Weak solutions to a nonlinear variational wave equation with general data
Zhang, Ping ; Zheng, Yuxi1
1 Departament of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. Research supported in part by the NSF-DMS grants 9703711, 0305497, 0305114 and by the Sloan Foundation.
Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 2, pp. 207-226.
DOI : 10.1016/j.anihpc.2004.04.001
Zhang, Ping  ; Zheng, Yuxi 1

1 Departament of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. Research supported in part by the NSF-DMS grants 9703711, 0305497, 0305114 and by the Sloan Foundation. 
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Zhang, Ping; Zheng, Yuxi. Weak solutions to a nonlinear variational wave equation with general data. Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 2, pp. 207-226. doi : 10.1016/j.anihpc.2004.04.001. http://www.numdam.org/articles/10.1016/j.anihpc.2004.04.001/

[1] Diperna R.J., Lions P.L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511-547. | MR | Zbl

[2] Diperna R.J., Majda A., Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987) 667-689. | MR | Zbl

[3] Evans L.C., Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conf. Ser. in Math., vol. 74, Amer. Math. Soc., Providence, RI, 1990. | MR | Zbl

[4] Gerard P., Microlocal defect measures, Comm. Partial Differential Equations 16 (1991) 1761-1794. | MR | Zbl

[5] Glassey R.T., Hunter J.K., Zheng Y., Singularities in a nonlinear variational wave equation, J. Differential Equations 129 (1996) 49-78. | MR | Zbl

[6] Glassey R.T., Hunter J.K., Zheng Y., Singularities and oscillations in a nonlinear variational wave equation, in: IMA, vol. 91, Springer, 1997. | MR | Zbl

[7] Grundland A., Infeld E., A family of nonlinear Klein-Gordon equations and their solutions, J. Math. Phys. 33 (1992) 2498-2503. | MR | Zbl

[8] Hunter J.K., Saxton R.A., Dynamics of director fields, SIAM J. Appl. Math. 51 (1991) 1498-1521. | MR | Zbl

[9] Hunter J.K., Zheng Y., On a nonlinear hyperbolic variational equation I and II, Arch. Rational Mech. Anal. 129 (1995) 305-353, and 355-383. | Zbl

[10] Jiang S., Zhang P., On the 3-D axi-symmetric solutions to the compressible Navier-Stokes equations, J. Math. Pures Appl. (9) 82 (2003) 949-973. | MR | Zbl

[11] Joly J.L., Métivier G., Rauch J., Focusing at a point and absorption of nonlinear oscillations, Trans. Amer. Math. Soc. 347 (1995) 3921-3970. | MR | Zbl

[12] Lions P.L., Mathematical Topics in Fluid Mechanics, vol. 1, Incompressible Models, Lecture Series in Mathematics and its Applications, vol. 3, Clarendon Press, Oxford, 1996. | MR | Zbl

[13] Lions P.L., Mathematical Topics in Fluid Mechanics, vol. 2, Compressible Models, Lecture Series in Mathematics and its Applications, vol. 6, Clarendon Press, Oxford, 1998. | MR | Zbl

[14] Saxton R.A., Dynamic instability of the liquid crystal director, in: Lindquist W.B. (Ed.), Current Progress in Hyperbolic Systems, Contemp. Math., vol. 100, Amer. Math. Soc., Providence, RI, 1989, pp. 325-330. | MR | Zbl

[15] Tartar L., Compensated compactness and applications to partial differential equations, in: Knops R.J. (Ed.), Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Res. Notes Math., vol. 39, Pitman, 1979. | MR | Zbl

[16] Tartar L., H-measures, a new approach for studying homogenisation oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburg Sect. A 115 (1990) 193-230. | MR | Zbl

[17] Young L.C., Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia, 1969. | MR | Zbl

[18] Zhang P., Zheng Y., Rarefactive solutions to a nonlinear variational wave equation, Comm. Partial Differential Equations 26 (2001) 381-420. | MR | Zbl

[19] Zhang P., Zheng Y., Existence and uniqueness of solutions to an asymptotic equation of a variational wave equation with general data, Arch. Rational Mech. Anal. 155 (2000) 49-83. | MR | Zbl

[20] Zhang P., Zheng Y., Singular and rarefactive solutions to a nonlinear variational wave equation, Chinese Ann. Math. Ser. B 22B (2000) 159-170. | MR | Zbl

[21] Zhang P., Zheng Y., Weak solutions to a nonlinear variational wave equation, Arch. Rational Mech. Anal. 166 (2003) 303-319. | MR | Zbl

[22] Zorski H., Infeld E., New soliton equations for dipole chains, Phys. Rev. Lett. 68 (1992) 1180-1183.

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