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On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

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Abstract

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L 2(ℝ) initial data.

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References

  1. Ping Zhang, Yuxi Zheng. On oscillations of an asymptotic equation of a nonlinear variational wave equation. Asymptotic Analysis, 1998, (in press)

  2. J K Hunter, Yuxi Zheng. On a Nonlinear hyperbolic variational equation I and II. Arch Rat Mech Anal, 1995, 129: 305–353, 355–383

    Article  MATH  Google Scholar 

  3. J K Hunter, R A Saxton. Dynamics of Director Fields. SIAM J Appl Math, 1991, 51: 1498–1521

    Article  MATH  Google Scholar 

  4. H Zorski, E Infeld. New soliton equations for dipole chains. Phys Rev Lett, 1992, 68: 1180–1183

    Article  Google Scholar 

  5. A Albers, R Camassa, D Holm, J Marsden. The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s. Lett Math Phys, 1994, 32: 137–151

    Article  Google Scholar 

  6. R Camassa, D Holm. An integrable shallow water equation with peaked solitons. Phy Rev Lett, 1993, 71: 1661–1664

    Article  MATH  Google Scholar 

  7. A Grundland, E Infeld. A family of nonlinear Klein-Gordon equations and their solutions. J Math Phys, 1992, 33: 2498–2503

    Article  MATH  Google Scholar 

  8. R T Glassey, J K Hunter, Yuxi Zheng. Singularities and oscillations in a nonlinear variational wave equation. in Singularities and Oscillations, ed by J Rauch, M Taylor, IMA, Springer 1997, 91: 37–60.

  9. R DiPerna, P L Lions. Ordinary Differential Equations, Transport Theory and Sobolev Spaces. Invent Math, 1989, 98: 511–547

    Article  MATH  Google Scholar 

  10. R DiPerna, P L Lions. On the Cauchy Problem for Boltzman Equations; Global Existence and Weak Stability. Annals of Math, 1989, 130: 321–366

    Article  Google Scholar 

  11. L C Evans. Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS 74, AMS, Providence, R I, 1990

    MATH  Google Scholar 

  12. P L Lions. Mathematical Topics in Fluid Mechanics. Vol 1, Incompressible Models, Lecture Series in Mathematics and its Application, Vol 3, Clarendon Press, Oxford, 1996

    Google Scholar 

  13. L C Evans, R F Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, Inc, 1992.

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Authors and Affiliations

  1. Institute of Mathematics, Chinese Academy of Sciences, 100080, Beijing, P. R. China

    Ping Zhang

  2. Department of Mathematics, Indiana University, 47405, Bloomington, Indiana, USA

    Yuxi Zheng

Authors
  1. Ping Zhang
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  2. Yuxi Zheng
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Additional information

The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.

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Zhang, P., Zheng, Y. On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation. Acta Math Sinica 15, 115–129 (1999). https://doi.org/10.1007/s10114-999-0063-7

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  • DOI: https://doi.org/10.1007/s10114-999-0063-7

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