Bipolar barotropic non-newtonian compressible fluids

ESAIM: M

Š ÁRKA M ATU Š U ^˚ -N E ˇ CASOVÁ

M ÁRIA M EDVIDOVÁ -L UKÁ ˇ COVÁ

Bipolar barotropic non-newtonian compressible fluids

ESAIM: Modélisation mathématique et analyse numérique, tome 34, n^o5 (2000), p. 923-934

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Modélisation Mathématique et Analyse Numérique Vol. 34, N° 5, 2000, pp. 923-934

BIPOLAR BAROTROPIC NON-NEWTONIAN COMPRESSIBLE FLUIDS

SÂRKA M A T U S Û - N E C A S O V Â1 AND MARIA M E D V I D O V Â - L U K A C O V Â2' 3

Abstract. We are interested in a barotropic motion of the non-Newtonian bipolar fluids . We consider a special case where the stress tensor is expressed in the form of potentials depending on e^%% and (-Q^-)-

We prove the asymptotic stability of the rest state under the assumption of the regularity of the potential forces.

Mathematics Subject Classification. 76N10, 76A05, 35Q35.

Received: June 3^; 1999. Revised: May 5, 2000.

INTRODUCTION

There are many substances which are capable of flowing but which exhibit flow characteristics that cannot be adéquately described by the classical linearly viscous fluid model. In order to describe some of the départures from Newtonian behaviour (rheological properties, elastic features such as yield stress, stress relaxation and non- zero normal stress différences) many idealized material models have been suggested. During the last decades, mathematicians have also started to pay attention to these models and several results concerning the existence, uniqueness and stability of solutions have appeared, see [1-3,5,8-19,22-24,26-28].

We will deal with compressible non-Newtonian fluids. The models which describe their rheological properties were studied from the mathematical point of view by Matusû-Necasovâ , Lukâcovâ-Medvid'ovâ see [14,16]. They proved the existence and uniqueness of a weak solution. All of these results were studied on bounded domains.

In the case of an isothermal process the stability of the rest state has been proved [16]. The global existence of équations of non-Newtonian compressible fluids when the coefficients of viscosity depend on the invariants of velocity field where the growth of these coefficients is not polynomial but exponential was proved in [12,13]. In the case of viscoelastic compressible fluids the existence of a classical solution of steady motion in the bounded domain was proved by Sy [27]. This result was extended to an exterior domain by Matusû-Necasovâ et al. [15].

Both results are valid for small data only. One of the very interest ing problem is the stability of the rest state. The crucial point of every proof of stability is the uniqueness of the steady state solutions. Beirâo da

Keywords and phrases. Non-Newtonian compressible fluids, global existence, uniqueness, asymptotic stability, the rest state.

1 Mathematical Institute of the Academy of Sciences, Zitnâ 25, 11567 Prague 1, Czech Republic.

2 Institute of Analysis and Numerics, Otto-von-Guericke-Universitât Magdeburg, Universitatsplatz 2, 39106 Magdeburg, Germany.

3 Institute of Mathematics, Faculty of Mechanical Engineering, Technical University of Brno, Technickâ 2, 616 39 Brno, Czech Republic.

© EDP Sciences, SMAI 2000

924 S. MATUSÛ-NECASOVÂ AND M MEDVIDOVÂ-LUKÂCOVÂ

Veiga [4] obtained a necessary and sufficient condition for the existence of a strictly positive solution of the following problem

pd£%, i = l , . . . , d ( 1 . 1 ) p > 0, / p(x)dx = m

Jn

where p is the density, m > 0 the total mass conserved by the flow, p(p) is the pressure, £ is the potential of external forces which is locally Lipschitz on O, Ü is the velocity. It is easy to show that such a solution is necessarily unique. On the other hand, this restriction excludes a class of solutions with vacuüm state (p — 0).

The optimal condition for the solution of (1.1) to be unique is shown by Feireisl and Petzeltovâ [6,7].

