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variational wave equation; weak solutions; $L^p$ Young measure; renormalized solutions
In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the $L^p$ Young measure theory and related compactness results, in the first section. Then we use the $L^p$ Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.
[1] A.  Bressan, Ping Zhang, and Yuxi Zheng: On the asymptotic variational wave equations. Archive for Rational Mechanics and Analysis, Online 2006. DOI 10.1007/s00205-006-0014-8 | MR 2259342
[2] R. Camassa, D. Holm: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993), 1661–1664. DOI 10.1103/PhysRevLett.71.1661 | MR 1234453
[3] J. S. Chapman, J. Rubinstein, and M. Schatzman: A mean-field model of superconducting vortices. Eur. J.  Appl. Math. 7 (1996), 97–111. DOI 10.1017/S0956792500002242 | MR 1388106
[4] A. Constantin, J. Escher: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181 (1998), 229–243. DOI 10.1007/BF02392586 | MR 1668586
[5] R. J. DiPerna: Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys. 91 (1983), 1–30. DOI 10.1007/BF01206047 | MR 0719807 | Zbl 0533.76071
[6] R. J. DiPerna, P.-L. Lions: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547. DOI 10.1007/BF01393835 | MR 1022305
[7] R. J. DiPerna, P.-L. Lions: On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130 (1989), 321–366. DOI 10.2307/1971423 | MR 1014927
[8] R. J. DiPerna, A. J. Majda: Oscillations and concentration in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108 (1987), 667–689. DOI 10.1007/BF01214424 | MR 0877643
[9] Qiang Du, Ping Zhang: Existence of weak solutions to some vortex density models. SIAM J.  Math. Anal. 34 (2003), 1279–1299. DOI 10.1137/S0036141002408009 | MR 2000970
[10] E. Weinan: Dynamics of vortex liquids in Ginsburg-Landau theories with application to superconductivity. Phys. Rev.  B 50 (1994), 1126–1135. DOI 10.1103/PhysRevB.50.1126
[11] L. C. Evans: Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS No. 74. AMS, Providence, 1990. MR 1034481
[12] E. Feireisl, A. Novotný, and H.  Petzeltová: On the existence of globally defined weak solutions to the Navier-Stokes equations. J.  Math. Fluid Mech. 3 (2001), 358–392. DOI 10.1007/PL00000976 | MR 1867887
[13] B. Fuchssteiner, A. S.  Fokas: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys.  D 4 (1981/1982), 47–66. DOI 10.1016/0167-2789(81)90004-X | MR 0636470
[14] R. T.  Glassey, J. K.  Hunter, and Yuxi Zheng: Singularities and oscillations in a nonlinear variational wave equation. In: Singularities and Oscillations. IMA, Vol.  91, J.  Rauch, M.  Taylor (eds.), Springer-Verlag, New York, 1997, pp. 37–60. MR 1601273
[15] R. T. Glassey, J. K. Hunter, and Yuxi Zheng: Singularities of a variational wave equation. J. Differ. Equations 129 (1996), 49–78. DOI 10.1006/jdeq.1996.0111 | MR 1400796
[16] A.  Grundland E.  Infeld: A family of nonlinear Klein-Gordon equations and their solutions. J.  Math. Phys. 33 (1992), 2498–2503. DOI 10.1063/1.529620 | MR 1167950
[17] J. K. Hunter, R. A. Saxton: Dynamics of director fields. SIAM J.  Appl. Math. 51 (1991), 1498–1521. DOI 10.1137/0151075 | MR 1135995
[18] J. K. Hunter, Yuxi Zheng: On a nonlinear hyperbolic variational equation  I and II. Arch. Ration. Mech. Anal. 129 (1995), 305–353, 355–383. DOI 10.1007/BF00379259
[19] R. L.  Jerrard, H. M.  Soner: Dynamics of Ginzburg-Landau vortices. Arch. Ration. Mech. Anal. 142 (1998), 99–125. DOI 10.1007/s002050050085 | MR 1629646
[20] J. L.  Joly, G.  Metivier, and J.  Rauch: Focusing at a point and absorption of nonlinear oscillations. Trans. Am. Math. Soc. 347 (1995), 3921–3969. DOI 10.1090/S0002-9947-1995-1297533-8 | MR 1297533
