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On the existence of solutions for non-stationary third-grade fluids
International Journal of Non-Linear Mechanics 34 (1999) 485—498
On the existence of solutions for non-stationary third-grade fluids Didier Bresch, Je´roˆme Lemoine* Laboratoire de Mathe& matiques Applique& es, CNRS UMR 6620, Blaise Pascal University (Clermont-Ferrand 2), 63177 Aubie% re cedex, France Received 7 June 1997; received in revised form 20 March 1998
Abstract This paper is devoted to non-stationary third-grade fluids. We present some existence and uniqueness results in ¸P where r'3 for open sets only of class C. The proof is based on a fixed point method. 1998 Elsevier Science Ltd. All rights reserved. Re´ sume´ Dans ce papier, on s’inte´resse aux fluides de grade 3. On pre´sente des re´sultats d’existence et d’unicite´ dans ¸P pour r'3 et pour un domaine seulement de classe C. La de´monstration est base´e sur une me´thode de point fixe. 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction Some fluids used in industry cannot be modelized by the ‘‘classical” Navier—Stokes equations which follow from the linear constitutive law p"!pI#l( u#R u), where l'0 is the viscosity. For example, liquid crystals, some oils and others (see, e.g., [1—4] and references therein) are modelized by constitutive laws taking into account non-linear characteristics which are shown by experiments. In the case of differential type fluids, there exists a number of
*Corresponding author.
constitutive laws. One of the most used actually has been introduced by Rivlin and Ericksen [5]. The fluids described by this law are called fluids of grade n. Here, we are interested more precisely in incompressible third-grade fluids, i.e. fluids obeying a non-linear constitutive law in which the stress is expressed in terms of the pressure and of some tensors of Rivlin—Ericksen type: p"!pI#lA #aA #bA#c(trA)A , where A " u#R u, d A " A #A u#R uA . dt
0020-7462/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 8 ) 0 0 0 3 4 - 1
(1)
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D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
For physical reasons, [6, Proposition 1.1] or [7], one assumes l'0,
c*0, a*0.
The stress is related to the velocity by the following equations: * u#(u ) )u! ) p"f, R
) u"0, u(0)"u , u" "0. f As usual, we define the potential P"P(u, p) (which has no physical meaning) by P"au ) *u# (2a#b)"A "! "u"!p. Thus, the previous equations can be rewritten in the following form: * (u!a*u)!l*u#( ;(u!a*u));u# P R "f#(a#b)N (u)#cN (u), (2a)
) u"0, u(0)"u , u" "0, (2b) f with N (u)"A *u#2 ) ( u (R u)), (3a) N (u)" ) ((tr A)A ). (3b) For c"0 and a"b"0, we find the classical Navier—Stokes equations. For c"0 and a"!b, we find the equations of second-grade fluids. A number of authors have studied the existence problem in this case for a simply-connected domain ) with a connected boundary and in the case r"2. Cioranescu and Ouazar [8], for instance, have given the local existence for a solution in a regular enough domain, and more recently Cioranescu and Girault [9], Galdi and Sequeira [10] have given the global existence in time (on [0,#R)) of a solution under suitable assumptions on the data and ) of class C , simply-connected. Let us go back to the third-grade case. With the supplementary hypothesis b*0, a'0 and "b#a")(24lc, Amrouche [3], Amrouche and Cioranescu [6] have given the local existence of a solution for ) of class C and simply-connected with a connected boundary; Sequeira and Videman [11] have given the global existence in time (on
[0,R)) of a solution for ) regular enough, simplyconnected, and for small enough data. Here, at first, we prove (Theorem 1) the local existence and the uniqueness of a solution of Eq. (2) for a'0 in any domain of class C. Next, we prove (Theorem 2) the global existence (on [0,R)) if in addition the data are ‘‘small enough” and f3¸(0,R;¸P())) or f3¸P(0,R; ¸P()))5¸(0,R;¸P())). We do not need *) to be connected and C , as in the previous works, since we solve directly the system Eq. (2) instead of using the image of this system by the curl operator. The results have been given in [12] with the supplementary assumptions a, l/a great enough to ensure the global existence. Here we have modified the given uncoupled system to get rid of these conditions. We use a more simple decomposition than in [13] or in [14]. We do not need to assume ) simply-connected to ensure the global existence as it was also supposed by Galdi [13]. Even though we consider homogeneous boundary conditions in this paper, let us note that the study of the inhomogeneous case u"u on * *);(0,¹) is of fundamental importance when we want to modelize the dynamics of the normal stress difference effect. But this boundary condition will not be sufficient to ensure well posedness of the problem. In [15,16], for instance, we can find elementary solutions in bounded and unbounded domains which do not possess the uniqueness property if the condition u ) n"0 is violated with * n the unit normal vector of the boundary. This last condition is not reasonable in a lot of interesting physical problems. Thus, Galdi et al. [17] have given an existence result with a suitable initial condition with the help of a fixed point on a multivalued map without imposing the condition u ) n"0. In [18,19], it has been proved in un* bounded domains that we can augment boundary conditions or use boundnedness requirements to obtain a well-posedness problem. The paper is organized as follows. In Section 2, the main results and the sketch of the proof are given. In Section 3, some functional spaces definitions and a result on a compact imbedding which is essential in our proof when we apply the Schauder Theorem are given. In Section 4, the local existence
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
Theorem is proved after the study of a transport equation and of a system which is practically a Stokes system. In Section 5, the uniqueness of the solution is proved. With all the previous results, the global existence of solutions is proved in Section 6.
Let D()) be the space of C functions with compact support in ) and D ()) the space of distributions on ). We denote by 1 , 2 the duality product between D()) and D ()). We denote by C ([0,R)) the space of bounded @ continuous functions on [0,R). For 1)s)#R, the Sobolev spaces are defined by ¼ Q())"+v3¸Q()): v3¸Q()),, ¼ Q())"closure of D()) in ¼ Q()),
# * v : v 3¸Q()), j"0,2,3 , G G H G all these spaces being endowed with their usual norms. We denote H())"¼ ()), H())"¼ ()), H\())"¼\ ()) and V"+v3D()) : ) v"0,, »"+v3H()) : ) v"0,. Let us recall that » coincides with the closure of V in H()). We give now the following result for which a proof can be found in [20, Corollary 4, p. 85]. Lemma 4. ¸et X, B and ½ be three Banach spaces such that XLBL½ where the imbedding X | B is compact and let s'1. ¹hen ¸(0,¹;X)5¼ Q(0,¹;½)LC([0,¹];B) with the corresponding compact imbedding.
In the sequel, c (or c ) denotes various real positG ive numbers.
3. Main results Let a'0 and let ) be a connected domain in 1 of class C. We denote Q ");(0,¹). 2 Throughout this paper, we assume that
2. Functional spaces and some preliminaries
¼\ Q())" v3D ()) : v"v
487
u 3¼ P())5», f3¸P(Q ), r'3. (4) 2 Without hypothesis on the size of the data, we have the following result of local existence. Theorem 1. ¹here exists ¹*, 0(¹*)¹, such that the system (2) has a unique solution (u, P) satisfying u3C([0,¹*];¼ P())), * u3¸P(0,¹*;¼ P())), R P3¸P(0,¹*;¸P())). Now we state the supplementary hypothesis which will ensure the existence of a global solution: u is small enough that is #u !a*u # P )c (a,l,b,c,r,)), (5) * and f is small enough in some suitable spaces. More precisely, we suppose, in a first case f3¸(0,R;¸P()))
(6)
with #f# (c (a,l,b,c,r,)). * d*P In a second case, we suppose
(7)
f3¸P(0,R;¸P()))5¸(0,R;¸P()))
(8)
with # f #P P ## f # (c (a,l,b,c,r,)). * d*P * d*P (9) For such small data, one has the following result of global existence. Theorem 2. ¼e assume Eqs. (4) and (5): (i) ¸et us suppose Eqs. (6) and (7) be satisfied. System (2) has a unique solution (u, P) satisfying u3C ([0,R);¼ P())), * u3¸(0,R;¼ P())), @ R P3¸(0,R;¸P())).