Hère, our objective is to study the stability of the rest state of the barotropic motion. This model will be mathematically formulated in Chapter 1. Chapter 2 is devoted to mathematical preliminaries and known results. In Chapter 3 we will prove the asymptotic stability of the rest state.

1. FORMULATION OF THE PROBLEM

The barotropic motion of the bipolar non-Newtonial fluids is governed by the following system of the équa- tions

(1-2)

d . d , v ^ v r \ ®P

dt dx3 dxj %3 dx%

w h e r e p d é n o t e s t h e density, v = (v±,..., Vd) is t h e velocity vector, b= (b±}..., ba) dénotes t h e density of external forces a n d p — p(p) is t h e pressure. We a s s u m e t h a t p G C2( 0 , + o o ) . T h e équations (1.2), (1.3) are solved in t h e t i m e - s p a c e cylinder Q T : = / X £Ï, ƒ = (0, T ) , where ft C WLd is a bounded domain with a smooth infinitely differentiable b o u n d a r y dfl. L e t u s define

rP ~'~^]-, p>0, (1.3')

o*

P(0) = hm P(p).

It may be verified (see [21]) that

pP'(p)-P(p)=p(p), p>0:

P(p) > -fci, P > 0.

We suppose that the body forces are given and satisfy

beL°°(QT). (1.4)

Now, we can specify the assumptions on the stress tensor r^. Namely:

d fdWjDe)-

Also, we consider the third stress tensor r^k, which has the following form

v dW{De)

Moreover, we will assume that the potentials V, W satisfy the following conditions:

d(l + \De\y-W < - W y M * ^ (1.8)

k )°\ dxkl )

C3(l + |Iïe|)«-2|^|2 < ^ ^ c „ f l l t x (1.9)

where C\, C2,C3,C4 are positive constants, q > d, and | • | is the usual Euclidean norm of a vector. Let W(0) = 0, V(0) = 0, (1.10) BW BV

_ ^ _ ( 0 ) = 0, ^ï-(O) = 0. (1.11) The system (1.2), (1.3) is completed by the initial conditions

v(0) = uo, /d(0) = po, Po > 0 in fi; (1.12) and the boundary conditions

^okVjVk = 0 on (0,T) x ôfi (1.13) (z^ is an outer normal to dSX) ,

v = 0 on (0,T) x du. (1.14)

9 2 6 S. MATUSU-NECASOVÂ AND M. MEDVIDOVÂ-LUKÂCOVÂ

2. MATHEMATICAL PRELIMINARIES

By LP(Q) and Wl>p(Q), 0 < p, l < oo, we dénote the Lebesgue and Sobolev spaces, respectively, equipped wit h the standard norm.

First, we give the well-known resuit s on the equilibrium solution, where we do not consider the cavitation of density. In the work of Beirâo da Veiga [4] the necessary and sufficient conditions for the existence of the equilibrium solutions for an arbitrary b G L°°(Q) was proved. Let p b e a continuously differentiable real function defined on R+ = {s G R : 5 > 0}, such that p'(s) > 0, Vs G M+. Let

0 < ess inf p(x), ess supp(x) < -foo (2-1) and

~ [ p(x)dx = m (2.2) l"l Jn

for a fixed m > 0, we define

7r(5) = / "~r~d^' ^e I^+- (2-3) Jo l

We dénote (a, ƒ) the range of TT, (a, ƒ) = ?r(M+).

Let us define </> = TT"1. Clearly, </>((a, ƒ)) = M+. Put 0(a) = 0 , <^(/) = +oo.

Définition 2.1. Let b G L°°(Q). A function p is called an equilibrium solution of (1.1) if p G L°°(Çl) and if

ir(p(x)) = £(x) + c a.e. in O and (2.1), (2.2) hold.

We set no = ess inf b in fi, JVQ = ess sup b in fi.

