Weak solutions to a nonlinear variational wave equation

Archive for Rational Mechanics and Analysis

Volumn 166, Issue 4, 2003, Pages 303-319

  Zhang, Ping ^a   Zheng, Yuxi ^b  

^a CHINESE ACADEMY OF SCIENCES   (China)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037934336     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00205-002-0232-7     Document Type: Article

References (24)

1. CALDERER, M.T.: private communications
2. de GENNES, P.G.: The Physics of Liquid Crystals. Clarendon Press, Oxford, 1974
3. DIPERNA, R.J., LIONS, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511-547 (1989)
4. DIPERNA, R.J., MAJDA, A.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108, 667-689 (1987)
5. ERICKSEN, J.L.: On the equations of motion for liquid crystals. Q. J. Mech. Appl. Math. 29, 202-208 (1976)
6. EVANS, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS 74, AMS, Providence, R. 1, 1990
7. GERARD, P.: Microlocal defect measures. Comm. in Partial Differential Equations 16, 1761-1794 (1991)
8. GLASSEY, R.T., HUNTER, J.K., ZHENG, Y.: Singularities in a nonlinear variational wave equation. J. Differential Equations 129, 49-78 (1996)
9. GLASSEY, R.T., HUNTER, J.K., ZHENG, Y.: Singularities and oscillations in a nonlinear variational wave equation. In Singularities and Oscillations, edited by J. RAUCH, M. TAYLOR, IMA, Vol. 91, Springer, 1997
10. GRUNDLAND, A., INFELD, E.: A family of nonlinear Klein-Gordon equations and their solutions. J. Math. Phys. 33, 2498-2503 (1992)
11. HUNTER, J.K., SAXTON, R.A.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498-1521 (1991)
12. HUNTER, J.K., ZHENG, Y.: On a nonlinear hyperbolic variational equation I and II. Arch. Rat. Mech. Anal. 129, 305-353 (1995)
13. HUNTER, J.K., ZHENG, Y.: On a nonlinear hyperbolic variational equation. I and II. Arch. Rat. Mech. Anal. 129, 355-383 (1995)
14. JOLY, J.L., MÉTIVIER, G., RAUCH, J.: Focusing at a point and absorption of nonlinear oscillations. Trans. Amer. Math. Soc. 347, 3921-3970 (1995)
15. LESLIE, F.M.: Theory of flow phenomena in liquid crystals. In Advances in Liquid Crystals, vol. 4, ed. G. H. Brown, Academic Press, New York, 1979
16. LIONS, P.L.: Mathematical Topics in Fluid Mechanics. Vol. 1, Incompressible Models, Lecture series in mathematics and its applications, V. 3 Clarendon Press, Oxford, 1996
17. LIONS, P.L.: Mathematical Topics in Fluid Mechanics. Vol. 2, Compressible Models, Lecture series in mathematics and its applications, V. 6, Clarendon Press, Oxford, 1998
18. SAXTON, R.A.: Dynamic instability of the liquid crystal director. In Contemporary Mathematics Vol. 100: Current Progress in Hyperbolic Systems, pp. 325-330, ed. W.B. LINDQUIST, AMS, Providence, 1989
19. TARTAR, L.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium (R. J. Knops, ed.), Research Notes in Math., Vol. 39, Pitman Press, 1979
20. YOUNG, L.C.: Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, Philadelphia, and London, 1969
21. ZHANG, P., ZHENG, Y.: Rarefactive solutions to a nonlinear variational wave equation. Comm. Partial Differential Equations 26, 381-419 (2001)
22. ZHANG, P., ZHENG, Y.: Existence and uniqueness of solutions to an asymptotic equation of a variational wave equation with general data. Arch. Rat. Mech. Anal. 155, 49-83 (2000)
23. ZHANG, P., ZHENG, Y.: Singular and rarefactive solutions to a nonlinear variational wave equation. Chinese Annals of Mathematics 22B, 1-12 (2001)
24. ZORSKI, H., INFELD, E.: New soliton equations for dipole chains. Phys. Rev. Lett. 68, 1180-1183 (1992)