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(ii) ¸et us suppose Eqs. (8) and (9) be satisfied. ¹herefore, system (2) has a unique solution (u, P) satisfying
(z,n) being defined by
u3C ([0,R);¼ P()))5¸(0,R;¼ P())), @
z" "0, f
* u3¸P(0,R;¼ P()))5¸(0,R;¼ P())), R P3¸P(0,R;¸P()))5¸(0,R;¸P())). Throughout the paper, for vector-valued functions u"(u ,u ,u ), v"(v ,v ,v ), we denote by u ) v and v ) u the vector-valued functions, respectively, defined by (u ) v) " u * v , ( v ) u) " * v u H G G G H H G H G G (note that u ) vO v ) u). Let u ) v" u v , u ) v" * u * v . G G G GH G H G H Proof (Sketch of the existence results). As it will be proved in Lemma 3, it suffices to find the (u,n,w) solution of the following system: l l * w# w#u ) w# u ) w! u R a a (10)
w"u!a*u# n,
(11)
n"0.
(12)
We will solve Eqs. (10)—(12) by linearization and use of the Schauder’s fixed-point Theorem. Given u satisfying Eq. (12), we define w by l * w# w#u ) w# u ) w R a l " u#f#(a#b)N (u)#cN (u), a w(0)"u !a * u .
) z"0,
n"0,
(13a) (13b)
We are still to prove that u may be chosen such that there exists n satisfying Eq. (11). This will be done by a fixed-point argument. More precisely,
(14a) (14b)
we will prove that the map u | z has a fixed point. For this we will prove (Proposition 5) that the system Eq. (13) has a unique solution w which satisfies #w#)G(t,#u#,c (# f ###u !a*u #))
(15)
for some convenient norms in functional spaces on );(0,¹), where G is an increasing continuous function of t such that G(0,x,y)"y. Next we will prove (Proposition 8) that the system Eq. (14) has a unique solution (z,n) such that #z#)c #w#.
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Finally, we will prove that, for a convenient M"M(u , f,a,l,b,c,),r), we can choose t'0 such that #u#)M implies c G(t,#u#,c (# f ###u !a*u #)))M.
"f#(a#b)N (u)#cN (u),
) u"0, u(0)"u , u" "0, f
z!a*z# n"w,
(17)
By Schauder’s fixed-point Theorem, together with some continuity properties, this will prove the existence of a fixed point which moreover satisfies u(0)"u . Therefore, we obtain the existence of a solution of Eqs. (10)—(12), that is the local existence. Using the additional hypothesis (5)—(7) (or (5), (8), (9)) on the data we will prove the existence of M for which condition Eq. (17) is satisfied for all positive t. This will prove the global existence and therefore ends the sketch of the proof. 䊐 Remark. It is important to note that n introduced in Eqs. (10)—(12) can be different from the potential P which is defined in Eq. (2). Throughout the paper, n is normalized by n"0. This ensures the uniqueness of n as soon as
n is unique. Let us now verify that any solution of Eqs. (10)—(12) provides a solution of Eq. (2).
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
Lemma 3. ¸et (u,n,w) be a solution of Eqs. (10)—(12), and l P"w ) u#* n# n. R a
489
4. Existence and uniqueness of a local solution In this section, we will prove Theorem 2.
(18)
¹hen (u,P) is solution of Eq. (2). Proof. Let (u,n,w) be a solution of Eqs. (10)—(12). Replacing w by u!a * u# n in Eq. (10) gives l * (u!a * u# n)# (u!a * u# n) R a #u ) (u!a * u# n) l # u ) (u!a * u# n)! u a "f#(a#b)N (u)#cN (u). This equation can be rewritten in the following form: * (u!a * u)!l*u#u ) (u!a * u# n) R ! ((u!a * u# n) ) u) # ((u!a * u# n) ) u) l # u ) (u!a * u# n)#* n# n R a "f#(a#b)N (u)#cN (u). Since u ) w! (w ) u)# u ) w"( ;w);u and ; n"0, we obtain * (u!a * u)!l* u#( ;(u!a*u));u R l # ((u!a * u# n) ) u)#* n# n R a "f#(a#b)N (u)#cN (u). With the definition (18) of P, this gives the first equation of Eq. (2). The other equations of Eq. (2) are given by Eq. (12). 䊐
4.1. Study of the transport Eqs. (13a,b) The goal of this part is to prove the following result. Proposition 5. ¸et u satisfy (12) and u3¸(0,¹,¼ P())).
(19)
¹hen the system (13) has a unique solution such that w3C([0,¹];¸P())), * w3¸P(0,¹;¼\ P())). R Denoting """u""""#u# , w satisfies, for * Rd5 P all t)¹, #* w# P ##w# R * Rd5\ P * Rd*P
R
# u# * ;(t P"""u"""## f # P ##u !a*u # P ). * / R * (20)
)c (1#t P (1#"""u""")) exp
The proof is given in several steps. In a first step we will see (Lemma 6) that for regular data (g,v ) there exists a solution v of the transport equation * v#u ) v"g, v(0)"v . (21) R In a second step, we will obtain, by approximation and density (Lemma 7), a solution of l * w# w#u ) w# u ) w"g, R a
w(0)"w .
(22)
For convenient g and w this will solve Eq. (13). Therefore, let us begin with the following Lemma. Lemma 6. ¸et u satisfy (12) and (19), v 3¼ P()) and g3¸P(0,¹;¼ P())). ¹hen the system (21) has a unique solution such that v3C([0,¹];¼ P())), * v3¸P(Q ). R 2 It satisfies, for all t, 0(t)¹, #v# * Rd5 P )c(¹,u)(#v # P ##g# ). 5 * Rd5 P
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D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
Proof. Since u" "0 and ) is C, there exists f a unique solution y3C([0,¹];)) of
R
The unique solution of Eq. (21) in C([0,¹]; ¼ P())) is given by v(t,x)"v (y(0,t,x))# R g(y(m,t,x),m) dm (see [21]). It satisfies * v3¸P(Q ). Moreover, one has for all t, q, and Rx such 2that 0(q)t)¹ and x3),
y(q,t,x)"x! u(y(m,t,x),m) dm. O
" y(q,t,x)")(3 exp
It satisfies, for all t)¹, a P\ r l
(23)
Therefore )sup#v (y(0,q, ) ))# P #v# * * Rd*P OXR O #sup g(y(m,q, ) ),m) dm . P * OXR Because y(m,t, ) ) is a one-to-one map on ) with a Jacobian determinant equal to 1 (cf. [22]), this implies
#v# )#v # P ##g# . * Rd*P * * Rd*P Moreover one has, for all i, * v(t,x)"* y(0,t,x) ) v (y(0,t,x)) G G
R # * y(m,t,x) ) g(y(m,t,x),m) dm. G
"g"P# "w "P /R
R ;exp r # u# ,
(24)
where r is given by 1/r#1/r "1. ¹he system (22) has no other solution such that w3¸(0,¹;¸P())),
R ""u"" . 5 O
"w"P (t))
* w3¸P(0,¹;¼\ P())). R
Proof. (1) Existence in the case of regular data: At first, we assume w 3¼ P()), g3¸P(0,¹;¼ P())).