Theorem 2.1. Let b G L°°(fi) be given. There exists an equilibrium solution p{x) if and only if there exists a constant

(2.4) such that

4 r [ <f>(c + Ç(x))dx = m. (2.5)

/ƒ 5^c/i a constant exists then the (unique) equilibrium solution is given by

n. (2.6)

Proof. see [4].

Theorem 2.2. Under the assumptions of Theorem 2.1 there exists an equilibrium solution p(x) if and only if

a - no< f-No, (2.7)

and

p ƒ <P(a - n

+ &x))dx <m< ^(f - N

+ £(x))dz. (2.8)

In this case the equüibrium solution p(x) is given by (2.6), where c is the (unique) solution of (2.4)-(2.5).

Proof. see [4].

We mention a weak formulation of the problem (1.2), (1.3), (1.12)^(1.14).

Définition 2.2. A pair (p,v) is said to be a weak solution of the problem (1.2), (1.3), (1.12)-(1.14), if the following conditions are satisfied

(i) peL^(I;^

(ii) §ee£°°(

(iii) v e L°°(I; W2'"{ü) n (iv) f 6 L²(Q^T),

(v) the continuity équation (1.2) is satisfied in the sense of distributions on

(vi) f ^d r ^ f ^d<Pi f / \^d<Pi f ^dV f m , ^ ( ^. dW(De(v)) de^%3 , . f ,

Jtt ot 7Q axj 7^ öxi Ja den d(^t) ÓXk -*&

holds for a.e. t G I and for every y = (tpu ..., y?d) E W2'g(^) H Wolï2(fi),

(vii) the initial conditions (1.12) with p0 G C1^ ) and v0 e W2^(£7) O W0lï2(^) are fulfilled.

In [14] authors proved the existence and uniqueness of weak solution of the similar problem except the nonlinear potential F , which dépends on the whole tensor e = (eij)fj⁼¹ instead of Tre = (eu)f^=l as in (1.5). Nevertheless the existence and uniqueness can be proved in the same way. In what follows we only point out the parts which differ from results stated in [14].

Theorem 2.3. (Existence of a weak solution).

Let po e Cl(Tl),po > 0 inTÏ, and vQ € W2>q(tï) H WQ'2(Q). Let the assumptions (1.8)-(1.11) hold. Then there is at least ont weak solution (p>v) to the problem (1*2), (1.3), (1.12)—(1.14) such that

^ e L°°(/; L«(n)), (2.9) v e L°°(J; W²'"(Q) n W⁰¹'²(fï))⁾ ^ e L²(Q^T).

Proof. First step is based on the modified Galerkin method and the method of characteristics, which give us the following identities see [141:

f f

/ pmd z = / Jnt Jn

/ / = m⁰,

n^t J

\S | ( p

K f ) + / §:(P(Pm))+ j ^(Tre(v™))e

(v

) +

^rr^

(v

)= f p^v?, (2.11)

f V(Tre(v

(O))) + W(De(v

(O)))+ f Pmh^f- • (2.12)

9 2 8 S. MATUSÜ-NECASOVA AND M. MEDVIDOVÂ-LUKÂCOVÂ

In the second step the limiting process is performed. We mention only the convergence in the nonlinear term which is different now and also was not clearly explained in [14]. It reads

Let us define

and

We set

X^sm= ['{G¹(v^m)-G¹{<p),e{v^m)-e(,<p))dt + ['(G²(v^m) - G²(<fi), De(v^m) - De(<p))dt +\\^Pmv^m{s)\².

Jo JO l

Choosing a subsequence v^nk which we dénote as vⁿ and using monotonicity of X^ we obtain

Thus, it follows from (2.11) that

Xsm= l\pmb,vm)àt + \\PomvZ\2- [S(G2(vm),De(v))dt- [* (G2(<p), De(vm - <p))àt

Jo z ^o Jo

(G^UWdt- f (GiM,e(</"-¥>))dt

Jo

pb,v) +hpovo\

- f\ti,De(<p))- f\Gz(<p),De(v-<p))- f {&,

^ Jo Jo Jo

- l\Gl{y),e{v-ip))>\\pv{s)\2-

Jo L

Finally, we get

/ Ki ~ G²(cp),De(v) - De&)) + (6 - G^ety - ^)) > 0

JO

which implies that

This concludes the proof. D Theorem 2.4. (Uniqueness of the weak solution).

