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Zbl 1178.65099
Holden, Helge; Raynaud, Xavier
Dissipative solutions for the Camassa-Holm equation.
(English)
[J] Discrete Contin. Dyn. Syst. 24, No. 4, 1047-1112 (2009). ISSN 1078-0947; ISSN 1553-5231/e

The authors study the Camassa-Holm equation $$u_t-u_{xxt}+2\kappa u_x+3 uu_x-2u_xu_{xx}-uu_{xxx}=0$$ on the real line with $\kappa=0$ and initial condition $u|_{t=0}=\bar u$. 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 $L^\infty$ 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 ${\mathcal G}_0$. Lastly, the two-direction mappings between the Eulerian variable $u\in H^1$ and the Lagrangian variable $X\in{\mathcal G}_0$ is defined.
[Rémi Vaillancourt (Ottawa)]
MSC 2000:
*65M06 Finite difference methods (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
35B10 Periodic solutions of PDE
35Q53 KdV-like equations

Keywords: Camassa-Holm equation; dissipative solutions; dichotomy; stability

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