(25)
Starting from w"0, we define by iteration wL3C([0,¹];¼ P())), * wL3¸P(Q ), R 2 to be the solution given by Lemma 6 of the following system: l * wL#u ) wL"g! u ) wL\! wL\, R a wL(0)"w . Let ¼L"wL!wL\. It is the unique solution of l * ¼L#u ) ¼L"! u ) ¼L\! ¼L\, R a
Then by the same technique as above, we get
¼L(0)"0,
#* v# )c(¹,u)(# v # P G * Rd*P *
and due to Lemma 6, it satisfies for all t, 0(t)¹,
## g# ), * Rd*P which implies the required inequality. 䊐 Now, we will prove the following Lemma. Lemma 7. ¸et u satisfy (12) and (19), w 3¸P()) and g3¸P(Q ). ¹hen the system (22) has a unique solu2 tion such that w3C([0,¹];¸P())), * w3¸P(0,¹;¼\ P())). R
#¼L"" * Rd5 P
l )c(¹,u) u ) ¼L\# ¼L\ a
. * Rd5 P
In addition, since u3¸(0,¹,¼ P())) and since ¼ P()) is a multiplicative algebra, one has, for all t, 0(t)¹, # u ) ¼L\#
* Rd5 P
)c(),r)# u# #¼L\# . * Rd5 P * Rd5 P
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
(2) Existence in the general case: Now we consider w 3¸P()), g3¸P(Q ). Let wL 3¼ P()) and 2 g 3¸P(0,¹;¼ P())) be such that L
Then, we have #¼L# * Rd5 P )c(¹,u,),r,l,a)#¼L\# . * Rd5 P
wL Pw in ¸P()),
Denoting by W (t) the left-hand side, we have L
w 3C([0,¹];¼ P())), * w 3¸P(Q ). L R L 2
which gives, since W (t) increases with t, tL\ W (t))cL\ W (t). L (n!1)! Then the series W (t) converge, and in conseL L quence wL converges to w in the space C([0,¹]; ¼ P())). In addition, w is a solution of Eq. (22) and * w3¸P(Q ). R 2 Moreover, by multiplying Eq. (22) by "w"P\w and integrating, we get
lr "w"P# "w"P)r "g" "w"P\ a #r
" u ) w" "w"P\.
(26)
r
"g" "w"P\)
a P\ r l
lr "g"P# "w"P, a
" u ) w" "w"P\)r# u# *
"w"P,
(27)
(28)
a P\ "w"P) "g"P#r# u# "w"P. * r l
Denoting t(s)"exp(!rQ # u# ) "w"P and * multiplying by exp(!r Q # u# ), the previous * inequality, we get since exp(!r Q # u# ))1, *
a P\ t (s)) r l
"(e(x)")1, ∀x3),
(e(x)"1 if d(x)*2c(e), " (e(x)")k e/d(x), ∀x3),
one has d dt
The inequality (24) applied to w !w implies that L K (w ) - is a Cauchy sequence in C([0,¹];¸P())). Its L LZ limit w3C([0,¹];¸P())) satisfies Eqs. (22) and (24). By Eq. (22), * w3¸P(0,¹;¼\ P())). R (3) ºniqueness: It follows after extension on 1L from a method due to R.J. Di Perna and P.-L. Lions [25]. This is a regularization method in which we use Hardy’s inequality. Let us give the proof here for the reader’s convenience. Let w be the difference of two eventual solutions of Eq. (22) such that w3¸(0,¹;¸P())) and * w3¸P(0,¹;¼\ P())). It satisfies Eq. (22) with R w "0 and g"0. Let us now recall that there exists a function (e3C()) such that
(e(x)"0 if d(x)(c(e)/2k ,
Since r
g Pg in ¸P(Q ). L 2
Then there exists a related solution w of Eq. (22) L such that
R W (t))c W (q) dq, L L\
d dt
491
"g"P.
Thus by integration from 0 to t, we obtain Eq. (24).
with c(e)"exp(!1/e), d(x)"dist (x,*)) and k , k two constants independent of ) and e (see, for instance, [24, pp. 166—168]). Denoting ve"w(e and ) the extension outside ) by 0, we verify that vJ e satisfies the following equation
l * vJ e#uJ ) vJ e# vJ e# uJ ) vJ e"! uJ * (e wJ , R G G a G vJ e(0)"0. One has vJ e3¸(0,¹;¸P(1)) and since u" "0, f uJ 3¸(0,¹;¼ (1)). By convolution of this equation by mollifiers o , and denoting vJ e "vJ e 夹 o , L L L
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D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
we get
Therefore, by definition of (e, one has
l * vJ e #uJ ) vJ e # vJ e # uJ ) vJ e L a L L R L
"r ! uJ * (ewJ L G G G
夹o
"R L L
(29)
r "!(uJ ) vJ e) 夹 o #uJ ) vJ e L L L
!(uJ ) vJ e) 夹 o #uJ ) vJ e P0 in ¸(0,¹;¸(1)). L L In addition ( uJ ) vJ e) 夹 o P uJ ) vJ e and uJ ) vJ e P L L
uJ ) vJ e in ¸(0,¹;¸(1)), then !( uJ ) vJ e) 夹 o # uJ ) vJ e P0 in ¸(0,¹;¸(1)). L L Multiplying Eq. (29) by vJ e and integrating on 1, L we obtain, since uJ has a compact support and satisfies ) uJ "0:
"vJ e " L 1
)
"R vJ e "#c # uJ # 1 * L L 1
"vJ e ". L 1
It follows that
# uJ # 1 *
u #w#P , * /2 d PY * /2
Moreover one has, see [25, Lemma 2.1, p. 516],
R
u d
!( uJ ) vJ e) 夹 o # uJ ) vJ e . L L
l "vJ e "# L a 1
)ce exp c
with r "2r/(r!2) and c independent of e. Now using Hardy’s inequality
where vJ e (0)"0 and L
1d 2 dt
"vJ e"(t)
1
)c#u# PY , 5 *PY
and letting e go to zero in the previous inequality, one obtains wJ "0, which proves the uniqueness. The proof of Lemma 7 is complete. 䊐 Proof of the Proposition 5. Let g denote the righthand side and w the initial data in Eq. (13), i.e.: l g" u#f#(a#b)N (u)#cN (u), a w "u !a*u .
(30)
Then g3¸P(Q ) and w 3¸P()). By Lemma 7, 2 there exists a unique solution w of Eq. (22) and therefore of Eq. (13). It remains to prove the inequality (20). Inequality (24) yields, denoting """u"""" #u# , * Rd5 P #w# * Rd*P
"vJ e "(t) L
)c # f #P P #t"""u"""P#t"""u"""P#t"""u"""P * /R
1
R
)c exp c
# uJ # 1 *
"R "(s) ds. L
1
R"
Since vJ e PvJ e in ¸P(0,¹;¸P(1)) and R P L L ! uJ * (ewJ in ¸(0,¹;¸(1)), by passing to G G G the limit in this inequality, we obtain
R"1
"u !a*u "P
P
exp
R
# u# . (31) *
#* w# P R * Rd5\ P )
uJ * (ewJ (s) ds. G G G
On the other hand, Eq. (22) gives
R "vJ e"(t))c exp c # uJ # 1 * 1 ;
#
l w a
## u ) w# P * Rd5\ P *P Rd5\ P
##u ) w# P ##g# P . * Rd5\ P * Rd5\ P (32)
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
)t P# ) # , we obtain Using # ) # P * Rd# * Rd# #* w# P R * Rd5\ P )c(#w# t P(1#"""u""")##g# P ). (33) * /R * Rd*P By Eqs. (30), (31) and (33), one has finally Eq. (20). 䊐 4.2. Study of the Stokes like equations Eq. (14) Let us now give some estimates of the solution of Eq. (14). Proposition 8. (i) ¸et 1(s)R and w3¸Q(0,¹; ¸P())), * w3¸Q(0,¹;¼\ P())). ¹hen the system R Eq. (14) has a unique solution (z,n) such that z3¸Q(0,¹;¼ P())), * z3¸Q(0,¹;¼ P())), R n3¸Q(0,¹;¼ P())), * n3¸Q(0,¹;¸P())). R (ii) If moreover w3C([0,¹];¸P())), then
and for all t, 0(t)¹, #z# ## n# * Rd5 P * Rd*P )c #w# , * Rd*P ##* n# Q #* z# Q R * Rd5\ P R * Rd5 P )c #* w# Q . R * Rd5\ P
(34a)
(34b)
Proof. This is a consequence of the following lemma which is related to similar results for Stokes equations, given for example in [26, Theorems 4.6, p. 128, p. 4.18, p. 136] or in [24, Theorem 6.1, p. 225]. Lemma 9. (i) ¸et w3¼\ P()). ¹hen the system (14) has a unique solution such that z3¼ P()), n3¸P()). It satisfies #z# P ## n# \ P )c #w# \ P . 5 5 5 (ii) ¸et w3¸P()). ¹hen z3¼ P()), n3¼ P()), and 1 #z# P # # n# P )c #w# P . 5 * * a