Let the assumptions of Theorem 2.3 be fulfiîled. Then the weak solution obtained in Theorem 2.3 is unique.

Proof. Is analogous to the proof of uniqueness in [14].

3. UNCONDITIONAL ASYMPTOTIC STABILITY OF THE REST STATE

The goal of this section is to prove the stability of the rest state (p, 0), characterized by (1.1). Let b = ~^- with

£ € W2>°°(tt), g > 0 and let (pOj^o) satisfy conditions pö e C1^ , p0 > 0 in fi, v0 e W ^ ' ^ î î ) n W0lï2(^)-We specify the class of perturbated flows where the rest state will be stable:

~^- e L°°(/,L⁹(ft)), - ^ e L²(Q^T),for any T, and there exist 6i,ö2 such that , O < Si < p < <52 uniformly in (0, oo) x Ù, (p,v) is a weak solution of (1.2), (1.3)}.

Muitiplying (1.3) by v we obtain

l d f ²

i_l ll^llw²-?^)) ' ^^{r O m} (^*^) W e n a V e t n a t

lim cr(t) = 0. (3.2)

t—»oo

Moreover, we would like to prove that

v(t) —»• 0 strongly in L²(^) as t —» oo.

It holds

Ö17 / PM2 + IMIw2.*(n) + l|V-ï;||L«(n) - / p(p)V • v = /

2 dt JQ Jfi Ja

By integrating the above relation with respect to t from s t o t , 0 < s < £ < o o and with respect to s from t — l to t we obtain

/

t pt pt pt

/ IMIi^(n) + / /

-1 Js Jt-lJs

ö / llvPW^WIIia/n) + / / / |pbv + p(p)V • v|da:drds. (3.3)

^ Jt-i Jt-i Js Jn

Further,

j ƒ ( n ) ) ( ) j ^ /

/ / / P(p)V • v < c sup p(r)a(t).

t/t—1 J s JQ d>i<r<52

Then

From (3.2), using the fact that p > 0, we get that

v(t) —• 0 strongly in L²({7) as t —> oo.

930 S. MATUSÛ-NECASOVÂ AND M. MBDVIDOVÂ-LUKÂCOVÂ

Now, we define a function w(t) in Qx as the solution of the following problem

Aw = V • (pb) in QT ï (3.4)

—— = pb • n on ffi x [0, oo), on

[ w

The goal is to prove that

p(t) —> p in L2(Q) as t ^ oo (3.5) where

Wp(p) = p6 in n, (3.6) / pdx = /

JQ JQ,

Firstly, we shall prove that

\J\p(p{t))-w{t)}(h-Mh)àx

with a constant c independent of w. We set

M/i-l^l"1 / Jn For this purpose we shall firstly estimate the intégral

<ca(t)\\h\\LHa), heL\Q), (3.7)

I(t)= [ tp(s-t) / b ( / 5 ( s ) ) - ^ ) ] M x d5 î ^€C0°°(~1,0), (3.8) where y? is a fixed function such that J_x (p(r)dr = 1.

Let ^ b e a solution to the following problem for arbitrary h E L2(Q)

= h - Mh in fi

= 0 on dn (3.9) an

/ = 0.

Jn

Applying the classical results on boundary-value problems for elliptic équations it yields

Q) < c\\h - Mh\\L2{n), (3.10)

\\h-Mh\\L2(iî)<\\h\\L,{n), (3.11)

with a constant c > 0 independent of h. Substituting (3.9) into (3.8) we find I(t) = - ƒ (p(s-t) [w(s) - p(p(s))]Aipdxds

Jt~i Jn

= f ip(s-t) f [p(s)b - Vp(p(s))]Vijdxds Jt-i Jn

= / tp(s — t) \p + p(s)v(s)Vv(s) — divrp tj Jt-i Jn l vs J

= f tp{s -t) f ^ W> + / <p(s-t) [ V • (p(s)v(s)v(s))Vi!>

Jt-i Jn a s Jt-i Jn

+ f <p(s-t) / r

V V ^

Jt-i Jn

= - ƒ (p'(s-t) / p{s)v(s)Vil; - /

- i

aV(Tre) ôe..

