-
J. Differential Equations 255 (2013) 12331253
Contents lists available at SciVerse ScienceDirect
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eakstrong uniqueness property for the compressible owf liquid
crystals
ong-Fu Yang a,c,, Changsheng Dou b,c, Qiangchang Ju c
epartment of Mathematics, College of Sciences, Hohai University,
Nanjing 210098, Jiangsu Province, PR Chinachool of Statistics,
Capital University of Economics and Business, Beijing 100070, PR
Chinanstitute of Applied Physics and Computational Mathematics,
P.O. Box 8009, Beijing 100088, PR China
r t i c l e i n f o a b s t r a c t
ticle history:ceived 30 March 2012vised 12 April 2013ailable
online 28 May 2013
ywords:uid crystalmpressible hydrodynamic owlative
entropyeakstrong uniqueness
Weakstrong uniqueness property in the class of nite energyweak
solutions is established for two different compressible
liquidcrystal systems by the method of relative entropy. To
overcome thediculties caused by the molecular direction with
inhomogeneousDirichlet boundary condition, new techniques are
introduced tobuild up the relative entropy inequalities.
2013 Elsevier Inc. All rights reserved.
Introduction
In physics, liquid crystals are states of matter which are
capable of ow and in which the molecularrangements give rise to a
preferred direction. Nematic liquid crystals are aggregates of
moleculeshich posses same orientational order and are made of
elongated, rod-like molecules. The continuumeory of compressible
(or incompressible) liquid crystals was rst developed by Ericksen
[5] andslie [15] during the period of 1958 through 1968. Since then
there have been remarkable researchvelopments in liquid crystals
from both theoretical and applied aspects.When the uid is an
incompressible, viscous uid, Lin [17] rst derived simplied
EricksenLeslieuations modeling liquid crystal ows in 1989.
Subsequently, the global existence of weak solutionsith large
initial data was proved under the condition that the orientational
conguration belongsH2, and the global existence of classical
solutions was also obtained if the viscosity coecient is
rge enough in the three-dimensional spaces by Lin and Liu
[1820]. While the uid is allowed to
Corresponding author at: Department of Mathematics, College of
Sciences, Hohai University, Nanjing 210098, Jiangsuovince, PR
China.E-mail addresses: [email protected] (Y.-F. Yang),
[email protected] (C. Dou), [email protected] (Q.
Ju).
22-0396/$ see front matter 2013 Elsevier Inc. All rights
reserved.
tp://dx.doi.org/10.1016/j.jde.2013.05.011
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1234 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
beto
liq
wmbakitam
wde
wF
w
bo
an
w
didecompressible, the corresponding EricksenLeslie system
becomes more complicated and dicultstudy mathematically due to the
compressibility.Our goal of this paper is to establish weakstrong
uniqueness property for the compressible ow ofuid crystals. To
begin with, we consider the following simplied version of
EricksenLeslie system:
t + div(u) = 0,(u)t + div(u u) + P = u div
(d d
(1
2|d|2 + F (d)I3
)),
dt + u d = (d f(d)),
(1.1)
here 0 denotes the density, u R3 the velocity, d R3 the
direction eld for the averagedacroscopic molecular orientations,
and P () = a the pressure with constant a > 0 and the adia-tic
exponent 1. The positive constants , , denote the viscosity, the
competition betweennetic energy and potential energy, and the
microscopic elastic relation time for the molecular orien-tion eld,
respectively. The symbol denotes the Kronecker tensor product, I3
is the 3 3 identityatrix, and d d denotes the 3 3 matrix whose i
j-th entry is xid, x jd. Indeed,
d d = (d)Td,
here (d)T stands for the transpose of the 33 matrix d. The
vector-valued smooth function f(d)notes the penalty function and
has the following form:
f(d) = dF (d),
here the scalar function F (d) is the bulk part of the elastic
energy. A typical example is to choose(d) as the GinzburgLandau
penalization thus yielding the penalty function f(d) as:
F (d) = 14 20
(|d|2 1)2, f(d) = 1 20
(|d|2 1)d,here 0 > 0 is a constant.Let R3 be a bounded smooth
domain. In this paper, we will consider the following initialundary
conditions:
(,u,d)t=0 =
(0(x),m0(x),d0(x)
), x , (1.2)
d
u| = 0, d| = d0(x), x , (1.3)