From Proposition 8, we deduce a possible value for the constant c defined in inequality (16) in Section 2. 4.3. Local existence of a solution of Eq. (2) This proof is based on the Schauder’s fixed-point Theorem that we can find, for example, in [27, Corollary 3.6.2, p. 163]. We define a non empty bounded and convex set by K *"+u3¸(0,¹*;¼ P()))5C([0,¹*];¼ P())): 2 u satisfies Eqs. (12) and (35), where 0(¹*)¹ will be chosen later and #u# * P )2M, * 2 d5 where
z3C([0,¹];¼ P())), n3C([0,¹];¼ P())),
493
(35)
M"c c (# f # P ##u !a*u # P ). * /2 * The constants c , c are given by Eqs. (20) and (34). It follows from Eqs. (20) and (34) that there exists ¹*3(0,¹] such that u | z maps K * into itself. 2 We provide K * with the topology of 2 X *"C([0,¹*];¼ P())). 2 Let us first prove that K * is closed in X *. 2 2 Let u 3K * with u Pu in X *. Let us prove that L 2 L 2 u3K *. Since u 3K *, u is bounded in ¸(0,¹*, 2 L 2 L ¼ P())) then, it converges to z in ¸(0,¹*, ¼ P())) 夹 weak and #z# * P )lim inf#u # * P . * 2 5 L * 2 5 By uniqueness of the limit in D (Q *), we obtain 2 that u3¸(0,¹*,¼ P())). Now, let us remark that u | z maps K * into 2 a relative compact set K * of X *. From the esti2 2 mates Eqs. (20) and (34), we deduce that K * 2 is bounded in ¸(0,¹*;¼ P()))5¼ P(0,¹*; ¼ P())). So using Lemma 4, we deduce that K * is relatively compact in X *. 2 2 Let us now prove that the map uPz is continuous provided with the topology of X * from 2 K * into itself. 2 At first let us prove that u | w is continuous from K * into C([0,¹*];¼\ P())), where K * is 2 2 endowed with the topology of X *. 2
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Let u and u in K *, u Pu in X *. Let w and w , L 2 L 2 L the solutions of Eq. (13), be associated with u and u . Let us denote ¼L"w !w. Then ¼L satisfies L L
fixed point, say u. This fixed point gives a solution (u,n,w) of Eqs. (10)—(12) on [0,¹*]. At most, from Propositions 5 and 8, one has
l * ¼L# ¼L#u ) ¼L#ºL ) w # u ) ¼L L R a
u3C([0,¹*];¼ P())),
# ºL ) w
n3¼ P(0,¹*;¸P()))5C([0,¹*];¼ P())).
L
Let us now define P3¸P(0,¹*;¸P())) by Eq. (18). By Lemma 4, (u,P) is the solution of Eq. (2). This gives the local existence result of Theorem 1.
l " ºL#(a#b)(N (u )!N (u)) L a #c(N (u )!N (u)), L
4.4. Uniqueness of a solution of Eq. (2)
¼L(0)"0, where ºL"u !u. L We have proved (cf. Eqs. (31) and (33)) that ¼L is bounded in ¸(0,¹*;¸P())) and that * ¼L is R bounded in ¸P(0,¹*;¼\ P())). Then there exists ¼K which converges to ¼ in ¸(0,¹*;¼\ P())) strongly where ¼3¸(0,¹*;¸P())), and * ¼3 R ¸P(0,¹*;¼\ P())). The functions N , N defined by Eq. (3) are con tinuous from K * endowed with the topology of 2 C([0,¹*];¼ P())) into C([0,¹*];¼\ ())), thus since ºLP0 in X *, ¼ satisfies 2
Considering the difference of two solutions of Eq. (2) denoted by (uG,PG) for i"1,2, we obtain the following inequality on º"u!u in D (0,¹*):
a * "º"# * " º"#l " º" R 2 R
1 2
"!a
#a
(u ) )º ) *º! (º ) )u ) º (º ) )u ) *º# (a#b)(N (u,º)
l * ¼# ¼#u ) ¼# u ) ¼"0, R a
(36a)
#N (º,u)) ) º#
¼(0)"0.
(36b)
#N (u,º,u)#N (u,u,º)) ) º,
Since ¼3¸(0,¹*;¸P())) and * ¼3¸P(0,¹*; R ¼\ P())), by uniqueness of a solution ¼ of Eq. (36) in this space (cf. Proposition 5), we obtain ¼"0. Therefore, the whole sequence ¼L goes to 0. This means that
(38)
where N (u,v)"( u#R u) ) *v#2 ) ( u ) (R v)), N (u,v,w) " ) (tr(( u#R u) ) ( v#R v))( w#R w)).
w Pw in C([0,¹*];¼\ P())). L On the other hand, by Proposition 8, the map w | z is continuous from C([0,¹*];¼\ P())) into X *. 2
c(N (º,u,u)
(37)
Therefore, from the continuity of u | w and w | z, we deduce that u | z is continuous from the bounded convex set K * into K *, and by 2 2 Schauder’s fixed point Theorem, it possesses a
Therefore, since uG3C([0,¹*];¼ P())), ) uG"0 and uG" "0, Eq. (38) gives
d dt
("º"#a" º")#l " º")R " º",
where R is a real number. This gives
d dt
R ("º"#a" º")) ("º"#a" º"), a
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
with º(0)"0. By the Gronwall lemma, we obtain º"0, which gives P"0. 䊐
5. Existence of a global solution
495
Since #N (u)# P )c (a,),r)#w#P , * * #N (u)# P )c (a,),r)#w#P , * * we obtain
In this Section, we prove Theorem 2 which states the existence of a global solution of Eq. (2) for small enough data. Proof of part (i). Let w be a solution of Eqs. (10)—(12) on [0,¹*], ¹*'0 given by Theorem 1. It satisfies the inequality (26) (this inequality was proved only for regular data satisfying Eq. (25); it extends to any data by a density argument). Instead of Eq. (27), let us use the following estimate:
"g" "w"P\)#g# P #w#P\ , *P *
to bound the right-hand side of Eq. (28). Together with Eq. (26), it gives d lr #w#P P # #w#P P * * dt a )r#g# P #w#P\ #r# u# #w#P P . (39) * *P * * Moreover, Lemma 9 gives # u# )Q #u# P )Q c #w# P , * 5 * where Q is the imbedding constant of ¼ P()) in ¼ ()). Therefore, with expression (30) of g, d lr #w#P P # #w#P P * * dt a
#c (a,b,),r) #w#P> #c (a,c,),r) #w#P> . *P *P (40) This estimate does not allow us to obtain the global existence of a solution for all a'0 (see [12]). Therefore we use the supplementary estimates on u and w. First, from the estimate (40), we deduce the following estimate on w: d 2l #w#P # #w#P * * a dt
l )2 # f # P # #u# P #w# P * * a * #c (a,b,),r) #w#P #c (a,c,),r) #w#P . * * (41) Now, mutiplying Eq. (2) by u and integrating, we obtain d l (#u# #a# u# )# # u# * * * dt 2
#c (a,c,),r)#w#P . *
#"c" #N (u)# P ) #w#P\ * *P
l #r # f # P # #u# P #w#P\ * *P a * #r Q c #w#P> . *P
l )r (# f # P # #u# P ) #w#P\ * *P a *
)c (l,),r) # f #P #c (a,b,),r)#w#P * *
)r("a#b" #N (u)# P *
d lr #w#P P # #w#P P * * dt a
(42)
Since, for all e'0 we have, using Young’s inequality and Lemma 9 #u# P )c (e,a,),r) # u# #e #w# P , * * * we obtain after some calculations and for a suitable choice of e, using Eqs. (40) and (41) the following
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estimates on w:
such that
l d #w#P P # #w#P P * * 2a dt
c F' on [0,k ]. 2
)c (a,l,),r) # f #P P #c (a,l,),r)# u# * * ##w#P\ #c #w#P> #c #w#P> . *P *P *P
(43)
Let us check that if y(0))k /2 and f satisfies c k c # f #P #c # f # ( , * d*P * d*P 2 (47)
and then, for all t3(0,¹*],
l d #w#P # #w#P * * 2a dt
y(t)(k .
