- Email: [email protected]
On uniqueness and continuous dependence for a linear micropolar fluid
/nr.J. Engng Sci.. 1973, Vol. 11, pp. 369-376.
Pergamon Press. Printed in Great Britain
ON UNIQUENESS AND FOR A LINEAR Heriot-Watt
CONTINUOUS MICROPOLAR
DEPENDENCE FLUID
R. N. HILLS University, Edinburgh, Scotland
Abstract-In this paper logarithmic convexity arguments are employed to establish the uniqueness and Halder continuous dependence on initial data of a certain class of solutions of an initial boundary value problem for a linear micropolar fluid. I. INTRODUCTION
paper Eringen[l] has presented a theory of micropolar fluids. Kinematically this theory differs from the classical theory of fluids in that at each point there is defined, in addition to the usual velocity vector Vi, a micro-rotational velocity vi which represents the intrinsic (rigid) rotation of the material point. The micropolar fluid can support body and surface couples and moreover, associated with the rotational field vi there is a micro-rotational inertia. Lee and Eringen[2-41 have discussed the nematic liquid crystal within the framework of the micropolar theory and it is thought that further physical applications are likely to lie in such areas as polymeric fluids and suspensions. In the present paper we shall consider the questions of uniqueness and continuous dependence upon the initial data for solutions of the equations governing the motion of a linear incompressible micropolar fluid under isothermal conditions. We assume that the fluid occupies a bounded region D of three space with a boundary dD which is smooth enough to allow applications of the divergence theorem. Then the velocity vector vi and the micro-rotational velocity vi satisfy the equations [l] IN A RECENT
zli,i
iii = 9i
0
=
_4,i +
7)Ui,jj
(1.1) +
(1.2)
AEijkUk,j
d*fli = 2’i - 2hvi + hEukVk,j + (a + p)
Yk,ki
+
(1.3)
Yui,kk
in the space-time cylinder D X [0, T), where T is a finite instant of time. We have employed a superposed dot to denote material time differentiation and a subscript k following a comma indicates partial differentiation with respect to the space variable xk. In equations (1.2) and (1.3) p denotes the (constant) density, Fi and _Yi are the prescribed body force and moment respectively, p is the unknown pressure and d* is the microinertia coefficient which is assumed to be positive. The alternating tensor is denoted by liik. The viscosity coefficients 7, (Y,/3, y, A are constants and moreover, the ClausiusDuhem inequality requires that these coefficients satisfy[2] 71 2
0,
A 3 0,
(3a+p+y)
3 0, 369
Y S 0,
-y
S p G y.
(1.4)
R. N. HILLS
370
We shall restrict our attention to classical solutions which are assumed to exist subject to prescribed initial conditions U~(X30) =f;(~i),
v~(x~,O) =F$(x~)
inDx0
(1 Sa)
and prescribed boundary conditions
4% tf = &b-j, t1,
~i(~jt t) = Ci(Xi, t)
on
SD
X
(0. 7’).
