$ and then transform it using a change of variable $\rho =\sqrt{{\eta }_x}$. The new variable removes the singularity of the CH equation, and we obtain both the global weak conservative solution and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter-Saxton equation using the geometric interpretation.

References in corpus (6)

• An integrable shallow water equation with peaked solitons
• Global solutions for the two-component Camassa-Holm system
• Lipschitz metric for the periodic Camassa-Holm equation
• Geometric investigations of a vorticity model equation
• Global conservative solutions of the Camassa-Holm equation for initial data nonvanishing asymptotics
• Global Existence and Blowup for Geodesics in Universal Teichmüller Spaces

Cited by in corpus (2)

• Vanishing distance phenomena and the geometric approach to SQG
• Generalized compressible flows and solutions of the H(div) geodesic problem