< sup

t€(0,T)

sup

te(o,T)

W

||v(s)

t \ 2/g

ll"(*)llw=».«(n)

-i 0

t-i l/S'

However, cp is an arbitrary fixed function, which implies that

I(t) < ca(t)\\h\\L2m, t>0.

Now, we go back and prove (3.7).

f \p{p{t)) ~ w{t)\{h - Mh)àx = / ip(s - t) f (Vp(p(t)) - p(t)b)Vipdxd, Jn Jt-i Ja.

<|/(t)|+ / ¥>(«-*) /b(pW)-p(^(s))][/i-M/i]da;ds

f ^(s - t) f \p{t) - p

t-i Jn

We have

~p(t) - ~p(s) - ƒ l ^ d r = " ƒ * V • (p(

(3.12)

(3.13)

(3.14)

932 Hence

S. MATUSÜ-NBCASOVA AND M. MEDVIDOVÂ-LUKÂCOVÂ

/ <p(s ~~ t) [ [p(t) - p{s)}b . W = I ƒ [ <p(*-t) [ p{r)u{r)V{b • Vtyàxdrds

Jt-i Jn \Jt-i Js Jn (3.15)

t - i

The second term of the RHS of (3.13) can be rewriten in the following form

f <p(s -t) f \p(p(t)) - p(p(s))](h - Mh)dxds = f ip(s-t) f f dTp(p(r))(h - Mh)drdsdx. (3.16) Jt-i Jn Jt-i Jn Js

From the continuity équation it follows that

dp(p) , „ „ , „ dt

Thus, we can estimât e (3.16) from above

ƒ <p(s-t) f f drp{p{r)){h - Mh)dxdrds < f ip(s - i) f f v(r)pf{p(r))Vp(r)(h-Mh)dxdrds

\Jt-\ Js Jn Jt-i Js Jn

-f / ip(s-t) / p(r)pf(p)V-v(h-Mh)dxdrds Jt-i Js Jn

Uoo(o,T) sup p^f{r)o{t){ \\Vp(T)\\l^H

51<r<Ô2 \Jt-l K

SUP

which together with (3.12), (3.13) and (3.15) leads to (3.7). But it follows from (3.7) that p(p(t)) - w(t) ~ Mp(p(t)) -> 0 inL2(H) t^oo.

\\h\\L2{n)

<ccr{t)\\h\\L2(Q), (3.17)

(3.18) In fact, there exists a subsequence tⁿ —> co such that p(tⁿ) —*• P weakly in L², where w(t) is a solution of

(3.4) for any t. The séquence w(tⁿ) is compact in L²(Q). Choosing a subsequence converging to WOQ G L²(Q) we find that Woo = A(pb), where A stands for the solution operator of (3.4). Thus, p(p(tn)) — Mp(p(tn)) —>

Woo in L2(Q) and a.e. in ft. Since p belongs to J, we get that Mp(p(tn)) —> Poo- This yields

Hence p —

which yields

+Poo) a.e. in Q and in L2( or p(p) = A(pb) +Poo- This is equivalent to

[Vp(p)-pb}V9dx = 0 V ^ C0° ° ( î î )

= (Id-R)(pb),

strongly. (3.19)

(3.20)

(3.21)

where R is the projection of L²(Q) onto the closure in L²{Çl) of the space of divergence free vector functions.