2a ) # f #P #c (a,l,),r)# u# * * l #c
#w#P #c #w#P . * *
For this, we consider t**0 such that (44)
Now multiplying Eq. (42) by c (a,l,),r)" (4/l)(c #c ) and summing to the inequalities (43) and (44), we obtain, if we denote y(t)"#w#P P ##w#P * * #c (#u# #a# u# ) * * since there exists c (a,)) such that c (#u# #a# u# ))(c #c )# u# * * * ((#u# #a# u# ) and # u# are two * * * equivalent norms on H())) y (t)#F(y)y4c # f #P P #c (a,l,),r)# f #P , * * (45) where F(y)"c (a,l,),r)!(c #c #c )y !(c #c #c )y !yP\
(48)
(46)
and c (a,l,),r)"inf (l/2a, c ). One has F(0)" c '0, therefore, since F is continuous, strictly decreasing on 1>, there exists k (a,l,b,c,),r)'0
y(t*)"k and y(t)(k for all t3[0,t*). Therefore F(y(t))y(t)*(c /2)y(t) for all t3[0,t*]. Then Eq. (45) gives on [0,t*] c y (t)# y(t))c # f #P P #c # f #P * * 2
(49)
and therefore y (t*)(0 which is not possible. So y(t)(k for all t3[0,t*]. Now we prove the global existence of a solution of Eq. (2) by induction. Choosing f satisfying Eq. (47), y(0) satisfying y(0))k /2 and M"c c (# f # #k P) * d*P in Eq. (35) there exists ¹*(1 (independent on y(0) provided that y(0))k ) such that Eqs. (10)—(12) has a solution (u,n,w) on ]0,¹*[. Moreover, y(t)" #w#P P ##w#P #c (#u# #a# u# ) * * * * satisfies Eq. (48) on [0,¹*] and therefore #w(¹*)# P (k P. Suppose now that we have * a solution on ]0,n ¹*[ and that #w(n ¹*)# P ( * k P. Then using the local existence results with w(n ¹*) as initial value, we obtain a solution (u,n,w) on ]n ¹*,(n#1) ¹*[. By uniqueness argument, we obtain a solution on ]0,(n#1) ¹*[ and therefore, from Eq. (48), y((n#1)¹*)(k (which implies that w((n#1) ¹*)(k ; Therefore (u,n,w) is a solu tion of Eqs. (10)—(12) on (0,R).
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
This solution satisfies w3C ([0,R);¸P())), @ u3C ([0,R);¼ P())) and n3C ([0,R);¼ P())). @ @ Then Eq. (13) gives * w3¸(0,R;¼\ P())). Since R * u!a** u# * n"* w, we deduce from PropR R R R osition 8 that * u3¸(0,R;¼ P())), * n3¸(0,R;¸P())). R R Lemma 3 gives us the global existence of (u, P) solution of the system (2) with the desired regularity. Proof of part (ii). Let us suppose now that f3¸P(0,R;¸P()))5¸(0,R;¸P())),
497
and * n3¸P(0,R;¸P()))5¸(0,R;¸P())). R Lemma 3 gives us the global existence of the (u,P) solution of the system (2) with the desired regularity. 䊐
Acknowledgements The authors would like to thank Professors Che´rif Amrouche and Jacques Simon for useful suggestions.
and satisfies #c # f # (k /2. c # f #P P * d*P * d*P Choosing M"c c (# f # P #k P), * d*P we can prove, as in the previous part, the existence of (u,n,w) on [0,R) such that w3C ([0,R);¸P())), @ u3C ([0,R);¼ P())) and n3C ([0,R);¼ P())). @ @ Since for all t3(0,R) the inequality (49) is satisfied,
R c y(t)# y(t) 2 )c # f #P P #c # f # #y(0). * Rd*P * Rd*P Since f3¸P(0,R;¸P()))5¸(0,R;¸P())), we will have y(t)(R. This means that w3¸P(0,R;¸P()))5¸(0,R; ¸P())). We deduce from Proposition 8 that u3¸P(0,R;¼ P()))5¸(0,R;¼ P())) and n3¸P(0,R;¼ P()))5¸(0,R;¼ P())). The Eq. (13) gives * w3¸P(0,R;¼\ P()))5¸(0,R;¼\ P())), R and therefore by Proposition 8 * u3¸P(0,R;¼ P()))5¸(0,R;¼ P())) R
References [1] M. Renardy, W.J. Hrusa, J.A. Nohel, Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Pitman, London, 1987. [2] J.E. Dunn, K.R. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, Int. J. Engng. Sci. 33(5) (1995) 689—729. [3] C. Amrouche, Etude globale des fluides non Newtoniens de troisie`me grade, Thesis, Pierre et Marie Curie University, 1986. [4] V. Coscia, A. Sequeira, J. Videman, Existence and uniqueness of classical solutions for a class of complexity 2 fluids, Int. J. Non-Linear Mech. 30 (1995) 531. [5] R.S. Rivlin, J.L. Ericksen, Stress—deformation relations for isotropic materials. J. Rat. Mech. Anal. 4 (1955) 323—425. [6] C. Amrouche, D. Cioranescu, On a class of fluids of grade 3. Int. J. Non-Linear Mech. 32 (1997) 73—88. [7] R.L. Fosdick, K.R. Rajagopal, Thermodynamics and stability of fluids of third-grade, Proc. Roy. Soc. London 339 (1980) 145—152. [8] D. Cioranescu, E.H. Ouazar, Existence and uniqueness for fluids of second grade, Research Notes in Mathematics, vol. 109, Pitman, Boston, 1984, pp. 178—197. [9] D. Cioranescu, V. Girault, Weak and classical solutions of a family of second grade fluids, Int. J. Non-Linear Mech. 32 (1997) 317—335. [10] G.P. Galdi, A. Sequeira, Further existence results for classical solutions of the equations of a second grade fluid, Arch. Rat. Mech. Anal. 128 (1994) 297—312. [11] A. Sequeira, J. Videman, Global existence of classical solutions for the equations of the third-grade fluids, J. Math. Phys. Sci. 29 (1995) 47—69. [12] D. Bresch, J. Lemoine, Sur l’existence et l’unicite´ de solution des fluides de grade 2 ou 3. C. R. Acad. Sci. Paris, se´rie I, 324 (1997) 605—610.
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[13] G.P. Galdi, Mathematical theory of second—grade fluids, Proc. CISM Course on ‘‘Stability and Wave Propagation in Fluids and Solids’’, Springer, Berlin, 1995. [14] G.P. Galdi, M. Grobbelaar-Van Dalsen, N. Sauer, Existence and uniqueness of classical solutions of the equations of motion for second-grade fluids, Arch. Rat. Mech. Anal. 124 (1993) 221—237. [15] A.S. Gupta, K.R. Rajagopal, An exact solution for the flow of a non Newtonian fluid past an infinite porous plate, Meccanica 19 (1984) 158—160. [16] P.N. Kaloni, K.R. Rajagopal, Some remarks on boundary conditions for flows of fluids of the differential type, in: Continuum Mechanics and its Applications, Hemisphere Press, New York, 1989. [17] G.P. Galdi, M. Grobbelaar-Van Dalsen, N. Sauer, Existence and uniqueness of solutions of the equations of motion for a fluid of second grade with non homogeneous boundary conditions, Int. J. Non-Linear Mech. 30(5) (1995) 701—709. [18] K.R. Rajagopal, A.Z Szeri, W.C. Troy, An existence theorem for the flow of a non newtonian fluid past an infinite porous plate, Int. J. Non-Linear Mech. 21 (1986) 279—289. [19] L. Garg, K.R. Rajagopal, Flow of a non newtonian fluid pas a wedge, Acta Mechanica 88 (1991) 113—123.
