A Quasilinear System Related with the Asymptotic Equation of the Nematic Liquid Crystal’s Director Field

Abstract

In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal’s asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math. Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal., 129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng, Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007, 163–185].

Keywords

Benney system / Conservation law / Schrödinger equation / Nematic liquid crystal / Director field

References

[1] Antontsev S, Dias J P, Figueira M, Oliveira F. Non-existence of global solutions for a quasilinear Benney system. J. Math. Fluid Mech., 2011, 13: 213-222

[2] Benney D. A general theory for interactions between short and long waves. Stud. Appl. Math., 1977, 56: 81-94

[3] Bressan A, Zhang P, Zheng Y. Asymptotic variational wave equations. Arch. Rat. Mech. Anal., 2007, 183: 163-185

[4] Cazenave, T., Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, Courant Institute of Math. Sciences, 2003.

[5] Dias J P, Figueira M. Existence of weak solutions for a quasilinear version of Benney equations. J. Hyperbolic Differ. Equ., 2007, 4: 555-563

[6] Dias J P, Figueira M, Frid H. Vanishing viscosity with short-long wave interactions for systems of conservation laws. Arch. Ration. Mech. Anal., 2010, 196: 981-1010

[7] Dias J P, Figueira M, Oliveira F. Existence of local strong solutions for a quasilinear Benney system. C. R. Acad. Sci. Paris I, 2007, 344: 493-496

[8] Dias J P, Oliveira F. On a quasilinear nonlocal Benney system. J. Hyperbolic Differ. Equ., 2017, 14: 135-156

[9] Glassey R T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys., 1977, 18: 1794-1797

[10] Hunter J K, Saxton R A. Dynamics of director fields. SIAM J. Appl. Math., 1991, 51: 1498-1521

[11] Hunter J K, Zheng Y. On a nonlinear hyperbolic variational equation I. Arch. Rat. Mech. Anal., 1995, 129: 305-353

[12] Hunter J K, Zheng Y. On a nonlinear hyperbolic variational equation II. Arch. Rat. Mech. Anal., 1995, 129: 355-383

[13] Kato T. Quasi-linear equations of evolution, with applications to partial differential equations. Lecture Notes in Mathematics, 1975, 448: 25-70 Springer

[14] Saxton R A. Lindquist W B. Dynamic instability of the liquid crystal director. Current Progress in Hyperbolic Systems, 1989, Providence, RI: Amer. Math. Soc. 325-330

[15] Zhang P, Zheng Y. On oscillation of an asymptotic equation of a nonlinear variational wave equation. Asymptotic Anal., 1998, 18: 307-327