here
0 L (), 0 0; d0 L() H1();
m0 L1(), m0 = 0 if 0 = 0; |m0|2
0 L1().
The global existence of weak solutions (,u,d) to the
initialboundary problem (1.1)(1.3) in threemensions with > 3/2
was obtained by Wang and Yu in [29] and Liu and Qing in [26],
indepen-
ntly. We should also remark that, when system (1.1) is
incompressible, Jiang and Tan [13] and Liu
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Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1235
antarenonaco
thfo
wthsybo
w
wocedoWth
emanAsexao
cationepooeqStsysufotoheWd Zhang in [25] proved the global weak
existence of solutions to the ow of nematic liquid crys-ls for uids
with non-constant density. Based on the existence result, Dai et
al. in [1] extended thegularity and uniqueness results of Lin and
Liu in [19] to the systems of nematic liquid crystals
withn-constant uid density. Since the global existence of the weak
solutions has been established, it istural to study their
uniqueness property. To our best knowledge, so far there are very
few resultsncerning uniqueness of weak solutions to the
initialboundary value problem (1.1)(1.3).When the OssenFrank energy
conguration functional reduces to the Dirichlet energy
functional,
e hydrodynamic ow equation of nematic compressible liquid
crystals can actually be written asllows:
t + div(u) = 0,(u)t + div(u u) + P = u div
(d d 1
2|d|2I3
),
dt + u d = (d+ |d|2d),
(1.4)
here d S2 and the other symbols have the same meanings with
those in system (1.1). We refer toe readers to consult the recent
papers [4] and [26] for the derivation of the system (1.4). For
thestem (1.4), we are concerned with the same initial conditions
(1.2) but with the following differentundary conditions:
u| = 0, d
= 0, x , (1.5)
here is the unit outer normal vector of .In contrast with system
(1.1), from the mathematical point of view, it is much more dicult
to deal
ith the nonlinear term |d|2d appearing in the third equation of
(1.4). Even for the incompressiblew, there have been no
satisfactory results concerning the global existence of weak
solutions. Re-ntly, Lin et al. in [21] proved the existence of
global-in-time weak solutions on a bounded smoothmain in R2. For
three-dimensional case, the problem is still open. We should
mention that Li andang in [16] have established its weakstrong
uniqueness principle in three dimensions, providedat the existence
of its weak solution is obtained.The compressible ow (1.4) of
liquid crystals is much more complicated and hard to study
math-atically due to the compressibility. For the one-dimensional
case, the global existence of smoothd weak solutions to the
compressible ow of liquid crystals was obtained by Ding et al. in
[2,3].for three-dimensional case, Huang et al. in [12] and Liu and
Zhang in [25] established the localistence of a unique strong
solution provided that the initial data are suciently regular and
satisfynatural compatibility condition. However, the global
existence of weak solution to the compressiblew of liquid crystals
in multi-dimensions is still open.We would like to point out that
the system (1.4) includes several important equations as
specialses: (i) When is constant, the rst equation in (1.4) reduces
to the incompressibility condi-n of the uid (divu = 0), and the
system (1.4) becomes the equation of incompressible ow ofmatic
liquid crystals provided that P is an unknown pressure function.
This was previously pro-sed by Lin [17] as a simplied
EricksenLeslie equation modeling incompressible liquid crystalws.
(ii) When d is a constant vector eld, the system (1.4) becomes a
compressible NavierStokesuation. There have been a number of papers
in the literature on the multi-dimensional Navierokes equations
(see [6,24,27] and the references therein). (iii) When both and d
are constants, thestem (1.4) becomes the incompressible
NavierStokes equation provided that P is an unknown pres-re
function, the fundamental equation to describe Newtonian uids (see
Lions [23] and Temam [28]r survey of important developments). (iv)
When is constant and u = 0, the system (1.4) reducesthe equation
for the heat ow of harmonic maps into S2. There have been extensive
studies on theat ow of harmonic maps in the past few decades (see,
for example, the monograph by Lin and
ang [22] and the references therein).