[20] J. Simon, Compact Sets in the Space ¸N(0,¹;B), Ann. Math. Pura Appl. IV CXLVI (1987) 65—96. [21] O.A. Ladyzenskaya, V.A. Solonnikov, Unique solvability of an initial—and boundary—value problem for viscous incompressible nonhomogeneous fluids, J. Sov. Math. 9 (1978) 697—749. [22] V.A. Solonnikov, Solvability of the initial-boundaryvalue problem for the equations of motion of a viscous compressible fluid, J. Sov. Math. 14(2) (1980) 1120—1133. [23] P.-L. Lions, Mathematical Topics in Fluids Mechanics, vol. I, Incompressible Models, Clarendon Press, Oxford, 1996. [24] G.P. Galdi, An introduction to the mathematical theory of the Navier—Stokes equations, vol. 1, Springer, Berlin, 1994. [25] R.J. Di Perna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511—547. [26] C. Amrouche, V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44(119) (1994) 109—140. [27] R.E. Edwards, Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, 1965. [28] H. Brezis, Analyse fonctionnelle et applications, Masson, Paris, 1987.
On the existence of solutions for non-stationary third-grade fluids Didier Bresch, Je´roˆme Lemoine* Laboratoire de Mathe& matiques Applique& es, CNRS UMR 6620, Blaise Pascal University (Clermont-Ferrand 2), 63177 Aubie% re cedex, France Received 7 June 1997; received in revised form 20 March 1998
Abstract This paper is devoted to non-stationary third-grade fluids. We present some existence and uniqueness results in ¸P where r'3 for open sets only of class C. The proof is based on a fixed point method. 1998 Elsevier Science Ltd. All rights reserved. Re´ sume´ Dans ce papier, on s’inte´resse aux fluides de grade 3. On pre´sente des re´sultats d’existence et d’unicite´ dans ¸P pour r'3 et pour un domaine seulement de classe C. La de´monstration est base´e sur une me´thode de point fixe. 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction Some fluids used in industry cannot be modelized by the ‘‘classical” Navier—Stokes equations which follow from the linear constitutive law p"!pI#l( u#R u), where l'0 is the viscosity. For example, liquid crystals, some oils and others (see, e.g., [1—4] and references therein) are modelized by constitutive laws taking into account non-linear characteristics which are shown by experiments. In the case of differential type fluids, there exists a number of
*Corresponding author.
constitutive laws. One of the most used actually has been introduced by Rivlin and Ericksen [5]. The fluids described by this law are called fluids of grade n. Here, we are interested more precisely in incompressible third-grade fluids, i.e. fluids obeying a non-linear constitutive law in which the stress is expressed in terms of the pressure and of some tensors of Rivlin—Ericksen type: p"!pI#lA #aA #bA#c(trA)A , where A " u#R u, d A " A #A u#R uA . dt
0020-7462/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 8 ) 0 0 0 3 4 - 1
(1)
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For physical reasons, [6, Proposition 1.1] or [7], one assumes l'0,
c*0, a*0.
The stress is related to the velocity by the following equations: * u#(u ) )u! ) p"f, R
) u"0, u(0)"u , u" "0. f As usual, we define the potential P"P(u, p) (which has no physical meaning) by P"au ) *u# (2a#b)"A "! "u"!p. Thus, the previous equations can be rewritten in the following form: * (u!a*u)!l*u#( ;(u!a*u));u# P R "f#(a#b)N (u)#cN (u), (2a)
) u"0, u(0)"u , u" "0, (2b) f with N (u)"A *u#2 ) ( u (R u)), (3a) N (u)" ) ((tr A)A ). (3b) For c"0 and a"b"0, we find the classical Navier—Stokes equations. For c"0 and a"!b, we find the equations of second-grade fluids. A number of authors have studied the existence problem in this case for a simply-connected domain ) with a connected boundary and in the case r"2. Cioranescu and Ouazar [8], for instance, have given the local existence for a solution in a regular enough domain, and more recently Cioranescu and Girault [9], Galdi and Sequeira [10] have given the global existence in time (on [0,#R)) of a solution under suitable assumptions on the data and ) of class C , simply-connected. Let us go back to the third-grade case. With the supplementary hypothesis b*0, a'0 and "b#a")(24lc, Amrouche [3], Amrouche and Cioranescu [6] have given the local existence of a solution for ) of class C and simply-connected with a connected boundary; Sequeira and Videman [11] have given the global existence in time (on
[0,R)) of a solution for ) regular enough, simplyconnected, and for small enough data. Here, at first, we prove (Theorem 1) the local existence and the uniqueness of a solution of Eq. (2) for a'0 in any domain of class C. Next, we prove (Theorem 2) the global existence (on [0,R)) if in addition the data are ‘‘small enough” and f3¸(0,R;¸P())) or f3¸P(0,R; ¸P()))5¸(0,R;¸P())). We do not need *) to be connected and C , as in the previous works, since we solve directly the system Eq. (2) instead of using the image of this system by the curl operator. The results have been given in [12] with the supplementary assumptions a, l/a great enough to ensure the global existence. Here we have modified the given uncoupled system to get rid of these conditions. We use a more simple decomposition than in [13] or in [14]. We do not need to assume ) simply-connected to ensure the global existence as it was also supposed by Galdi [13]. Even though we consider homogeneous boundary conditions in this paper, let us note that the study of the inhomogeneous case u"u on * *);(0,¹) is of fundamental importance when we want to modelize the dynamics of the normal stress difference effect. But this boundary condition will not be sufficient to ensure well posedness of the problem. In [15,16], for instance, we can find elementary solutions in bounded and unbounded domains which do not possess the uniqueness property if the condition u ) n"0 is violated with * n the unit normal vector of the boundary. This last condition is not reasonable in a lot of interesting physical problems. Thus, Galdi et al. [17] have given an existence result with a suitable initial condition with the help of a fixed point on a multivalued map without imposing the condition u ) n"0. In [18,19], it has been proved in un* bounded domains that we can augment boundary conditions or use boundnedness requirements to obtain a well-posedness problem. The paper is organized as follows. In Section 2, the main results and the sketch of the proof are given. In Section 3, some functional spaces definitions and a result on a compact imbedding which is essential in our proof when we apply the Schauder Theorem are given. In Section 4, the local existence
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
Theorem is proved after the study of a transport equation and of a system which is practically a Stokes system. In Section 5, the uniqueness of the solution is proved. With all the previous results, the global existence of solutions is proved in Section 6.
Let D()) be the space of C functions with compact support in ) and D ()) the space of distributions on ). We denote by 1 , 2 the duality product between D()) and D ()). We denote by C ([0,R)) the space of bounded @ continuous functions on [0,R). For 1)s)#R, the Sobolev spaces are defined by ¼ Q())"+v3¸Q()): v3¸Q()),, ¼ Q())"closure of D()) in ¼ Q()),
# * v : v 3¸Q()), j"0,2,3 , G G H G all these spaces being endowed with their usual norms. We denote H())"¼ ()), H())"¼ ()), H\())"¼\ ()) and V"+v3D()) : ) v"0,, »"+v3H()) : ) v"0,. Let us recall that » coincides with the closure of V in H()). We give now the following result for which a proof can be found in [20, Corollary 4, p. 85]. Lemma 4. ¸et X, B and ½ be three Banach spaces such that XLBL½ where the imbedding X | B is compact and let s'1. ¹hen ¸(0,¹;X)5¼ Q(0,¹;½)LC([0,¹];B) with the corresponding compact imbedding.
In the sequel, c (or c ) denotes various real positG ive numbers.
