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Dissipative solutions for the Camassa-Holm equation. (English) Zbl 1178.65099

The authors study the Camassa-Holm equation

u t -u xxt +2κu x +3uu x -2u x u xx -uu xxx =0

on the real line with κ=0 and initial condition u| t=0 =u ¯. This equation admits two distinct classes of solutions, and the dichotomy between the two classes is associated with wave breaking, which takes place in finite time such that the H 1 and L norms of the solution remain finite while the spatial derivative u x becomes pointwise unbounded. This equation is reformulated by means of a different set of variables from Eulerian to Lagrangian coordinates to produce a system of semilinear ordinary differential equations. The existence of solutions, short-time stability and global stability are established. The system is shown to be invariant with respect to a relabeling set 𝒢 0 . Lastly, the two-direction mappings between the Eulerian variable uH 1 and the Lagrangian variable X𝒢 0 is defined.

65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35B10Periodic solutions of PDE
35Q53KdV-like (Korteweg-de Vries) equations