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1236 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
toisfeintiesexticoatglrestdaentitoneNunThclcoww
LebaCow(fanmththeffrisde
soqueq
th
thIt is well known that the uniqueness of weak solutions is
another fundamental and importantpic in mathematical theory of the
compressible NavierStokes equations. In contrast with the ex-tence
theory of weak solutions, it seems that uniqueness issue is more
dicult and there are onlyw results dealing with uniqueness. With
the aid of concept of the relative entropy, Germain [11]troduced a
class of (weak) solutions to the compressible NavierStokes system
satisfying a rela-ve entropy inequality with respect to a
(hypothetical) strong solution of the same problem, andtablished
the weakstrong uniqueness property within this class (see also
[10]). Unfortunately, theistence of weak solutions belonging to
this class, where, in particular, the density possesses a spa-al
gradient in a suitable Lebesgue space, is not known. Recently,
Feireisl et al. [7] introduced thencept of suitable weak solution
for the compressible NavierStokes system, satisfying a general
rel-ive entropy inequality with respect to any suciently regular
pair of functions. They showed theobal-in-time existence of the
suitable weak solutions for any nite energy initial data. Moreover,
thelative entropy inequality can be used to show that suitable weak
solutions comply with the weakrong uniqueness principle, meaning a
weak and a strong solution emanating from the same initialta
coincide as long as the latter exists. More recently, Feireisl et
al. [8] further showed that any niteergy weak solution satises a
relative entropy inequality with respect to any couple of smooth
func-ons satisfying relevant boundary conditions. Based on the
relative entropy inequality, they succeededprove the weakstrong
uniqueness and to provide a satisfactory answer to the weakstrong
unique-ss problem initially related to the fundamental questions of
the well-posedness for the compressibleavierStokes system. In
addition, Feireisl and Novotn [9] also established the similar
weakstrongiqueness property of the variational solutions for the
compressible NavierStokesFourier system.e authors of the present
paper [30] have established the weakstrong uniqueness property in
theass of nite energy weak solutions to the magnetohydrodynamic
equations of three-dimensionalmpressible isentropic ows with the
adiabatic exponent > 1 and constant viscosity coecients,hich
improves the corresponding results for the compressible isentropic
NavierStokes equationsith > 32 by Feireisl, Jin, and Novotn
[8].This paper aims to establish the weakstrong uniqueness property
for two simplied Ericksenslie systems (1.1) and (1.4),
respectively, in the spirit of Feireisl et al. [8]. Our method is
essentiallysed on the relative entropy, the modied relative entropy
inequality, and a Gronwall type argument.mpared with the existence
result of weak solution in [26,29], where both of them require >
3/2,e are going to make use of the techniques, established in [30],
to estimate the remainder R (or R1)or the denition see (3.10) (or
(4.10))), so as to establish the weakstrong uniqueness property
forimproved lower bound for any adiabatic exponent > 1. When
dealing with the compressible ne-
atic liquid crystal ow (1.1) or (1.4), the main diculty lies in
the coupling and interaction betweene velocity eld u and the
direction eld d. In particular, we should emphasize here that, as
far ase weakstrong uniqueness property for the initialboundary
problem (1.1)(1.3) is concerned, moreforts are required to build up
the relative entropy inequality so as to overcome the effects
arisingom inhomogeneous boundary condition d| = d0(x), which is
quite different from the case of theentropic compressible
NavierStokes equations investigated by Feireisl et al. in [8].
According to thenition of weak solutions for the compressible ow of
liquid crystals, it is easy to see that
d L2(0, T ; H2()),the weak solution d actually solves the third
equation of (1.1) (or (1.4)) in the strong sense. Conse-ently, we
can combine the arguments of Feireisl et al. in [8] with the energy
estimates for parabolicuation to obtain the desired relative
entropy inequality.The sizes of the positive constants , , and do
not play important roles in our proofs, we shall
erefore assume, for simplicity, that
= = = 1, (1.6)roughout this paper. In addition, to simplify the
notations, we always set
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Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1237
in
anmTh
2.
leA (B)C = ((C )B) A,which A, B, and C are vectors in R3.The
rest of the paper is organized as follows. In the next section, we
recall the denitions of weakd strong solutions to the compressible
ow of liquid crystals for two kinds of models and state theain
results. Section 3 is devoted to the derivation of the relative
entropy inequality and the proof ofeorem 2.1. Finally, we prove
Theorem 2.2 in Section 4.
Main results
We say that a triplet {,u,d} is a nite energy weak solution to
the initialboundary value prob-m (1.1)(1.3), if, for any T >
0,
0, L(0, T ; L ()), u L2(0, T ; H10()), u C0([0, T ], L2+1 ()
w),
d L((0, T ) ) L(0, T ; H1()) L2(0, T ; H2()),with (,u,d)(0, x) =
(0(x),m0(x),d0(x)) for x .
The rst equation in (1.1) is replaced by a family of integral
identities
( , )( , )dx
0(0, )dx =
0
(t + u )dxdt (2.1)
for any C1([0, T ] ), and any [0, T ]. Momentum equations (1.1)2
are satised in the sense of distributions, specically,
u( , )( , )dx
0u0 (0, )dx
=
0
(u t + u u : + P ()div u :
)dxdt
+
0
(d d
(1
2|d|2 + F (d)
)I3
): dxdt (2.2)
for any C1([0, T ] ), | = 0, and any [0, T ]. Eqs. (1.1)3 are
replaced by a family of integral identities
d( , ) ( , )dx
d0 (0, )dx
=
0
(d t d : (d)u f(d)
)dxdt (2.3)
1for any C ([0, T ] ), | = 0, and any [0, T ].
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1238 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
lea
(0
anre
Thththda
pr The energy inequality
E(t) +
0
(|u|2 + d f(d)2)dxdt E(0) (2.4)holds for a.e. [0, T ], where
E(t) =
(1
2|u|2 + a
1 + 1
2|d|2 + F (d)
)dx,
and
E(0) =
(1
2
|m0|20
+ a 1
0 +
1
2|d0|2 + F (d0)
)dx.
