Archive for Rational Mechanics and Analysis

, Volume 129, Issue 4, pp 305–353 | Cite as

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

  • John K. Hunter
  • Yuxi Zheng
Article

Abstract

We study the nonlinear hyperbolic partial differential equation, (ut+uux)x=1/2u x 2 . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.

Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.

There are no a priori estimates on the second derivatives in any Lp space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.

We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.

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References

  1. [C]
    Coron, J. M., Ghidaglia, J. M. & Hélein, F. (eds.), Nematics, Kluwer Academic Publishers, 1991.Google Scholar
  2. [D1]
    Dafermos, C. M., The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Eqs., 14 (1973), pp. 202–212.Google Scholar
  3. [D2]
    Dafermos, C. M., Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana U. Math. J., 26 (1977), pp. 1097–1119.Google Scholar
  4. [E]
    Ericksen, J. L. On the equations of motion for liquid crystals, Q. J. Mech. Appl. Math., 29 (1976), pp. 202–208.Google Scholar
  5. [EK]
    Ericksen, J. L. & Kinderlehrer, D., Theory and applications of liquid crystals, Springer-Verlag, 1987.Google Scholar
  6. [EG]
    Evans, L. C. & Gariepy, R. F., Lecture notes on measure theory and fine properties of functions, CRC Press, 1991.Google Scholar
  7. [HK]
    Hunter, J. K. & Keller, J. B., Weakly nonlinear high frequency waves, Comm. Pure Appl. Math., 36 (1983), pp. 547–569.Google Scholar
  8. [HS]
    Hunter, J. K. & Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), pp. 1498–1521.Google Scholar
  9. [HZ1]
    Hunter, J. K. & Zheng, Yuxi, On a nonlinear hyperbolic variational equation: II. The zero viscosity and dispersion limits, Arch. Rational Mech. Anal, 129 (1995), pp. 355–383.Google Scholar
  10. [HZ2]
    Hunter, J. K. & Zheng, Yuxi, On a completely integrable nonlinear hyperbolic variational equation, to appear in Physica D.Google Scholar
  11. [L]
    Leslie, F. M., Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), pp. 265–283.Google Scholar
  12. [LS]
    Lusternik, L. A. & Sobolev, V. J., Elements of Functional Analysis, 3rd ed., Wiley, New York (1974).Google Scholar
  13. [S]
    Saxton, R. A., Dynamic instability of the liquid crystal director, in Contemporary Mathematics, 100: Current Progress in Hyperbolic Systems, 325–330, Ed. W. B. Lindquist, Amer. Math. Soc., Providence, 1989.Google Scholar
  14. [SM]
    Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.Google Scholar
  15. [T]
    Tartar, L., Compensated Compactness and Applications, Heriot-Watt Symposium, IV, Ed. R. J. Knops, Research Notes in Math., No. 39, Pitman, London, 1979.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • John K. Hunter
    • 1
    • 2
  • Yuxi Zheng
    • 1
    • 2
  1. 1.Department of Mathematics and Institute for Theoretical DynamicsUniversity of California at DavisUSA
  2. 2.Department of MathematicsIndiana University at BloomingtonUSA

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