Now, we want to prove that R(pb) = 0. To this purpose let us take an arbitrary 6 E Co°(Q),^ = R0. Then we have

R(p(s)b)0= f \p(s)^+fts)v(s)Vv(s)-divT^%3\r)dx. (3.22)

Jn Jn L ös \

In the same way as before we can prove

\ <p{s-t) R{p{s)b)6dxds

\Jt-i Jn which implies that

R(p(t)b)0dx

JQ

Since ||?7||^/x,2^\ < ||ö||^i³2^\^} we will obtain that

0.

R(p(t)b) -> 0 in W-lt2(Q) strongly. (3.23) Then R{p{tn)b) -^ 0 in W-^2(Ü) and R(p(tn)b) -> #(p6) in L2(Q). Thus, B(pb) = 0 and

S7p(p) = pb in fi. (3.24)

Now, if 7T is such a function that 7r'(r) = r~1p/(r) then, V?r(p) = V^. Hence, we have ir(p) = g + h with some constant h ox p ~ ir~^l(g -\-h). It holds that

/ p(x,tn)dx= / podx, (3.25)

f ~ f

/ p(x)dx = / po(^)dx.

i n i n

Since the function a(/i) = J^Q^^¹(g + /i)dx is increasing, the function p is uniquely determined by a unique constant h* satisfying fn n~1(g + /i*)dx = Jn podx. This implies

p ^ p i n L2( ^ ) . (3.26)

We have proved the following result.

Theorem 3.1. Let b^z = J|-, Ç e W²'°°(n)^; Ç small enough, & > 0 and ^ &e tfte se^ of all solutions (p,u) of the problem (1.2)-(1.13) with the initial conditions v⁰ G W²^^q{ü) H WQ'²(Ü), p⁰ € C^l(Ü). Then the rest state (p, v) characterized by équations (IA) is uncondttionally asymptotically stable in the class J in the sense that there exists a subsequence {tn},tn —> -hoo such that

p{tⁿ) ~* P inl²(ü), v(tn) ->0 mL2(fi), and

Vp(p) — pb in ft.

Remark 3.1. The behaviour of incompressible Newtonian fluids under assumptions of sufficient regularity and p > 0 was studied by Salvi and Straskraba [25]. Semigroup approach we can find in the work of Neustupa [21].

934 S. MATUSÛ-NECASOVÂ AND M. MEDVIDOVÂ-LUKÂCOVÂ

Acknowledgements. The présent research of S. Matusû-Necasovâ has been supported under Grant No. 201/98/1450 of the Grant Agency of the Czech Republic. The research of M. Lukâcovâ-Medvid'ovâ has been supported by Grant No. 201/97/0153 and No. 201/00/0577 of the Grant Agency of the Czech Republic. The authors express their thanks for this support.

REFERENCES

[1] G. Amrouche and D. Cioranescu, On a class of Ûuids of grade 3, Laboratoire d'analyse numérique de l'université Pierre et Marie Curie, rapport 88006 (1988).

[2] C. Amrouche, Sur une classe de fluides non newtoniens ; les solutions aqueuses de polymère» Quart. Appl Math, L(4) (1992) 779-791.

[3] H. Bellout, F. Bloom and J. Necas. Young measure-valued solutions for non-Newtonian incompressible fluids. Commun. Partial Differential Equations 19 (1994) 1763-1803.

[4] Beirao da Veiga, An Lp - theory for the n-dimensional stationary compressible Navier-Stokes équations and the incompressible limit for compressible fluids. The equilibrium solutions. Comm. Math. Phys. 109 (1987) 229-248.

[5] D. Cioranescu and E.H. Quazar, Existence and uniqueness for fluids of second grade. Collège de France Seminars, Pîtman Res. Notes Math. Ser. 109 (1984) 178-197.

[6] E. Feireisl and H. Petzeltovâ, On the steady state solutions to the Navier-Stokes équations of compressible flow. Manuscripta Math. 97 (1998) 109-116.

[7] E. Feireisl and H. Petzeltovâ, The zéro - velocity limit solutions of the Navier-Stokes équations of compressible fluid revisited, in Proc. of Navier-Stokes équations and the Related Problem, (1999).

