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Global conservative solutions of the generalized hyperelastic-rod wave equation. (English) Zbl 1116.35115
Summary: We prove existence of global and conservative solutions of the Cauchy problem for the nonlinear partial differential equation \[ u_t-u_{xxt}+ f(u)_x-f(u)_{xxx}+(g(u)+\frac 12 f''(u)(u_x)^2)_x=0 \] where \(f\) is strictly convex or concave and \(g\) is locally uniformly Lipschitz. This includes the Camassa-Holm equation \((f(u)=u^2/2\) and \(g(u)=\kappa u+u^2)\) as well as the hyperelastic-rod wave equation \((f(u)=\gamma u^2/2\) and \(g(u)=(3-\gamma)u^2/2)\) as special cases. It is shown that the problem is well-posed for initial data in \(H^1(\mathbb{R})\) if one includes a Radon measure that corresponds to the energy of the system with the initial data. The solution is energy preserving. Stability is proved both with respect to initial data and the functions \(f\) and \(g\). The proof uses an equivalent reformulation of the equation in terms of Lagrangian coordinates.

35Q72 Other PDE from mechanics (MSC2000)
35B35 Stability in context of PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
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