(1Sb)
In what follows we show that if the solutions of f I. I )-f 1S) are suitably restricted then they are unique and depend Holder continuously on the initial data in the sense of John[5]. In particular we require that the translat~onai and rotational velocities and velocity gradients of the base flow and the translational and rotational velocities of the perturbed flow remain uniformly bounded in D X [0, T]. These results are established by employing the well-known arguments of io~rithmic convexity. This technique has been successfulty applied to discuss the uniqueness and continuous dependence of a wide variety of well and non-well posed problems. See for example [6-101 and the references cited therein. 2. UNIQUENESS
AND STABILITY
Throughout this paper we shall say that ( ui, vi) are solutions to problem jai if they satisfy (l.l)-( 1.3) and the designated initial and boundary conditions ( 1.5). Furthermore, a solution pair (vi(xi* t) , z.q(Xi+ t) ) of & is said to be of class A#’if
and is of class N if
for prescribed constants M and n. Let ( ui, vi> be a solution pair of problem d with initial data A and Fi respectively and the pair (u:, $1 be a solution corresponding to the same body force .Fi and moment 55’ibut with initial datafi and Ff. If we set
then it is easily verified that these difference velocities satisfy
wi,t= d2&
= -li”vj*~*j
-VfWi,j-
WjVi,j-bP.i
- d%jyl?j, -
2h-h
+ qWi,jj+
+
hjkWk.j
f
(2.3)
XEijk&Lk,j
(a + @)pk.ki
+
Y&,kk
(2.4)
where P denotes the pressure difference p-p” and a comma fotlowed by t indicates partial differentiation with respect to time holding the spatial variable fixed. We are now in a position to establish the theorem:
371
On uniqueness and continuous dependence for a linear micropolar fluid
theorem I. If (vi, vi)EA’, (vi*, v?)eX all &[O, T) the function
+
KI
JD,(WiWi+ d’pipi)
and K1, Kz are positive cons~a~r~ then fir
dx + K,I I,, (WiWi,t+ d2pipi,l) dX1
(2.5)
satisfies an inequality of the form FF”-
(F’)2
> -kIFF’-kzF2
(2.6)
fur some ~~rn~u~able c~~s~a~ts k, and k,. In (2.5) the symbols D,, D, denote integration
over the region D at time T and 0 respectively and in (2.6) the prime indicates time differentiation. To prove this theorem we write
F(t) = j; where
P is
I,, (wfWf+d2pijLi)
dxdT+
a constant. Differentiation
(T-t)
~Du(W~Wi+d2~~~i)
dx+ r2
(2.7)
yields
(2.8) and
+
2 S,,(wiwi,t+ d2pipi,t)dx*
We can now use the differential equations (2.3) and (2.4) to substitute for w$,~and pi,v. Then, on integrating by parts and again using the equations (2.3), (2.4), we obtain after some algebra F’(t) = 4 I,” I,, fWi,7(Wi,7+aJ +~i,~(~<,~+bi)l +
dxd7
2 J,” JDT CWjWi,jvi,s + d2WjPi,jvi,rI dx d7-t 2 JDo (WiWi,t + d2pipi,t)dx
where? ai =
Wi,jVj* f
W##i,~~
-$d2~jvj,i
bi = /Li,jV~’ + *WjVi,j.
tHere and later [ 1and f ) enclosing indices are used to denote
a tensor respectively.
the aRtisymmetric and symmetric parts of
372
R. N. HILLS
Noting that IS: Dr (wiai + d”pibi)
dx dr
vanishes, we can write (2.8) as F’(t) = 2 Jd jr], [wi(wi,7+~ai)$-d2~i(~i.~+)bi)]
dxdr
and then on forming FF” - ( F’)2 we obtain
-( su’ jD, [wi(wi,,+&)
+@/Q(J++&)]
dxdT}’ 2 0
by Schwarz’s inequality. We now proceed to obtain an estimate for the second term on the right hand side of (2.9). From an integration by parts we find that
: D, (WiU’jUi,j+ d’/JiM’jVi,j) dx dT IS
=-
(2.10)
+7
-w,: [ Dr
It is convenient
(2WjUci.j)
+
d’rUjVj,i)Ui
+ d’WjVi.jbi]
dx dr.
to consider this expression in stages. Let
J, = ISd ,), (WiWjZ)i,j+ d’piWjvi.j) .7dx d7 =
J;,
(wiwjui.j
+
d2/&wjvf.j)
BY applyingthe arithmetic-geometric
dx-
I,,
( w[w~u~,~
+
d”~~w~v~,~)
dx.
mean and Schwarz’s inequalities we deduce that
25, =GJo, [(1+M2+d2)~‘i~i+t2d2~~~~] =Gc,F’ + 2c, I ,)” (wiwi + d”~ipi) dx
dx+jDO [(1+M2+d2)~i~‘i+Mzd2~i~i]
dx
373
On uniqueness and continuous dependence for a linear micropolar fluid
where c1 = 1 + M2 + d2. Next we define J2 E jf JD, [ (2wju,i,j,+ d2pjvj,i) (wi,,+h)
+ d2Wjvi,j(pi,r+ibi)]
then by Schwarz’s and the arithmetic-geometric
dx d7
mean inequalities
where c; = 2M2 +iM2d2. Since, for real numbers aI and a,,
c Iall + la,1
(+ta,Z)“2 we have
IJZIs 2~2rS-f IF’I] c 2~2[S+F’+2
S,, (wiwi+d2pipi)
(2.1 I)
dx].