[8] G.P. Galdi, Mathematical theory of second grade fluids, Stability and Wave Propagation in Fluids, G.P. Galdi Ed., CISM Course and Lectures 344, Springer, New York (1995) 66-103.

[9] G.P. Galdi and A. Sequeira, Further existence results for classical solutions of the équations of a second grade fluid. Arch.

Ration. Mech. Anal 28 (1994) 297-321.

[10] D.D. Joseph, Fluid Dynamics of Viscoetastic Liquids. Springer Verlag, New York (1990)

[11] J. Mâlek , J. Necas, M. Rokyta and R. Rûzicka, Weak and Measure-valued solutions to evolutionary partial differential équations, Chapman and Hall (1996).

[12] A.E. Mamontov, Global solvability of the multidimensional Navier-Stokes équations of a compressible fluid with nonlinear viscosity L Siberian Math. J. 40 (1999) 351-362.

[13] A.E. Mamontov, Global solvability of the multidimensional Navier-Stokes équations of a compressible fluid with nonlinear viscosity IL Siberian Math. J. 40 (1999) 541-555.

[14] S Matusû-Necasovâ and M. Medvid'ovâ, Bipolar barotropic nonnewtonian fluid, Comment, Math. Univ. Carolin 35 (1994) 467-483.

[15] S. Matusû-Necasovâ, A. Sequeira and J.H. Videman, Existence of Classical solutions for compressible viscoelastic fluids of Oldroyd type past an obstacle. Math. Methods Appl Sci. 22 (1999) 449-460.

[16] S. Matusû-Necasovâ and M. Medvidbvâ-Lukâcovâ, Bipolar Isothermal non-Newtonian compressible fluids. J. Math. Anal.

Appl 225 (1998) 168-192.

[17] J. Necas and M. Silhavy, Multipolar viscous fluids. Quart. Appl Math. XLIX (1991) 247-266.

[18] J. Necas, A. Novotny and M. Silhavy, Global solutions to the viscous compressible barotropic multipolar gas. Theoret. Comp.

Fluid Dynamics 4 (1992) 1-11.

[19] J. Necas, Theory of multipolar viscous fluids, in The Mathematics of Finite Eléments and Applications VII MAFELAP 1990, J.R. Whitemann Ed., Académie Press, New York (1991) 233-244.

[20] J. Neustupa, A semigroup generated by the linearized Navier-Stokes équations for compressible fluid and its uniform growth bound in Hôlder spaces, in Proc. of the International Conférence on the Navier-Stokes équations, Theory and Numerical Methods, Varenna, June 1997, R. Salvi Ed., Piiman Res. Notes Math. Ser. 388 (1998) 86-100.

[21] J. Neustupa, The global existence of solutions to the équations of motion of a viscous gas with an artificial viscosity. Math.

Methods Appl Sci. 14 (1991) 93-119.

[22] J.G. Oldroyd, On the formulation of rheological équations of state. Proc. Roy. Soc. London A200 (1950) 523-541.

[23] K.R. Rajagopal, Mechanics of non-Newtonian fluids, in Récent Developments in Theoretical Fluid Mechanics Séries 291, Longman Scientific & Technical Reports (1993).

[24] M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical problems in Viscoelasticity} Longman, New York (1987).

[25] R. Salvi and I. Straskraba, Global existence for viscous compressible fluids and their behaviour as t —*• oo. J. Faculty Sci.

Univ. Tokyo, Sect. I, A40 (1993) 17-51.

[26] W.R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, New York (1978).

[27] M.H. Sy, Contributions à l'étude mathématique des problèmes isssus de la mécanique des fluides viscoélastiques. Lois de comportement de type intégral ou différentiel Thèse d'université de Paris-Sud, Orsay (1996).

[28] C. Truesdell and W. Noll, The Nonlinear Field Théories of Mechanics, 2nd edn. Springer, Berlin (1992).