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On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions. II: The zero-viscosity and dispersion limits. (English) Zbl 0834.35085

In these two papers the authors study in great detail the nonlinear hyperbolic equation ${\left({u}_{t}+u{u}_{x}\right)}_{x}={u}_{x}^{2}/2$; it arises as the canonical asymptotic equation for weakly nonlinear solutions of some hyperbolic equations derived from variational principles (e.g., in the dynamics of liquid crystals).

In the first paper two kinds of weak solutions are defined: they are called dissipative and conservative solutions, according to energy decays rapidly or is conserved; they agree for smooth solutions. In both classes it is proved the global existence of Hölder continuous solutions (in spite of the fact that derivatives may blow up) provided that the derivative of the initial data has bounded variation and compact support. Proofs are obtained by means of a careful analysis of some approximate explicit solutions. It is also shown that conservative solutions can be obtained as limits of solutions constructed with the so-called regularized method of characteristics; moreover dissipative solutions are asymptotic, for large time, to a special piecewise linear solution. The paper also includes detailed comparaisons with Burgers’ equation.

The second paper deals with the convergence of solutions obtained by viscous or dispersive regularizations. The authors consider as initial data a simple step function giving blow up of the derivative of the solution and show that solutions obtained by viscous regularization tend to a unique global weak solution of the equation without viscosity. At last, some numerical computations are given which suggest that solutions obtained by dispersive regularization converge, for vanishing dispersion, to a conservative solution which is different from the preceding one.

##### MSC:
 35L70 Nonlinear second-order hyperbolic equations 35Q53 KdV-like (Korteweg-de Vries) equations 35B40 Asymptotic behavior of solutions of PDE 35D05 Existence of generalized solutions of PDE (MSC2000)
##### References:
 [1] Coron, J. M., Ghidaglia, J. M. & Hélein, F. (eds.), Nematics, Kluwer Academic Publishers, 1991. [2] Dafermos, C. M., The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Eqs., 14 (1973), pp. 202-212. · Zbl 0262.35038 · doi:10.1016/0022-0396(73)90043-0 [3] Dafermos, C. M., Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana U. Math. J., 26 (1977), pp. 1097-1119. · Zbl 0377.35051 · doi:10.1512/iumj.1977.26.26088 [4] Ericksen, J. L. On the equations of motion for liquid crystals, Q. J. Mech. Appl. Math., 29 (1976), pp. 202-208. · Zbl 0328.76002 · doi:10.1093/qjmam/29.2.203 [5] Ericksen, J. L. & Kinderlehrer, D., Theory and applications of liquid crystals, Springer-Verlag, 1987. [6] Evans, L. C. & Gariepy, R. F., Lecture notes on measure theory and fine properties of functions, CRC Press, 1991. [7] Hunter, J. K. & Keller, J. B., Weakly nonlinear high frequency waves, Comm. Pure Appl. Math., 36 (1983), pp. 547-569. · Zbl 0547.35070 · doi:10.1002/cpa.3160360502 [8] Hunter, J. K. & Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), pp. 1498-1521. · Zbl 0761.35063 · doi:10.1137/0151075 [9] 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. · doi:10.1007/BF00379260 [10] Hunter, J. K. & Zheng, Yuxi, On a completely integrable nonlinear hyperbolic variational equation, to appear in Physica D. [11] Leslie, F. M., Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), pp. 265-283. · Zbl 0159.57101 · doi:10.1007/BF00251810 [12] Lusternik, L. A. & Sobolev, V. J., Elements of Functional Analysis, 3rd ed., Wiley, New York (1974). [13] 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. [14] Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. [15] Tartar, L., Compensated Compactness and Applications, Heriot-Watt Symposium, IV, Ed. R. J. Knops, Research Notes in Math., No. 39, Pitman, London, 1979.