The existence of global-in-time nite energy weak solutions to
the initialboundary prob-m (1.1)(1.3) with the adiabatic exponent
> 32 was established in [26,29], provided there existspositive
constant C0 such that d f(d) 0 for all |d| C0 > 0.{, u, d} is
called a classical (strong) solution to the initialboundary problem
(1.1)(1.3) in
, T ) if{
C1([0, T ] ), (t, x) > 0 for all (t, x) (0, T ) ,u, t u,2u
C
([0, T ] ), d, t d,2d C([0, T ] ) (2.5)d , u, d satisfy Eq.
(1.1), together with the boundary conditions (1.3). Observe that
hypothesis (2.5)quires the following regularity properties of the
initial data:
{(0, ) = 0 C1(), 0 > 0,u(0, ) = u0 C2(), d(0, ) = d0
C2().
(2.6)
We are now ready to state the rst result of this paper.
eorem 2.1. Let R3 be a bounded domain with a boundary of class
C2+ , > 0, and > 1. Supposeat {,u,d} is a nite energy weak
solution to the initialboundary problem (1.1)(1.3) in (0, T ) ine
sense specied above, and suppose that {, u, d} is a strong solution
emanating from the same initialta (2.6).Then
, u u, d d.
In a similar way, we dene the nite energy weak solution
{1,u1,d1} to the initialboundary valueoblem (1.4), (1.2), and (1.5)
in the following sense: for any T > 0,
1 0, 1 L(0, T ; L ()), u1 L2(0, T ; H10()),
d1 L((0, T ) ) L(0, T ; H1()) L2(0, T ; H2()),with (1,1u1,d1)(0,
x) = (0(x),m0(x),d0(x)) for x .
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Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1239
Re
Rethet
(1
ansi
Thth(0sa Similarly to (2.1)(2.3), Eqs. (1.4) hold in D((0, T )
). The energy inequality
E1(t) +
0
(|u1|2 + d1 + |d1|2d12)dxdt E1(0) (2.7)
holds for a.e. [0, T ], where
E1(t) =
(1
21|u1|2 + a
11 +
1
2|d1|2
)dx,
and
E1(0) =
(1
2
|m0|20
+ a 1
0 +
1
2|d0|2
)dx.
mark 2.1. In the derivation of energy inequality (2.7), we have
used the fact that |d1| = 1 to get
(td1 + u1 d1) |d1|2d1 = 12|d1|2
(t |d1|2 + u1 |d1|2
)= 0.mark 2.2. We should remark that the global-in-time
renormalized nite energy weak solutions toe initialboundary value
problem (1.4), (1.2) and (1.5) is still open. For one-dimensional
case, Dingal. in [2,3] obtained the global-in-time existence of
weak solutions.
Similarly, {1, u1, d1} is called a classical (strong) solution
to the initialboundary value problem.4), (1.2) and (1.5) in (0, T )
if
{1 C1
([0, T ] ), 1(t, x) > 0 for all (t, x) (0, T ) ,u1, t u1,2u1
C
([0, T ] ), d1, t d1,2d1 C([0, T ] ) (2.8)d 1, u1, d1 satisfy
Eqs. (1.4), together with the boundary conditions (1.5). Observe
that hypothe-s (2.8) requires the following regularity properties
of the initial data:
{1(0, ) = 0 C1(), 0 > 0,u1(0, ) = u0 C2(), d1(0, ) = d0
C2().
(2.9)
We now end up this section with another result of this
paper.
eorem 2.2. Let R3 be a bounded domain with a boundary of class
C2+ , > 0, and > 1. Supposeat {1,u1,d1} is a nite energy weak
solution to the initialboundary value problem (1.4), (1.2) and
(1.5) in, T ) in the sense specied above, and suppose that {1, u1,
d1} is a strong solution emanating from theme initial data
(2.9).Then
1 1, u1 u1, d1 d1.
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1240 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
3.
E
w
tofuso
wreovarin
th
an
SiProof of Theorem 2.1
Motivated by the concept of relative entropy in [8], we rst dene
relative entropy E =([,u,d]|[, u, d]), with respect to {, u, d},
as
E =
(1
2|u u|2 + () ()( ) () + 1
2|d d|2
)dx, (3.1)
here
() = a 1
. (3.2)
In this section, we are going to deduce a relative entropy
inequality satised by any weak solutionthe initialboundary value
problem (1.1)(1.3). To this end, consider a triplet {, u, d} of
smoothnctions, bounded away from zero in [0, T ] , u| = 0, and d| =
d0(x). In addition, u and dlve the third equation of (1.1).Noticing
that the boundary conditions for d and d are inhomogeneous, i.e. d|
= d| = d0(x),
e should adapt and modify the arguments in [8] to build up the
relative entropy inequality. Moregularity of d allows us to make
use of energy estimates for parabolic equation, which help usercome
the diculty due to the inhomogeneous boundary conditions.