We now write 2Ja s Jo’JDTC(2wjuci,j)+ d2pjuj,i) ( Wt,/&‘k* + WkV[i,kl
Then on using the arithmetic-geometric
-Md2~k~k,i)
] dx
d7
mean and Schwarz’s inequalities we deduce
45, c J’: JD, [2M2(3+2d2)wiw~+M2d2(3+d2)~~~i + N2(2 + d2) wi,jwi,j + N”d2pi,jpi,j] dx dr.
(2.12)
To obtain estimates of the integrals D Wi,jWi,jdx d7 SI0t 7
and
d 4 Pi.+i,j dx d7 J-s
we return to the equations for the difference velocities. On multiplying (2.3) by IQ, (2.4) by pi and then integrating the sum over space-time, we obtain the identity J, jD, (wiwi,T+ d2pipi,7) dx dr = - s,’ S,, [ wiwjuf,j+ Twi,jwi,j + d’piWjVi,j+ 2Apipi $- (a!+P)~i,kE*k,i+YCLi,k~i,k
$- 2AEijkwi,jpk]
If we introduce new variables j&) defined by /-h,k)
=
1
z/-b,raik
+
pCi.k)r
&,i
=
0,
dx
dr-
(Z-13)
374
R. N. HILLS
where 8ik is the Kronecker +F’ = -
delta. then we can rewrite (2.3) as d2 Piwjvi,j
1: jD, [wiwjvi,j+
+ (b + ?‘)&i,k&,k)+
+
7)wi,jwi,j
(Y -P)PIL~IP~~.~II
+
2Q4Pi
+-6(3a + P + Y)/&,rPs.s
dx d7.
Using the inequalities (1.4) we infer
7
s,’ fD,
Wi,jWi,j
dx d7 s :F’ - ji S,, (WiWjVi,j + d2piWjvi,j) dx d7.
Thus, by again applying the arithmetic-geometric find
mean and Schwarz’s inequalities, we
Wi,jWi,jdx dr s -iF’+:F
(2.14)
where cg = (1 +M2 + d2+ (8X2/vd2)). Using the result (2.14) we may similarly determine estimates for each of the integrals
in turn and thus obtain (2.15) where
c,=;+;,co=&+ Y -P
++ rl Consequently
1 3cu+p+y.
we can combine these results to deduce I 2 -~c~S-~C,F-_C,F’-~(C,
where c7 = [3M2+2d2+$N2c,(2+d2)
+4C2)
jD,, (wiwi+d21.LiPi)
+3V2d2c,c6]
and c8 = [4c,+c,
dx
(2.16)
- ( 1/273)N2(2+
d “) - 4c5c6d 2N2].
We can employ similar arguments to estimate the third term on the right hand side of (2.9). From the arithmetic-geometric mean and Schwarz’s inequalities we obtain
<2
t ff0
Do
[M2
( 1+
id’)
WiWi
+
iM2d “pipi + N2Wi,jWi.j+ d 2NZ~i,j~~,j] dx d7.
Hence on using (2.14) and (2.15) we have J, =s cgF-ccl,,F’
(2.17)
375
On uniqueness and continuous dependence for a linear micropolar fluid
Q4cg) and cl0 = 2N2 [ ( l/q) + d*c&. where~,=2M~(1+4~~)+2N*[(ll~)c,+d employing (2.16) and (2.17) in (2.9) it is found that FF” - (F’)*
1
(c,+c,)F*-
4S*-‘k,FS--
-2(cl+4~*)F
Thus,
Cc,-c,o)FF’
fDo (“‘i”‘i+d*PiE.Li) dx+2F
Jb, (wiwi,,+d*piki,t)
dx.
