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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.

MSC:
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
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[1] A. Bressan, A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., in press · Zbl 1105.76013
[2] Bressan, A.; Fonte, M., An optimal transportation metric for solutions of the camassa – holm equation, Methods appl. anal., 12, 191-200, (2005) · Zbl 1133.35054
[3] Brezis, H., Analyse fonctionnelle, (1983), Masson Paris · Zbl 0511.46001
[4] Camassa, R.; Holm, D.D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 11, 1661-1664, (1993) · Zbl 0972.35521
[5] Camassa, R.; Holm, D.D.; Hyman, J., A new integrable shallow water equation, Adv. appl. mech., 31, 1-33, (1994) · Zbl 0808.76011
[6] Coclite, G.M.; Holden, H.; Karlsen, K.H., Well-posedness for a parabolic – elliptic system, Discrete contin. dyn. syst., 13, 659-682, (2005) · Zbl 1082.35056
[7] Coclite, G.M.; Holden, H.; Karlsen, K.H., Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. math. anal., 37, 1044-1069, (2005) · Zbl 1100.35106
[8] Constantin, A.; Escher, J., Global existence and blow-up for a shallow water equation, Ann. sc. norm. super Pisa cl. sci. (4), 26, 2, 303-328, (1998) · Zbl 0918.35005
[9] Constantin, A.; Molinet, L., Global weak solutions for a shallow water equation, Comm. math. phys., 211, 1, 45-61, (2000) · Zbl 1002.35101
[10] Dai, H.-H., Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods, Wave motion, 28, 4, 367-381, (1998) · Zbl 1074.74541
[11] Dai, H.-H., Model equations for nonlinear dispersive waves in a compressible mooney – rivlin rod, Acta mech., 127, 1-4, 193-207, (1998) · Zbl 0910.73036
[12] Dai, H.-H.; Huo, Y., Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, Proc. R. soc. lond. ser. A math. phys. eng. sci., 456, 1994, 331-363, (2000) · Zbl 1004.74046
[13] Evans, L.C.; Gariepy, R.F., Measure theory and fine properties of functions, Stud. adv. math., (1992), CRC Press Boca Raton, FL · Zbl 0626.49007
[14] Folland, G.B., Real analysis, (1999), Wiley New York · Zbl 0671.58036
[15] Fonte, M., Conservative solution of the camassa – holm equation on the real line
[16] H. Holden, X. Raynaud, Global conservative solutions of the Camassa-Holm equation—a Lagrangian point of view, Comm. Partial Differential Equations, in press · Zbl 1136.35080
[17] H. Holden, X. Raynaud, Global conservative multipeakon solutions of the Camassa-Holm equation, J. Hyperbolic Differ. Equ., in press · Zbl 1128.65065
[18] Johnson, R.S., Camassa – holm, korteweg – de Vries and related models for water waves, J. fluid mech., 455, 63-82, (2002) · Zbl 1037.76006
[19] Lieb, E.H.; Loss, M., Analysis, (2001), Amer. Math. Soc. Providence, RI · Zbl 0966.26002
[20] Málek, J.; Nečas, J.; Rokyta, M.; Růžička, M., Weak and measure-valued solutions to evolutionary pdes, (1996), Chapman & Hall London · Zbl 0851.35002
[21] Xin, Z.; Zhang, P., On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. partial differential equations, 27, 1815-1844, (2002) · Zbl 1034.35115
[22] Yosida, K., Functional analysis, (1995), Springer-Verlag Berlin, reprint of the sixth (1980) edition · Zbl 0152.32102
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