Consequently, combining theguments in [8] with the energy estimates
for parabolic equations yields the desired relative
entropyequality.To begin with, we take u as a test function in the
momentum equation (2.2) to obtain
u u( , )dx
0u0 u(0, )dx
=
0
(u t u+ u u : u+ P ()div u u : u
)dxdt
0
((d f(d)) (d)u)dxdt. (3.3)
Second, we can use the scalar quantity = 12 |u|2 and = (),
respectively, as test functions ine continuity equation (2.1) to
get
1
2|u|2( , )dx =
1
20|u|2(0, )dx+
0
(u t u+ u (u)u
)dxdt (3.4)
d
()( , )dx =
0()(0, )dx+
0
(t
() + u ())dxdt. (3.5)nce (u,d) solves the third equation of
(1.1) in the strong sense, it is easy to see that
( )
t(d d) + u d u d = (d d) f(d) f(d) , a.e.
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Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1241
M
Fiultiplying the above equation by (d d) and integrating over
(0, ), we have
1
2|d d|2( , )dx+
0
|d d|2 dxdt
=
1
2
d0 d(0, )2 dx+
0
(d d) (d)udxdt
0
(d d) (d)udxdt +
0
(f(d) f(d)) (d d)dxdt. (3.6)
nally, multiplying the third equation of (1.1) by d f(d)
gives
(1
2|d|2 + F (d)
)( , )dx+
0
d f(d)2 dxdt
=
(1
2|d0|2 + F (d0)
)dx+
0
(d f(d)) (d)udxdt. (3.7)
Summing up the relations (3.3)(3.6) with the energy inequality
(2.4), we infer that
(1
2|u u|2 + () () + 1
2|d d|2
)( , )dx
+
0
|u u|2 dxdt +
0
|d d|2 dxdt
(1
20u0 u(0, )2 + (0) 0 ((0, ))+ 1
2
d0 d(0, )2)dx
+
0
(t u+ u u) (u u)dxdt +
0
u : (u u)dxdt
0
(t
() + u ())dxdt
0
P ()div udxdt
0
(d f(d)) (d)udxdt +
0
(d f(d)) (d)udxdt
+
0
(d d) ((d)u (d)u)dxdt +
0
(d d) (f(d) f(d))dxdt,
(3.8)
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1242 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
w
an
A
an
w
an
itprhere we have used (3.7). By virtue of the denition (3.2) of
, it is easy to see that
() () = P ()d
(t
() + () u+ P ()div u)dx =
t P ()dx.
s a consequence, we deduce from the identities
P ()( , )dx
P ()(0, )dx =
0
t P ()dxdt
d (3.8) that the desired relative entropy inequality holds:
E( ) +
0
|u u|2 dxdt +
0
|d d|2 dxdt E(0) +
0
R(,u,d, , u, d)dt,
(3.9)
here
R = R(,u,d, , u, d) :=Rd +Rc, (3.10)Rd :=
(u u) (t u+ (u)u)dx+
u : (u u)dx
+
(( )t () + () (u u)
)dx
div u(P () P ())dx,
(3.11)
d
Rc : =
0
(d f(d)) (d)udxdt +
0
(d f(d)) (d)udxdt
+
0
(d d) ((d)u (d)u)dxdt +
0
(d d) (f(d) f(d))dxdt.(3.12)
In what follows, we shall nish the proof of Theorem 2.1 by
applying the relative entropy inequal-y (3.9) to {, u, d}, where {,
u, d} is a classical (smooth) solution of the initialboundary
valueoblem (1.1), (2.6), and (1.3), such that(0, ) = 0, u(0, ) =
u0, d(0, ) = d0.
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Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1243
w
Oby
tew
Af
CoAccordingly, the integrals depending on the initial data on
the right-hand side of (3.9) vanish, ande apply a Gronwall type
argument to deduce the desired result, namely,
, u u, d d.ur purpose is to examine all terms in the remainder
(3.10) and to show that they can be absorbedthe left-hand side of
(3.9).Compared with [8], we should remark that the main diculty
comes from the coupling and in-
raction between the velocity eld u and the direction eld d.