(2.18)
Let us now take r* of equation (2.7) to be
where K1, K, are positive constants. Then FF”-
(F’)* 2 4(S-fc,F)*-kIFF’-k2F2
where k, = cg-cIo and k2 = [ci+c7+c9+ (l/K,)(2c,+8c2) + (2/K,)] and the theorem is firoved. We are now in a position to apply the standard arguments of logarithmic convexity to the inequality (2.6). We first establish the following uniqueness result. Corollary 1. Zf (ai, Yi)eA and (v:, wf)eV are solutions to problem zf with the same initial and boundary data, then Vi = v,+, vi = vi* for all te[O, T]. By definition F(0) = 0. It can be shown (see for example [lo]) that if F satisfies an inequality (2.6) and if F(t,) = 0 for some t,e[O, T) then F = 0 on [0, T]. Therefore, since Wiand pi are continuous on D, they both vanish identically on D. The next corollary establishes that the solutions to problem & depend Holder continuously upon the initial data. Corollary 2. Zf (Vi, Vi)E& and (I$, vP)EN are solutions to problem & then for jinite time it is possible to determine constants L, ((Y = 0, 1,2,3) such that (wiwi+d*PW)
dx dT c LoLf’(L2 I,, (wiwi + d2pipi) dx +L31 JD, (wiwi,t+d*pipi,t)
dxJ) I-’
(2.19)
where e-kzt 6=
_
1 ee-kzT
e-kzT *
It follows from (2.6) and Jensen’s inequality that (see [lo]) F(t)
G L,[F(T)]*[F(O>]‘-”
where 6 is as given in the corollary. If the initial velocities and their time derivatives are square integrable then, since (vi, vi)eA and (UC, vT)EJ~T,there exists a finite constant L, such that F(T) G Lt. The inequality (2.19) then follows immediately so that under perturbations (ui, vi)e.,& any solution ( vi*, vT)EX of & is Holder stable on compact subsets of [0, T) in the sense of [5].
316
R. N. HILLS
REFERENCES Marh. Mech. 16, 1(1966). 121 J. D. LEE and A. C. ERINGEN,J. Chem. Phys. 54,5027 (1971). [31 J. D. LEE and A. C. ER1NGEN.J. Chem. Phys. 55,4504 (197 1). 141 J. D. LEE and A. C. ER1NGEN.J. Chem. Phys. 55,4509 (1971). 151 F. JOHN, Comm. pure uppl. Math. 13.55 1 (1960). 161R. J. KNOPS and L. E. PAYNE. Arch. rution Mech. Amdysis 29.331 (1968). [71 R. J. KNOPS and L. E. PAYNE, Proc. Cumb. Phil. Sot. 66,481 (1969). PI R. J. KNOPS and L. E. PAYNE, 1nr.J. SolidsSrruct. 6. 1173 (1970). 191 R. N. HILLS,Actu Mech. (To appear). [lOI H. A. LEV1NE.J. di$erentiu/Equnrions8,34 (1970).
[II A. C. ERINGEN,J.
f Received 2 June 1972)
Resume-Dans cet article, des arguments de convexite logarithmique sont utilises pour Ctablir I’unicite et la dependance continue de Holder sur des don&es initiales dune certaine classe de solutions d’un probleme de valeurs limite initiales pour un fluide lineaire micropolaire. Zusammenfassung- In dieser Arbeit werden logarithmische Konvexizittitsargumente verwendet, urn die Eindeutigkeit und Holder’sche kontinuierliche AbhGrgigkeit von Anfangsdaten einer bestimmten Liisungsklasse eines Anfangsgrenzwertproblemes fir eine lineare mikropolaire Fliissigkeit festzustellen. Sommario-
In quest0 articolo 1’A. fa uso di argomenti di convessita logaritmica per stabilire la singolarita e la dipendenza continua Holderiana dai dati iniziali di una certa classe di soluzioni di un problema di valore limite iniziale nei riguardi di un fluid0 micropolare lineare. A~CT~BICT - Ha OII~@JlCHHOrO IlOJI~pHOk
OCHOBe
JtOr2JJEi~MWECKHX
KJIaCCa ~LUCHkifi
mHJ,KOCTlr,
a TBKxe
NIR
,&3HHOk
HC~,,CpblBHaSI
apr)‘MCHTOB ACXOLlHOfi PaBHCAMOCTb
BblLIYKnOCTM
KpaCBOk
KIpO6JIeMbI
CC OT ACXO,llHblX
yCTaHOBfleHa HeKOTOPOii AaHHbIX
C~HHCTBCHHOCTb JlHH&iHO&
rOeJIAt?pa.