3. Main results Let a'0 and let ) be a connected domain in 1 of class C. We denote Q ");(0,¹). 2 Throughout this paper, we assume that
2. Functional spaces and some preliminaries
¼\ Q())" v3D ()) : v"v
487
u 3¼ P())5», f3¸P(Q ), r'3. (4) 2 Without hypothesis on the size of the data, we have the following result of local existence. Theorem 1. ¹here exists ¹*, 0(¹*)¹, such that the system (2) has a unique solution (u, P) satisfying u3C([0,¹*];¼ P())), * u3¸P(0,¹*;¼ P())), R P3¸P(0,¹*;¸P())). Now we state the supplementary hypothesis which will ensure the existence of a global solution: u is small enough that is #u !a*u # P )c (a,l,b,c,r,)), (5) * and f is small enough in some suitable spaces. More precisely, we suppose, in a first case f3¸(0,R;¸P()))
(6)
with #f# (c (a,l,b,c,r,)). * d*P In a second case, we suppose
(7)
f3¸P(0,R;¸P()))5¸(0,R;¸P()))
(8)
with # f #P P ## f # (c (a,l,b,c,r,)). * d*P * d*P (9) For such small data, one has the following result of global existence. Theorem 2. ¼e assume Eqs. (4) and (5): (i) ¸et us suppose Eqs. (6) and (7) be satisfied. System (2) has a unique solution (u, P) satisfying u3C ([0,R);¼ P())), * u3¸(0,R;¼ P())), @ R P3¸(0,R;¸P())).
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(ii) ¸et us suppose Eqs. (8) and (9) be satisfied. ¹herefore, system (2) has a unique solution (u, P) satisfying
(z,n) being defined by
u3C ([0,R);¼ P()))5¸(0,R;¼ P())), @
z" "0, f
* u3¸P(0,R;¼ P()))5¸(0,R;¼ P())), R P3¸P(0,R;¸P()))5¸(0,R;¸P())). Throughout the paper, for vector-valued functions u"(u ,u ,u ), v"(v ,v ,v ), we denote by u ) v and v ) u the vector-valued functions, respectively, defined by (u ) v) " u * v , ( v ) u) " * v u H G G G H H G H G G (note that u ) vO v ) u). Let u ) v" u v , u ) v" * u * v . G G G GH G H G H Proof (Sketch of the existence results). As it will be proved in Lemma 3, it suffices to find the (u,n,w) solution of the following system: l l * w# w#u ) w# u ) w! u R a a (10)
w"u!a*u# n,
(11)
n"0.
(12)
We will solve Eqs. (10)—(12) by linearization and use of the Schauder’s fixed-point Theorem. Given u satisfying Eq. (12), we define w by l * w# w#u ) w# u ) w R a l " u#f#(a#b)N (u)#cN (u), a w(0)"u !a * u .
) z"0,
n"0,
(13a) (13b)
We are still to prove that u may be chosen such that there exists n satisfying Eq. (11). This will be done by a fixed-point argument. More precisely,
(14a) (14b)
we will prove that the map u | z has a fixed point. For this we will prove (Proposition 5) that the system Eq. (13) has a unique solution w which satisfies #w#)G(t,#u#,c (# f ###u !a*u #))
(15)
for some convenient norms in functional spaces on );(0,¹), where G is an increasing continuous function of t such that G(0,x,y)"y. Next we will prove (Proposition 8) that the system Eq. (14) has a unique solution (z,n) such that #z#)c #w#.
(16)
Finally, we will prove that, for a convenient M"M(u , f,a,l,b,c,),r), we can choose t'0 such that #u#)M implies c G(t,#u#,c (# f ###u !a*u #)))M.
"f#(a#b)N (u)#cN (u),
) u"0, u(0)"u , u" "0, f
z!a*z# n"w,
(17)
By Schauder’s fixed-point Theorem, together with some continuity properties, this will prove the existence of a fixed point which moreover satisfies u(0)"u . Therefore, we obtain the existence of a solution of Eqs. (10)—(12), that is the local existence. Using the additional hypothesis (5)—(7) (or (5), (8), (9)) on the data we will prove the existence of M for which condition Eq. (17) is satisfied for all positive t. This will prove the global existence and therefore ends the sketch of the proof. 䊐 Remark. It is important to note that n introduced in Eqs. (10)—(12) can be different from the potential P which is defined in Eq. (2). Throughout the paper, n is normalized by n"0. This ensures the uniqueness of n as soon as
n is unique. Let us now verify that any solution of Eqs. (10)—(12) provides a solution of Eq. (2).
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
Lemma 3. ¸et (u,n,w) be a solution of Eqs. (10)—(12), and l P"w ) u#* n# n. R a
489
4. Existence and uniqueness of a local solution In this section, we will prove Theorem 2.
(18)
¹hen (u,P) is solution of Eq. (2). Proof. Let (u,n,w) be a solution of Eqs. (10)—(12). Replacing w by u!a * u# n in Eq. (10) gives l * (u!a * u# n)# (u!a * u# n) R a #u ) (u!a * u# n) l # u ) (u!a * u# n)! u a "f#(a#b)N (u)#cN (u). This equation can be rewritten in the following form: * (u!a * u)!l*u#u ) (u!a * u# n) R ! ((u!a * u# n) ) u) # ((u!a * u# n) ) u) l # u ) (u!a * u# n)#* n# n R a "f#(a#b)N (u)#cN (u). Since u ) w! (w ) u)# u ) w"( ;w);u and ; n"0, we obtain * (u!a * u)!l* u#( ;(u!a*u));u R l # ((u!a * u# n) ) u)#* n# n R a "f#(a#b)N (u)#cN (u). With the definition (18) of P, this gives the first equation of Eq. (2). The other equations of Eq. (2) are given by Eq. (12). 䊐
4.1. Study of the transport Eqs. (13a,b) The goal of this part is to prove the following result. Proposition 5. ¸et u satisfy (12) and u3¸(0,¹,¼ P())).
(19)
¹hen the system (13) has a unique solution such that w3C([0,¹];¸P())), * w3¸P(0,¹;¼\ P())). R Denoting """u""""#u# , w satisfies, for * Rd5 P all t)¹, #* w# P ##w# R * Rd5\ P * Rd*P
R
# u# * ;(t P"""u"""## f # P ##u !a*u # P ). * / R * (20)
)c (1#t P (1#"""u""")) exp
The proof is given in several steps. In a first step we will see (Lemma 6) that for regular data (g,v ) there exists a solution v of the transport equation * v#u ) v"g, v(0)"v . (21) R In a second step, we will obtain, by approximation and density (Lemma 7), a solution of l * w# w#u ) w# u ) w"g, R a
w(0)"w .
(22)
For convenient g and w this will solve Eq. (13). Therefore, let us begin with the following Lemma. Lemma 6. ¸et u satisfy (12) and (19), v 3¼ P()) and g3¸P(0,¹;¼ P())). ¹hen the system (21) has a unique solution such that v3C([0,¹];¼ P())), * v3¸P(Q ). R 2 It satisfies, for all t, 0(t)¹, #v# * Rd5 P )c(¹,u)(#v # P ##g# ). 5 * Rd5 P
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D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
Proof. Since u" "0 and ) is C, there exists f a unique solution y3C([0,¹];)) of
R
The unique solution of Eq. (21) in C([0,¹]; ¼ P())) is given by v(t,x)"v (y(0,t,x))# R g(y(m,t,x),m) dm (see [21]). It satisfies * v3¸P(Q ). Moreover, one has for all t, q, and Rx such 2that 0(q)t)¹ and x3),
y(q,t,x)"x! u(y(m,t,x),m) dm. O
" y(q,t,x)")(3 exp
It satisfies, for all t)¹, a P\ r l
(23)
Therefore )sup#v (y(0,q, ) ))# P #v# * * Rd*P OXR O #sup g(y(m,q, ) ),m) dm . P * OXR Because y(m,t, ) ) is a one-to-one map on ) with a Jacobian determinant equal to 1 (cf. [22]), this implies
#v# )#v # P ##g# . * Rd*P * * Rd*P Moreover one has, for all i, * v(t,x)"* y(0,t,x) ) v (y(0,t,x)) G G
R # * y(m,t,x) ) g(y(m,t,x),m) dm. G
"g"P# "w "P /R
R ;exp r # u# ,
(24)
where r is given by 1/r#1/r "1. ¹he system (22) has no other solution such that w3¸(0,¹;¸P())),
R ""u"" . 5 O
"w"P (t))
* w3¸P(0,¹;¼\ P())). R
Proof. (1) Existence in the case of regular data: At first, we assume w 3¼ P()), g3¸P(0,¹;¼ P())).