Moreover, in the context of theeakstrong uniqueness, we only assume
the adiabatic exponent > 1, but not > 3/2 as in [8].Similarly
to the proof of Theorem 2.1 in [30] (see also [8]), we use (3.11)
to nd that
Rd =
(u u) ((u)(u u))dx+
( )
u (u u)dx
div u(P () P ()( ) P ())dx
div
(d d
(1
2|d|2 + F (d)
)I3
) (u u)dx
=
(u u) ((u)(u u))dx
div u(P () P ()( ) P ())dx
+
( )
[u div
(d d
(1
2|d|2 + F (d)
)I3
)] (u u)dx
(d f(d)) (d)(u u)dx
=:Rd
(d f(d)) (d)(u u)dx. (3.13)
ter a tedious but straightforward computation, it follows from
(3.12) that
Rc :=Rc
(d f(d)) (d)(u u)dx
=
(d d) (f(d) f(d))dx+
(d d) (d d)udx
+
d (d d)(u u)dx+
f(d) (d d)(u u)dx
+
(f(d) f(d)) d(u u)dx. (3.14)
nsequently, it follows from the denitions of Rd and Rc
thatR(,u,d, , u, d) =Rd +Rc . (3.15)
-
1244 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
foth
H
O
Si
wth
foIn order to prove the weakstrong uniqueness property, we have
to estimate the remainder R. Asr Rd , since the procedures are
almost same as that in [30] (see also [8]), we just follow [30],
liste outlines, and skip the details. In the sequel, we are going
to focus on the estimation of Rc .From (2.6), it is clear to see
that
(u u) ((u)(u u))dx
div u(P () P ()( ) P ())dx
CuL()E([,u,d][, u, d]). (3.16)
ere and hereafter C stands for a generic constant, which may
change from line to line.Let
g= g(u, d,d,d) = u div(d d
(1
2|d|2 + F (d)
)I3
).
bviously, we have
1
( )g (u u)dx =
{ 2
-
Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1245
fo
an
w
an
fo
for any > 0. On the other hand, noticing that
() ()( ) () C , as 2 2d
12 2 C, as 2 2 and > 1,e conclude from the denition of
relative entropy E([,u,d]|[, u, d]) that
{2}
1
( )g (u u)dx
=
{2}
12 g 12 |u u|dx
=
{2}
( 12 2
)
2 g 12 |u u|dx
CgL()(
dx
) 12(
2|u u|2 dx
) 12
CgL()E([,u,d][, u, d]). (3.20)
Next, we continue to estimate Rc . Noticing that
dL((0,T )),dL((0,T )) C (3.21)d the fact that f is smooth, we
have
(d d) (f(d) f(d))dx C(
|d d|2 dx) 1
2(
|d d|2 dx) 1
2
d d2L2() + C()d d2L2() d d2L2() + C()d d2L2() d d2L2() +
C()E
([,u,d][, u, d]) (3.22)r any > 0, where we have used Sobolevs
inequality. It follows from Hlders inequality that
(d d) (d d)udx uL()d dL2()d dL2() d d2L2() + C()u2L()d d2L2() d
d2L2() + C()u2L()E
([,u,d][, u, d])(3.23)r any > 0. It is clear that
-
1246 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
Fi
de
w
Iteq
4.
valiklintahaplthE1
d (d d)(u u)dx+
f(d) (d d)(u u)dx
(dL() + f(d)L())u uL2()d dL2()
u u2L2() + C()(dL() + f(d)L())2d d2L2()
u u2L2() + C()(dL() + f(d)L())2E([,u,d][, u, d]). (3.24)
nally, (3.21) and the fact that f is smooth imply that
(f(d) f(d)) (d)(u u)dx
CdL()u uL2()d dL2() u u2L2() + C()
(dL())2d d2L2() u u2L2() + C()
(dL())2E([,u,d][, u, d]). (3.25)Summing up relations
(3.9)(3.25), we conclude that the relative entropy inequality
yields thesired conclusion
E([,u,d][, u, d])( )
0
h(t)E([,u,d][, u, d])(t)dt,
here
h(t) = C{uL() +
g2
L3()+ gL() + u2L()
+ (dL() + f(d)L())2 + d2L() + 1}.
follows from (3.21) that h L1(0, T ). Thus, Theorem 2.1
immediately follows from Gronwalls in-uality.
Proof of Theorem 2.2
This section is devoted to proving the weakstrong uniqueness
property for the initialboundarylue problem (1.4), (1.2), and
(1.5). Compared with the problem discussed in Section 3, we woulde
to point out two points: (i) The term |d1|2d1 in the third equation
of (1.4) has higher non-earity than f(d) in the third equation of
(1.1), so that more regularity results should be ob-ined to
overcome the diculty caused by |d1|2d1. (ii) Different from d| =
d0(x), we nowve homogeneous Neumann boundary condition for d1 at
hand, namely,
d1 | = 0, which im-
ies that a modied relative entropy is required to prove the
weakstrong uniqueness property fore initialboundary value problem
(1.4), (1.2), and (1.5). To be precise, we dene relative
entropy
= E1([1,u1,d1]|[1, u1, d1]), with respect to {1, u1, d1}, as
-
Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1247
wva1th
m
Setin
an
Nw
MwE1 =
(1
21|u1 u1|2 +
((1) (1)(1 1) (1)
)
+ 12
(|d1 d1|2 + |d1 d1|2))dx, (4.1)
here is dened by (3.2), {1,u1,d1} is the nite energy weak
solution to the initialboundarylue problem (1.4), (1.2), and (1.5)
in the sense of Section 2, and {1, u1, d1} are smooth
functions,bounded away from zero in [0, T ] , u1| = 0, and d1 | =
0. In addition, u1 and d1 solve
e third equation of (1.4).We rst establish the relative entropy
inequality. For this purpose, take u1 as a test function in the
omentum equation to obtain
1u1 u1( , )dx
0u0 u1(0, )dx (4.2)
=
0
(1u1 t u1 + 1u1 u1 : u1 + P (1)div u1 u1 : u1
)dxdt
0
d1 (d1)u1 dxdt.