MHKPO-
Pergamon Press. Printed in Great Britain
ON UNIQUENESS AND FOR A LINEAR Heriot-Watt
CONTINUOUS MICROPOLAR
DEPENDENCE FLUID
R. N. HILLS University, Edinburgh, Scotland
Abstract-In this paper logarithmic convexity arguments are employed to establish the uniqueness and Halder continuous dependence on initial data of a certain class of solutions of an initial boundary value problem for a linear micropolar fluid. I. INTRODUCTION
paper Eringen[l] has presented a theory of micropolar fluids. Kinematically this theory differs from the classical theory of fluids in that at each point there is defined, in addition to the usual velocity vector Vi, a micro-rotational velocity vi which represents the intrinsic (rigid) rotation of the material point. The micropolar fluid can support body and surface couples and moreover, associated with the rotational field vi there is a micro-rotational inertia. Lee and Eringen[2-41 have discussed the nematic liquid crystal within the framework of the micropolar theory and it is thought that further physical applications are likely to lie in such areas as polymeric fluids and suspensions. In the present paper we shall consider the questions of uniqueness and continuous dependence upon the initial data for solutions of the equations governing the motion of a linear incompressible micropolar fluid under isothermal conditions. We assume that the fluid occupies a bounded region D of three space with a boundary dD which is smooth enough to allow applications of the divergence theorem. Then the velocity vector vi and the micro-rotational velocity vi satisfy the equations [l] IN A RECENT
zli,i
iii = 9i
0
=
_4,i +
7)Ui,jj
(1.1) +
(1.2)
AEijkUk,j
d*fli = 2’i - 2hvi + hEukVk,j + (a + p)
Yk,ki
+
(1.3)
Yui,kk
in the space-time cylinder D X [0, T), where T is a finite instant of time. We have employed a superposed dot to denote material time differentiation and a subscript k following a comma indicates partial differentiation with respect to the space variable xk. In equations (1.2) and (1.3) p denotes the (constant) density, Fi and _Yi are the prescribed body force and moment respectively, p is the unknown pressure and d* is the microinertia coefficient which is assumed to be positive. The alternating tensor is denoted by liik. The viscosity coefficients 7, (Y,/3, y, A are constants and moreover, the ClausiusDuhem inequality requires that these coefficients satisfy[2] 71 2
0,
A 3 0,
(3a+p+y)
3 0, 369
Y S 0,
-y
S p G y.
(1.4)
R. N. HILLS
370
We shall restrict our attention to classical solutions which are assumed to exist subject to prescribed initial conditions U~(X30) =f;(~i),
v~(x~,O) =F$(x~)
inDx0
(1 Sa)
and prescribed boundary conditions
4% tf = &b-j, t1,
~i(~jt t) = Ci(Xi, t)
on
SD
X
(0. 7’).
(1Sb)
In what follows we show that if the solutions of f I. I )-f 1S) are suitably restricted then they are unique and depend Holder continuously on the initial data in the sense of John[5]. In particular we require that the translat~onai and rotational velocities and velocity gradients of the base flow and the translational and rotational velocities of the perturbed flow remain uniformly bounded in D X [0, T]. These results are established by employing the well-known arguments of io~rithmic convexity. This technique has been successfulty applied to discuss the uniqueness and continuous dependence of a wide variety of well and non-well posed problems. See for example [6-101 and the references cited therein. 2. UNIQUENESS
