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

João-Paulo DIAS

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

1 Introduction and Main Results

Motivated by the study of an asymptotic equation for the director field in a simplified model of a nematic liquid crystal,introduced by Saxton,R.A.and Hunter,J.K.in[10,14]and studied by Hunter,J.K.,Zheng,Y.and Zhang,P.in[11-12,15],we study the interaction with a short wave induced by an electron beam(cf.also[3]).

This interaction can be described by the following coupled time dependent system of the Benney type(cf.[2,5-8]for some previous examples):

the solution

In Section 3,we start by deducing some identities for the flow(1.1)and,with some additional requirements on the initial data,and in particular forv≥0 anda<0,we derive a blow-up in time result for the local strong solution of the IBV problem(1.1)-(1.2)obtained in Theorem 1.1.We will apply a virial technique developed in the seminal work of R.T.Glassey(cf.[9])concerning nonlinear Schrödinger equations(see[8]for a related result and[10-12],for blow-up results concerning the equation(1.3)).The function

2 Local Existence and Uniqueness

This lemma implies Theorem 1.1.Indeed,if(F,v)is a solution of the IBV problem(2.2)we obtainut=Fandu(x,0)=u0(x).We derive

and

3 A Blow-up Result

and so we obtain(3.2)by integrating int.Similarly,we derive

To prove the blow-up result Theorem 1.2,we begin to establish the following lemma.

and(by(1.1))

and Lemma 3.1 is proved.

AcknowledgementThe author is indebted to the referee for valuable suggestions and corrections.