cond, we can use the scalar quantity 12 |u1|2 and (1),
respectively, as test functions in the con-uity equation to get
1
21|u1|2( , )dx =
1
20|u1|2(0, )dx+
0
(1u1 t u1 + 1u1 (u1)u1
)dxdt
(4.3)
d
1(1)( , )dx =
0(1)(0, )dx+
0
(1t
(1) + 1u1 (1))dxdt.
(4.4)
otice that (u1,d1) solves the third equation of (1.4) in the
strong sense, we have for almost every-here
t(d1 d1) + u1 d1 u1 d1 = (d1 d1) +(|d1|2d1 |d1|2d1).
ultiplying the above equation by (d1 d1) and (d1 d1),
respectively, and integrating by parts,
e obtain
-
1248 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
an
Si
1
2|d1 d1|2 dx+
0
|d1 d1|2 dxdt
=
1
2
d0 d1(0, )2 dx+
0
(|d1|2d1 |d1|2d1) (d1 d1)dxdt
0
((d1)u1 (d1)u1
) (d1 d1)dxdt (4.5)
d
1
2|d1 d1|2 dx+
0
|d1 d1|2 dxdt
=
1
2
d0 d1(0, )2 dx
0
(|d1|2d1 |d1|2d1) (d1 d1)dxdt
+
0
((d1)u1 (d1)u1
) (d1 d1)dxdt. (4.6)
milarly, we multiply the third equation of (1.4) by d1 + |d1|2d1
to obtain
1
2|d1|2( , )dx+
0
d1 + |d1|2d12 dxdt
=
1
2|d0|2 dx+
0
d1 (d1)u1 dxdt. (4.7)
Summing up the relations (4.2)(4.6) with the energy inequality
(2.7), we infer that
(1
2|u1 u1|2 + (1) 1 (1) + 1
2
(|d1 d1|2 + |d1 d1|2))( , )dx
+
0
|u1 u1|2 dxdt +
0
|d1 d1|2 dxdt +
0
|d1 d1|2 dxdt
(1
20u0 u1(0, )2 + (0) 0 (1(0, ))
+ 1 (d d (0, )2 + d d (0, )2))dx
2
0 1 0 1
-
Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1249
wob
w
an
R+
0
1(t u1 + u1 u1) (u1 u1)dxdt +
0
u1 : (u1 u1)dxdt
0
(1t
(1) + 1u1 (1))dxdt
0
P (1)div u1 dxdt
+
0
d1 (d1)(u1 u1)dxdt +
0
(d1 d1) ((d1)u1 (d1)u1
)dxdt
0
(d1 d1) (|d1|2d1 |d1|2d1)dxdt
+
0
(d1 d1) (|d1|2d1 |d1|2d1)dxdt
0
(d1 d1) ((d1)u1 (d1)u1
)dxdt, (4.8)
here we have used (4.7). Thus, we proceed the similar procedures
as in the previous section totain the desired relative entropy
inequality
E1( ) +
0
|u1 u1|2 dxdt +
0
(|d1 d1|2 + |d1 d1|2)dxdt
E1(0) +
0
R1(1,u1,d1, 1, u1, d1)dt, (4.9)
here
R1 = R1(1,u1,d1, 1, u1, d1) :=R1d +R1c, (4.10)R1d :=
1(u1 u1) ((u1)(u1 u1)
)dx
div u1(P (1) P (1)(1 1) P (1)
)dx
+
(1 1)1
[u1 div
(d1 d1 1
2|d1|2I3
)] (u1 u1)dx (4.11)
d
1c := (
d1 (d1) d1 (d1))(u1 u1)dx+
(d1 d1)
((d1)u1 (d1)u1
)dx
-
1250 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
Sitiin
Aap
Toca
Le
O
Si
fo
(d1 d1) (|d1|2d1 |d1|2d1)dx+
(d1 d1) (|d1|2d1 |d1|2d1)dx
(d1 d1) ((d1)u1 (d1)u1
)dx. (4.12)
milar to the proof of Theorem 2.1, we shall nish the proof of
Theorem 2.2 by applying the rela-ve entropy inequality (4.9) to {1,
u1, d1}, where {1, u1, d1} is a classical (smooth) solution of
theitialboundary value problem (1.4), (2.6), and (1.5), such
that
1(0, ) = 0, u1(0, ) = u0, d1(0, ) = d0.s a result, the integrals
depending on the initial data on the right-hand side of (4.9)
vanish, and weply a Gronwall type argument to deduce the desired
result, namely,
1 1, u1 u1, d1 d1.complete the proof, we have to examine all
terms in the remainder (4.10) and to show that theyn be absorbed by
the left-hand side of (4.9).From (2.6), it is clear to see that
1(u1 u1) ((u1)(u1 u1)
)dx
div u1(P (1) P (1)(1 1) P (1)
)dx
Cu1L()E1. (4.13)
t
g1 = g1(u1,d1,d1) = u1 div(d1 d1 1
2|d1|2I3
).