AND STABILITY
Throughout this paper we shall say that ( ui, vi) are solutions to problem jai if they satisfy (l.l)-( 1.3) and the designated initial and boundary conditions ( 1.5). Furthermore, a solution pair (vi(xi* t) , z.q(Xi+ t) ) of & is said to be of class A#’if
and is of class N if
for prescribed constants M and n. Let ( ui, vi> be a solution pair of problem d with initial data A and Fi respectively and the pair (u:, $1 be a solution corresponding to the same body force .Fi and moment 55’ibut with initial datafi and Ff. If we set
then it is easily verified that these difference velocities satisfy
wi,t= d2&
= -li”vj*~*j
-VfWi,j-
WjVi,j-bP.i
- d%jyl?j, -
2h-h
+ qWi,jj+
+
hjkWk.j
f
(2.3)
XEijk&Lk,j
(a + @)pk.ki
+
Y&,kk
(2.4)
where P denotes the pressure difference p-p” and a comma fotlowed by t indicates partial differentiation with respect to time holding the spatial variable fixed. We are now in a position to establish the theorem:
371
On uniqueness and continuous dependence for a linear micropolar fluid
theorem I. If (vi, vi)EA’, (vi*, v?)eX all &[O, T) the function
+
KI
JD,(WiWi+ d’pipi)
and K1, Kz are positive cons~a~r~ then fir
dx + K,I I,, (WiWi,t+ d2pipi,l) dX1
(2.5)
satisfies an inequality of the form FF”-
(F’)2
> -kIFF’-kzF2
(2.6)
fur some ~~rn~u~able c~~s~a~ts k, and k,. In (2.5) the symbols D,, D, denote integration
over the region D at time T and 0 respectively and in (2.6) the prime indicates time differentiation. To prove this theorem we write
F(t) = j; where
P is
I,, (wfWf+d2pijLi)
dxdT+
a constant. Differentiation
(T-t)
~Du(W~Wi+d2~~~i)
dx+ r2
(2.7)
yields
(2.8) and
+
2 S,,(wiwi,t+ d2pipi,t)dx*
We can now use the differential equations (2.3) and (2.4) to substitute for w$,~and pi,v. Then, on integrating by parts and again using the equations (2.3), (2.4), we obtain after some algebra F’(t) = 4 I,” I,, fWi,7(Wi,7+aJ +~i,~(~<,~+bi)l +
dxd7
2 J,” JDT CWjWi,jvi,s + d2WjPi,jvi,rI dx d7-t 2 JDo (WiWi,t + d2pipi,t)dx
where? ai =
Wi,jVj* f
W##i,~~
-$d2~jvj,i
bi = /Li,jV~’ + *WjVi,j.
tHere and later [ 1and f ) enclosing indices are used to denote
a tensor respectively.
the aRtisymmetric and symmetric parts of
372
R. N. HILLS
Noting that IS: Dr (wiai + d”pibi)
dx dr
vanishes, we can write (2.8) as F’(t) = 2 Jd jr], [wi(wi,7+~ai)$-d2~i(~i.~+)bi)]
dxdr
and then on forming FF” - ( F’)2 we obtain
-( su’ jD, [wi(wi,,+&)
+@/Q(J++&)]
dxdT}’ 2 0
by Schwarz’s inequality. We now proceed to obtain an estimate for the second term on the right hand side of (2.9). From an integration by parts we find that
: D, (WiU’jUi,j+ d’/JiM’jVi,j) dx dT IS
=-
(2.10)
+7
-w,: [ Dr
It is convenient
(2WjUci.j)
+
d’rUjVj,i)Ui
+ d’WjVi.jbi]
dx dr.
to consider this expression in stages. Let
J, = ISd ,), (WiWjZ)i,j+ d’piWjvi.j) .7dx d7 =
J;,
(wiwjui.j
+
d2/&wjvf.j)
BY applyingthe arithmetic-geometric
dx-
I,,
( w[w~u~,~
+
d”~~w~v~,~)
dx.
mean and Schwarz’s inequalities we deduce that
25, =GJo, [(1+M2+d2)~‘i~i+t2d2~~~~] =Gc,F’ + 2c, I ,)” (wiwi + d”~ipi) dx
dx+jDO [(1+M2+d2)~i~‘i+Mzd2~i~i]
dx
373
On uniqueness and continuous dependence for a linear micropolar fluid
where c1 = 1 + M2 + d2. Next we define J2 E jf JD, [ (2wju,i,j,+ d2pjvj,i) (wi,,+h)
+ d2Wjvi,j(pi,r+ibi)]
then by Schwarz’s and the arithmetic-geometric
dx d7
mean inequalities
where c; = 2M2 +iM2d2. Since, for real numbers aI and a,,
c Iall + la,1
(+ta,Z)“2 we have
IJZIs 2~2rS-f IF’I] c 2~2[S+F’+2
S,, (wiwi+d2pipi)
(2.1 I)
dx].
