Weak solutions to a nonlinear variational wave equation with general data
Zhang, Ping ; Zheng, Yuxi
Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005), p. 207-226 / Harvested from Numdam
@article{AIHPC_2005__22_2_207_0,
     author = {Zhang, Ping and Zheng, Yuxi},
     title = {Weak solutions to a nonlinear variational wave equation with general data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {22},
     year = {2005},
     pages = {207-226},
     doi = {10.1016/j.anihpc.2004.04.001},
     mrnumber = {2124163},
     zbl = {1082.35129},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2005__22_2_207_0}
}
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) pp. 207-226. doi : 10.1016/j.anihpc.2004.04.001. http://gdmltest.u-ga.fr/item/AIHPC_2005__22_2_207_0/

[1] Diperna R.J., Lions P.L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511-547. | MR 1022305 | Zbl 0696.34049

[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 877643 | Zbl 0626.35059

[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 1034481 | Zbl 0698.35004

[4] Gerard P., Microlocal defect measures, Comm. Partial Differential Equations 16 (1991) 1761-1794. | MR 1135919 | Zbl 0770.35001

[5] Glassey R.T., Hunter J.K., Zheng Y., Singularities in a nonlinear variational wave equation, J. Differential Equations 129 (1996) 49-78. | MR 1400796 | Zbl 0879.35107

[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 1601273 | Zbl 0958.35084

[7] Grundland A., Infeld E., A family of nonlinear Klein-Gordon equations and their solutions, J. Math. Phys. 33 (1992) 2498-2503. | MR 1167950 | Zbl 0764.35088

[8] Hunter J.K., Saxton R.A., Dynamics of director fields, SIAM J. Appl. Math. 51 (1991) 1498-1521. | MR 1135995 | Zbl 0761.35063

[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 0834.35085

[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 2005201 | Zbl 1109.35088

[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 1297533 | Zbl 0857.35087

[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 1422251 | Zbl 0866.76002

[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 1637634 | Zbl 0908.76004

[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 1033527 | Zbl 0702.35180

[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 584398 | Zbl 0437.35004

[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 1069518 | Zbl 0774.35008

[17] Young L.C., Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia, 1969. | MR 259704 | Zbl 0177.37801

[18] Zhang P., Zheng Y., Rarefactive solutions to a nonlinear variational wave equation, Comm. Partial Differential Equations 26 (2001) 381-420. | MR 1842038 | Zbl 0989.35112

[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 1799274 | Zbl 0982.35062

[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 1835396 | Zbl 0980.35137

[21] Zhang P., Zheng Y., Weak solutions to a nonlinear variational wave equation, Arch. Rational Mech. Anal. 166 (2003) 303-319. | MR 1961443 | Zbl 1029.35173

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