[21] P. Gérard: Microlocal defect measures. Commun. Partial Differ. Equations 16 (1991), 1761–1794. DOI 10.1080/03605309108820822 | MR 1135919
[22] Fanghua Lin: Some dynamical properties of Ginzburg-Landau vortices. Commun. Pure Appl. Math. 49 (1996), 323–359. DOI 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E | MR 1376654
[23] Fanghua Lin, Ping Zhang: On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete Contin. Dyn. Syst. 6 (2000), 121–142. DOI 10.3934/dcds.2000.6.121 | MR 1739596
[24] P.-L. Lions: Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible Models. Oxford Lecture Series in Mathematics and Its Applications. Clarendon Press, Oxford, 1998. MR 1637634
[25] P.-L.  Lions, N. Masmoudi: Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math., Ser.  B 21 (2000), 131–146. DOI 10.1142/S0252959900000170 | MR 1763488
[26] N. Masmoudi, Ping Zhang: Weak solutions to the vortex density equations arising from sup-conductivity. Ann. Inst. Henri  Poincaré, Anal. Non Linéaire 22 (2005), 441–458. DOI 10.1016/j.anihpc.2004.07.002 | MR 2145721
[27] H. Mckean: Breakdown of shallow water equation. Asian J.  Math. 2 (1998), 867–874. DOI 10.4310/AJM.1998.v2.n4.a10 | MR 1734131
[28] F. Murat: Compacité par compensation. Ann. Sc. Norm. Super. Pisa, Cl.  Sci, IV 5 (1978), 489–507. MR 0506997 | Zbl 0399.46022
[29] R. A.  Saxton: Dynamic instability of the liquid crystal director. In: Contemp. Math. Vol.  100: Current Progress in Hyperbolic Systems, W. B. Lindquist (ed.), AMS, Providence, 1989, pp. 325–330. MR 1033527 | Zbl 0702.35180
[30] L. Tartar: Compensated compactness and applications to partial differential equations. Nonlinear Anal. Mech. Heriot-Watt Symposium. Research Notes in Math., Vol.  39,, R. J. Knops (ed.), Pitman Press, , 1979. MR 0584398
[31] L. Tartar: $H$-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edinb., Sect. A 115 (1990), 193–230. DOI 10.1017/S0308210500020606 | MR 1069518 | Zbl 0774.35008
[32] Zhouping Xin, Ping Zhang: On the weak solutions to a shallow water equation. Comm. Pure. Appl. Math. LIII (2000), 1411–1433. MR 1773414
[33] L. C. Young: Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, Philadelphia-London-Toronto, 1969. MR 0259704 | Zbl 0177.37801
[34] Ping Zhang, Yuxi Zheng: On oscillations of an asymptotic equation of a nonlinear variational wave equation. Asymptotic Anal. 18 (1998), 307–327. MR 1668954
[35] Ping Zhang, Yuxi Zheng: Existence and uniqueness of solutions to an asymptotic equation arising from a variational wave equation with general data. Arch. Ration. Mech. Anal. 155 (2000), 49–83. DOI 10.1007/s205-000-8002-2 | MR 1799274
[36] Ping Zhang, Yuxi Zheng: Rarefactive solutions to a nonlinear variational wave equation. Commun. Partial Differ. Equations 26 (2001), 381–419. DOI 10.1081/PDE-100002240 | MR 1842038
[37] Ping Zhang, Yuxi Zheng: Singular and rarefactive solutions to a nonlinear variational wave equation. Chin Ann. Math., Ser. B 22 (2001), 159–170. DOI 10.1142/S0252959901000152 | MR 1835396
[38] Ping Zhang, Yuxi Zheng: Weak solutions to nonlinear variational wave equation. Arch. Ration. Mech. Anal. 166 (2003), 303–319. DOI 10.1007/s00205-002-0232-7 | MR 1961443
[39] Ping Zhang, Yuxi Zheng: Weak solutions to a nonlinear variational wave equation with general data. Ann. Inst. Henri  Poincaré, Anal. Non Linéaire 22 (2005), 207–226. DOI 10.1016/j.anihpc.2004.04.001 | MR 2124163
[40] H.  Zorski, E.  Infeld: New soliton equations for dipole chains. Phys. Rev. Lett. 68 (1992), 1180–1183. DOI 10.1103/PhysRevLett.68.1180
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