We now write 2Ja s Jo’JDTC(2wjuci,j)+ d2pjuj,i) ( Wt,/&‘k* + WkV[i,kl
Then on using the arithmetic-geometric
-Md2~k~k,i)
] dx
d7
mean and Schwarz’s inequalities we deduce
45, c J’: JD, [2M2(3+2d2)wiw~+M2d2(3+d2)~~~i + N2(2 + d2) wi,jwi,j + N”d2pi,jpi,j] dx dr.
(2.12)
To obtain estimates of the integrals D Wi,jWi,jdx d7 SI0t 7
and
d 4 Pi.+i,j dx d7 J-s
we return to the equations for the difference velocities. On multiplying (2.3) by IQ, (2.4) by pi and then integrating the sum over space-time, we obtain the identity J, jD, (wiwi,T+ d2pipi,7) dx dr = - s,’ S,, [ wiwjuf,j+ Twi,jwi,j + d’piWjVi,j+ 2Apipi $- (a!+P)~i,kE*k,i+YCLi,k~i,k
$- 2AEijkwi,jpk]
If we introduce new variables j&) defined by /-h,k)
=
1
z/-b,raik
+
pCi.k)r
&,i
=
0,
dx
dr-
(Z-13)
374
R. N. HILLS
where 8ik is the Kronecker +F’ = -
delta. then we can rewrite (2.3) as d2 Piwjvi,j
1: jD, [wiwjvi,j+
+ (b + ?‘)&i,k&,k)+
+
7)wi,jwi,j
(Y -P)PIL~IP~~.~II
+
2Q4Pi
+-6(3a + P + Y)/&,rPs.s
dx d7.
Using the inequalities (1.4) we infer
7
s,’ fD,
Wi,jWi,j
dx d7 s :F’ - ji S,, (WiWjVi,j + d2piWjvi,j) dx d7.
Thus, by again applying the arithmetic-geometric find
mean and Schwarz’s inequalities, we
Wi,jWi,jdx dr s -iF’+:F
(2.14)
where cg = (1 +M2 + d2+ (8X2/vd2)). Using the result (2.14) we may similarly determine estimates for each of the integrals
in turn and thus obtain (2.15) where
c,=;+;,co=&+ Y -P
++ rl Consequently
1 3cu+p+y.
we can combine these results to deduce I 2 -~c~S-~C,F-_C,F’-~(C,
where c7 = [3M2+2d2+$N2c,(2+d2)
+4C2)
jD,, (wiwi+d21.LiPi)
+3V2d2c,c6]
and c8 = [4c,+c,
dx
(2.16)
- ( 1/273)N2(2+
d “) - 4c5c6d 2N2].
We can employ similar arguments to estimate the third term on the right hand side of (2.9). From the arithmetic-geometric mean and Schwarz’s inequalities we obtain
<2
t ff0
Do
[M2
( 1+
id’)
WiWi
+
iM2d “pipi + N2Wi,jWi.j+ d 2NZ~i,j~~,j] dx d7.
Hence on using (2.14) and (2.15) we have J, =s cgF-ccl,,F’
(2.17)
375
On uniqueness and continuous dependence for a linear micropolar fluid
Q4cg) and cl0 = 2N2 [ ( l/q) + d*c&. where~,=2M~(1+4~~)+2N*[(ll~)c,+d employing (2.16) and (2.17) in (2.9) it is found that FF” - (F’)*
1
(c,+c,)F*-
4S*-‘k,FS--
-2(cl+4~*)F
Thus,
Cc,-c,o)FF’
fDo (“‘i”‘i+d*PiE.Li) dx+2F
Jb, (wiwi,,+d*piki,t)
dx.
