Weak solutions to a nonlinear variational wave equation and some related problems

Ping Zhang

Applications of Mathematics (2006)

  • Volume: 51, Issue: 4, page 427-466
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Zhang, Ping. "Weak solutions to a nonlinear variational wave equation and some related problems." Applications of Mathematics 51.4 (2006): 427-466. <http://eudml.org/doc/33260>.

@article{Zhang2006,
abstract = {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.},
author = {Zhang, Ping},
journal = {Applications of Mathematics},
keywords = {variational wave equation; weak solutions; $L^p$ Young measure; renormalized solutions; variational wave equation; weak solutions; Young measure; renormalized solutions},
language = {eng},
number = {4},
pages = {427-466},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Weak solutions to a nonlinear variational wave equation and some related problems},
url = {http://eudml.org/doc/33260},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Zhang, Ping
TI - Weak solutions to a nonlinear variational wave equation and some related problems
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 4
SP - 427
EP - 466
AB - 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.
LA - eng
KW - variational wave equation; weak solutions; $L^p$ Young measure; renormalized solutions; variational wave equation; weak solutions; Young measure; renormalized solutions
UR - http://eudml.org/doc/33260
ER -

References

top
  1. 10.1007/s00205-006-0014-8, Archive for Rational Mechanics and Analysis, Online 2006. MR2259342DOI10.1007/s00205-006-0014-8
  2. 10.1103/PhysRevLett.71.1661, Phys. Rev. Lett. 71 (1993), 1661–1664. (1993) MR1234453DOI10.1103/PhysRevLett.71.1661
  3. 10.1017/S0956792500002242, Eur. J.  Appl. Math. 7 (1996), 97–111. (1996) MR1388106DOI10.1017/S0956792500002242
  4. 10.1007/BF02392586, Acta Math. 181 (1998), 229–243. (1998) MR1668586DOI10.1007/BF02392586
  5. 10.1007/BF01206047, Comm. Math. Phys. 91 (1983), 1–30. (1983) Zbl0533.76071MR0719807DOI10.1007/BF01206047
  6. 10.1007/BF01393835, Invent. Math. 98 (1989), 511–547. (1989) MR1022305DOI10.1007/BF01393835
  7. 10.2307/1971423, Ann. Math. 130 (1989), 321–366. (1989) MR1014927DOI10.2307/1971423
  8. 10.1007/BF01214424, Commun. Math. Phys. 108 (1987), 667–689. (1987) MR0877643DOI10.1007/BF01214424
  9. 10.1137/S0036141002408009, SIAM J.  Math. Anal. 34 (2003), 1279–1299. (2003) MR2000970DOI10.1137/S0036141002408009
  10. 10.1103/PhysRevB.50.1126, Phys. Rev.  B 50 (1994), 1126–1135. (1994) DOI10.1103/PhysRevB.50.1126
  11. Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS No. 74, AMS, Providence, 1990. (1990) MR1034481
  12. 10.1007/PL00000976, J.  Math. Fluid Mech. 3 (2001), 358–392. (2001) MR1867887DOI10.1007/PL00000976
  13. 10.1016/0167-2789(81)90004-X, Phys.  D 4 (1981/1982), 47–66. (1981/1982) MR0636470DOI10.1016/0167-2789(81)90004-X
  14. 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. (1997) MR1601273
  15. 10.1006/jdeq.1996.0111, J. Differ. Equations 129 (1996), 49–78. (1996) MR1400796DOI10.1006/jdeq.1996.0111
  16. 10.1063/1.529620, J.  Math. Phys. 33 (1992), 2498–2503. (1992) MR1167950DOI10.1063/1.529620
  17. 10.1137/0151075, SIAM J.  Appl. Math. 51 (1991), 1498–1521. (1991) MR1135995DOI10.1137/0151075
  18. 10.1007/BF00379259, Arch. Ration. Mech. Anal. 129 (1995), 305–353, 355–383. (1995) DOI10.1007/BF00379259
  19. 10.1007/s002050050085, Arch. Ration. Mech. Anal. 142 (1998), 99–125. (1998) MR1629646DOI10.1007/s002050050085
  20. 10.1090/S0002-9947-1995-1297533-8, Trans. Am. Math. Soc. 347 (1995), 3921–3969. (1995) MR1297533DOI10.1090/S0002-9947-1995-1297533-8
  21. 10.1080/03605309108820822, Commun. Partial Differ. Equations 16 (1991), 1761–1794. (1991) MR1135919DOI10.1080/03605309108820822
  22. 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E, Commun. Pure Appl. Math. 49 (1996), 323–359. (1996) MR1376654DOI10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E
  23. 10.3934/dcds.2000.6.121, Discrete Contin. Dyn. Syst. 6 (2000), 121–142. (2000) MR1739596DOI10.3934/dcds.2000.6.121
  24. Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible Models. Oxford Lecture Series in Mathematics and Its Applications, Clarendon Press, Oxford, 1998. (1998) MR1637634
  25. 10.1142/S0252959900000170, Chin. Ann. Math., Ser.  B 21 (2000), 131–146. (2000) MR1763488DOI10.1142/S0252959900000170
  26. 10.1016/j.anihpc.2004.07.002, Ann. Inst. Henri  Poincaré, Anal. Non Linéaire 22 (2005), 441–458. (2005) MR2145721DOI10.1016/j.anihpc.2004.07.002
  27. 10.4310/AJM.1998.v2.n4.a10, Asian J.  Math. 2 (1998), 867–874. (1998) MR1734131DOI10.4310/AJM.1998.v2.n4.a10
  28. Compacité par compensation, Ann. Sc. Norm. Super. Pisa, Cl.  Sci, IV 5 (1978), 489–507. (1978) Zbl0399.46022MR0506997
  29. 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. (1989) Zbl0702.35180MR1033527
  30. 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. (1979) MR0584398
  31. 10.1017/S0308210500020606, Proc. R. Soc. Edinb., Sect. A 115 (1990), 193–230. (1990) Zbl0774.35008MR1069518DOI10.1017/S0308210500020606
  32. On the weak solutions to a shallow water equation, Comm. Pure. Appl. Math. LIII (2000), 1411–1433. (2000) MR1773414
  33. Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia-London-Toronto, 1969. (1969) Zbl0177.37801MR0259704
  34. On oscillations of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal. 18 (1998), 307–327. (1998) MR1668954
  35. 10.1007/s205-000-8002-2, Arch. Ration. Mech. Anal. 155 (2000), 49–83. (2000) MR1799274DOI10.1007/s205-000-8002-2
  36. 10.1081/PDE-100002240, Commun. Partial Differ. Equations 26 (2001), 381–419. (2001) MR1842038DOI10.1081/PDE-100002240
  37. 10.1142/S0252959901000152, Chin Ann. Math., Ser. B 22 (2001), 159–170. (2001) MR1835396DOI10.1142/S0252959901000152
  38. 10.1007/s00205-002-0232-7, Arch. Ration. Mech. Anal. 166 (2003), 303–319. (2003) MR1961443DOI10.1007/s00205-002-0232-7
  39. 10.1016/j.anihpc.2004.04.001, Ann. Inst. Henri  Poincaré, Anal. Non Linéaire 22 (2005), 207–226. (2005) MR2124163DOI10.1016/j.anihpc.2004.04.001
  40. 10.1103/PhysRevLett.68.1180, Phys. Rev. Lett. 68 (1992), 1180–1183. (1992) DOI10.1103/PhysRevLett.68.1180

NotesEmbed ?

top

You must be logged in to post comments.