bviously, we have
1
1(1 1)g1 (u1 u1)dx
=
{01
-
Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1251
w
an
Fo
fo
an
foNow we are going to estimate R1c . From (4.12), a direct
calculation gives
R1c =Ra1c +Rb1c (4.17)
here
Ra1c =
(d1 d1) (d1 d1)u1 dx+
d1 (d1 d1)(u1 u1)dx
(|d1| + |d1|)d1 (d1 d1)(|d1| |d1|)dx+
(d1 d1) (d1)(u1 u1)dx (4.18)
d
Rb1c =
(d1 d1) (d1 d1)u1 dx+
(|d1| + |d1|)d1 (d1 d1)(|d1| |d1|)dx+
|d1 d1|2|d1|2 dx. (4.19)
r the rst term on the right-hand side of (4.18), we have
(d1 d1) (d1 d1)u1 dx
u1L()d1 d1L2()d1 d1L2() d1 d12L2() + C()u12L()d1 d12L2() d1
d12L2() + C()u12L()E1, (4.20)
r any > 0. In a similar way, we also obtain
d1 (d1 d1)(u1 u1)dx u1 u12L2() + C()d12L()E1, (4.21)
(|d1| + |d1|)d1 (d1 d1)(|d1| |d1|)dx
d1 d12L2() + C()(d12L() + d12L())d12L()E1, (4.22)
d
(d1 d1) (d1)(u1 u1)dx u1 u12L2() + C()d12L()E1, (4.23)r any >
0.
-
1252 Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253
Li
an
Se
w
w
Resoapliq
A
thw(GtoNFinally, the Cauchy inequality and the denition of the
relative entropy E1 give
(d1 d1) (d1 d1)u1 dx u1L()d1 d1L2()d1 d1L2() u1L()E
121 E
121
u1L()E1. (4.24)
kewise, we get
(|d1| + |d1|)d1 (d1 d1)(|d1| |d1|)dx
d1L()(d1L() + d1L())d1 d1L2()d1 d1L2()
d1L()(d1L() + d1L())E1 (4.25)
d
|d1 d1|2|d1|2 dx d12L()d1 d12L2() d12L()E1. (4.26)
Moreover, by applying the quasilinear equations of parabolic
type estimates (see [14, Chapter VI,ction 3]) to the third equation
of (1.4) and a density argument, we have
d1L() C . (4.27)
Summing up relations (4.10)(4.27) with the denition of nite
energy weak solution {1,u1,d1},e conclude from the relative entropy
inequality (4.9) that
E1([1,u1,d1][1, u1, d1])( )
0
h1(t)E1([1,u1,d1][1, u1, d1])(t)dt,
here h1 L1(0, T ). Thus, Theorem 2.1 immediately follows from
Gronwalls inequality.
mark 4.1. For one-dimensional case, Ding et al. in [2,3]
established the existence of global classicallutions and the
existence of global weak solutions. Based on the existence results,
one can directlyply Theorem 2.2 to obtain the weakstrong uniqueness
principle for one-dimensional compressibleuid crystal system.
cknowledgments
The authors are grateful to the referee for valuable
suggestions. The authors also would like toank Prof. Song Jiang for
helpful discussions and sustained encouragement. Yong-Fu Yangs
researchas in part supported by NSFC (Grant No. 11201115) and China
Postdoctoral Science Foundationrant No. 2012M510365). Changsheng
Dous research was partially supported by China Postdoc-ral Science
Foundation (Grant No. 2012M520205). Ju Qiangchang was supported by
NSFC (Grant
o. 11171035).
-
Y.-F. Yang et al. / J. Differential Equations 255 (2013)
12331253 1253
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Weak-strong uniqueness property for the compressible ow of
liquid crystals1 Introduction2 Main results3 Proof of Theorem 2.14
Proof of Theorem 2.2AcknowledgmentsReferences