(2.18)
Let us now take r* of equation (2.7) to be
where K1, K, are positive constants. Then FF”-
(F’)* 2 4(S-fc,F)*-kIFF’-k2F2
where k, = cg-cIo and k2 = [ci+c7+c9+ (l/K,)(2c,+8c2) + (2/K,)] and the theorem is firoved. We are now in a position to apply the standard arguments of logarithmic convexity to the inequality (2.6). We first establish the following uniqueness result. Corollary 1. Zf (ai, Yi)eA and (v:, wf)eV are solutions to problem zf with the same initial and boundary data, then Vi = v,+, vi = vi* for all te[O, T]. By definition F(0) = 0. It can be shown (see for example [lo]) that if F satisfies an inequality (2.6) and if F(t,) = 0 for some t,e[O, T) then F = 0 on [0, T]. Therefore, since Wiand pi are continuous on D, they both vanish identically on D. The next corollary establishes that the solutions to problem & depend Holder continuously upon the initial data. Corollary 2. Zf (Vi, Vi)E& and (I$, vP)EN are solutions to problem & then for jinite time it is possible to determine constants L, ((Y = 0, 1,2,3) such that (wiwi+d*PW)
dx dT c LoLf’(L2 I,, (wiwi + d2pipi) dx +L31 JD, (wiwi,t+d*pipi,t)
dxJ) I-’
(2.19)
where e-kzt 6=
_
1 ee-kzT
e-kzT *
It follows from (2.6) and Jensen’s inequality that (see [lo]) F(t)
G L,[F(T)]*[F(O>]‘-”
where 6 is as given in the corollary. If the initial velocities and their time derivatives are square integrable then, since (vi, vi)eA and (UC, vT)EJ~T,there exists a finite constant L, such that F(T) G Lt. The inequality (2.19) then follows immediately so that under perturbations (ui, vi)e.,& any solution ( vi*, vT)EX of & is Holder stable on compact subsets of [0, T) in the sense of [5].
316
R. N. HILLS
REFERENCES Marh. Mech. 16, 1(1966). 121 J. D. LEE and A. C. ERINGEN,J. Chem. Phys. 54,5027 (1971). [31 J. D. LEE and A. C. ER1NGEN.J. Chem. Phys. 55,4504 (197 1). 141 J. D. LEE and A. C. ER1NGEN.J. Chem. Phys. 55,4509 (1971). 151 F. JOHN, Comm. pure uppl. Math. 13.55 1 (1960). 161R. J. KNOPS and L. E. PAYNE. Arch. rution Mech. Amdysis 29.331 (1968). [71 R. J. KNOPS and L. E. PAYNE, Proc. Cumb. Phil. Sot. 66,481 (1969). PI R. J. KNOPS and L. E. PAYNE, 1nr.J. SolidsSrruct. 6. 1173 (1970). 191 R. N. HILLS,Actu Mech. (To appear). [lOI H. A. LEV1NE.J. di$erentiu/Equnrions8,34 (1970).
[II A. C. ERINGEN,J.
f Received 2 June 1972)
Resume-Dans cet article, des arguments de convexite logarithmique sont utilises pour Ctablir I’unicite et la dependance continue de Holder sur des don&es initiales dune certaine classe de solutions d’un probleme de valeurs limite initiales pour un fluide lineaire micropolaire. Zusammenfassung- In dieser Arbeit werden logarithmische Konvexizittitsargumente verwendet, urn die Eindeutigkeit und Holder’sche kontinuierliche AbhGrgigkeit von Anfangsdaten einer bestimmten Liisungsklasse eines Anfangsgrenzwertproblemes fir eine lineare mikropolaire Fliissigkeit festzustellen. Sommario-
In quest0 articolo 1’A. fa uso di argomenti di convessita logaritmica per stabilire la singolarita e la dipendenza continua Holderiana dai dati iniziali di una certa classe di soluzioni di un problema di valore limite iniziale nei riguardi di un fluid0 micropolare lineare. A~CT~BICT - Ha OII~@JlCHHOrO IlOJI~pHOk
OCHOBe
JtOr2JJEi~MWECKHX
KJIaCCa ~LUCHkifi
mHJ,KOCTlr,
a TBKxe
NIR
,&3HHOk
HC~,,CpblBHaSI
apr)‘MCHTOB ACXOLlHOfi PaBHCAMOCTb
BblLIYKnOCTM
KpaCBOk
KIpO6JIeMbI
CC OT ACXO,llHblX
yCTaHOBfleHa HeKOTOPOii AaHHbIX
C~HHCTBCHHOCTb JlHH&iHO&
rOeJIAt?pa.
MHKPO-