(25)
Starting from w"0, we define by iteration wL3C([0,¹];¼ P())), * wL3¸P(Q ), R 2 to be the solution given by Lemma 6 of the following system: l * wL#u ) wL"g! u ) wL\! wL\, R a wL(0)"w . Let ¼L"wL!wL\. It is the unique solution of l * ¼L#u ) ¼L"! u ) ¼L\! ¼L\, R a
Then by the same technique as above, we get
¼L(0)"0,
#* v# )c(¹,u)(# v # P G * Rd*P *
and due to Lemma 6, it satisfies for all t, 0(t)¹,
## g# ), * Rd*P which implies the required inequality. 䊐 Now, we will prove the following Lemma. Lemma 7. ¸et u satisfy (12) and (19), w 3¸P()) and g3¸P(Q ). ¹hen the system (22) has a unique solu2 tion such that w3C([0,¹];¸P())), * w3¸P(0,¹;¼\ P())). R
#¼L"" * Rd5 P
l )c(¹,u) u ) ¼L\# ¼L\ a
. * Rd5 P
In addition, since u3¸(0,¹,¼ P())) and since ¼ P()) is a multiplicative algebra, one has, for all t, 0(t)¹, # u ) ¼L\#
* Rd5 P
)c(),r)# u# #¼L\# . * Rd5 P * Rd5 P
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
(2) Existence in the general case: Now we consider w 3¸P()), g3¸P(Q ). Let wL 3¼ P()) and 2 g 3¸P(0,¹;¼ P())) be such that L
Then, we have #¼L# * Rd5 P )c(¹,u,),r,l,a)#¼L\# . * Rd5 P
wL Pw in ¸P()),
Denoting by W (t) the left-hand side, we have L
w 3C([0,¹];¼ P())), * w 3¸P(Q ). L R L 2
which gives, since W (t) increases with t, tL\ W (t))cL\ W (t). L (n!1)! Then the series W (t) converge, and in conseL L quence wL converges to w in the space C([0,¹]; ¼ P())). In addition, w is a solution of Eq. (22) and * w3¸P(Q ). R 2 Moreover, by multiplying Eq. (22) by "w"P\w and integrating, we get
lr "w"P# "w"P)r "g" "w"P\ a #r
" u ) w" "w"P\.
(26)
r
"g" "w"P\)
a P\ r l
lr "g"P# "w"P, a
" u ) w" "w"P\)r# u# *
"w"P,
(27)
(28)
a P\ "w"P) "g"P#r# u# "w"P. * r l
Denoting t(s)"exp(!rQ # u# ) "w"P and * multiplying by exp(!r Q # u# ), the previous * inequality, we get since exp(!r Q # u# ))1, *
a P\ t (s)) r l
"(e(x)")1, ∀x3),
(e(x)"1 if d(x)*2c(e), " (e(x)")k e/d(x), ∀x3),
one has d dt
The inequality (24) applied to w !w implies that L K (w ) - is a Cauchy sequence in C([0,¹];¸P())). Its L LZ limit w3C([0,¹];¸P())) satisfies Eqs. (22) and (24). By Eq. (22), * w3¸P(0,¹;¼\ P())). R (3) ºniqueness: It follows after extension on 1L from a method due to R.J. Di Perna and P.-L. Lions [25]. This is a regularization method in which we use Hardy’s inequality. Let us give the proof here for the reader’s convenience. Let w be the difference of two eventual solutions of Eq. (22) such that w3¸(0,¹;¸P())) and * w3¸P(0,¹;¼\ P())). It satisfies Eq. (22) with R w "0 and g"0. Let us now recall that there exists a function (e3C()) such that
(e(x)"0 if d(x)(c(e)/2k ,
Since r
g Pg in ¸P(Q ). L 2
Then there exists a related solution w of Eq. (22) L such that
R W (t))c W (q) dq, L L\
d dt
491
"g"P.
Thus by integration from 0 to t, we obtain Eq. (24).
with c(e)"exp(!1/e), d(x)"dist (x,*)) and k , k two constants independent of ) and e (see, for instance, [24, pp. 166—168]). Denoting ve"w(e and ) the extension outside ) by 0, we verify that vJ e satisfies the following equation
l * vJ e#uJ ) vJ e# vJ e# uJ ) vJ e"! uJ * (e wJ , R G G a G vJ e(0)"0. One has vJ e3¸(0,¹;¸P(1)) and since u" "0, f uJ 3¸(0,¹;¼ (1)). By convolution of this equation by mollifiers o , and denoting vJ e "vJ e 夹 o , L L L
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D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
we get
Therefore, by definition of (e, one has
l * vJ e #uJ ) vJ e # vJ e # uJ ) vJ e L a L L R L
"r ! uJ * (ewJ L G G G
夹o
"R L L
(29)
r "!(uJ ) vJ e) 夹 o #uJ ) vJ e L L L
!(uJ ) vJ e) 夹 o #uJ ) vJ e P0 in ¸(0,¹;¸(1)). L L In addition ( uJ ) vJ e) 夹 o P uJ ) vJ e and uJ ) vJ e P L L
uJ ) vJ e in ¸(0,¹;¸(1)), then !( uJ ) vJ e) 夹 o # uJ ) vJ e P0 in ¸(0,¹;¸(1)). L L Multiplying Eq. (29) by vJ e and integrating on 1, L we obtain, since uJ has a compact support and satisfies ) uJ "0:
"vJ e " L 1
)
"R vJ e "#c # uJ # 1 * L L 1
"vJ e ". L 1
It follows that
# uJ # 1 *
u #w#P , * /2 d PY * /2
Moreover one has, see [25, Lemma 2.1, p. 516],
R
u d
!( uJ ) vJ e) 夹 o # uJ ) vJ e . L L
l "vJ e "# L a 1
)ce exp c
with r "2r/(r!2) and c independent of e. Now using Hardy’s inequality
where vJ e (0)"0 and L
1d 2 dt
"vJ e"(t)
1
)c#u# PY , 5 *PY
and letting e go to zero in the previous inequality, one obtains wJ "0, which proves the uniqueness. The proof of Lemma 7 is complete. 䊐 Proof of the Proposition 5. Let g denote the righthand side and w the initial data in Eq. (13), i.e.: l g" u#f#(a#b)N (u)#cN (u), a w "u !a*u .
(30)
Then g3¸P(Q ) and w 3¸P()). By Lemma 7, 2 there exists a unique solution w of Eq. (22) and therefore of Eq. (13). It remains to prove the inequality (20). Inequality (24) yields, denoting """u"""" #u# , * Rd5 P #w# * Rd*P
"vJ e "(t) L
)c # f #P P #t"""u"""P#t"""u"""P#t"""u"""P * /R
1
R
)c exp c
# uJ # 1 *
"R "(s) ds. L
1
R"
Since vJ e PvJ e in ¸P(0,¹;¸P(1)) and R P L L ! uJ * (ewJ in ¸(0,¹;¸(1)), by passing to G G G the limit in this inequality, we obtain
R"1
"u !a*u "P
P
exp
R
# u# . (31) *
#* w# P R * Rd5\ P )
uJ * (ewJ (s) ds. G G G
On the other hand, Eq. (22) gives
R "vJ e"(t))c exp c # uJ # 1 * 1 ;
#
l w a
## u ) w# P * Rd5\ P *P Rd5\ P
##u ) w# P ##g# P . * Rd5\ P * Rd5\ P (32)
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
)t P# ) # , we obtain Using # ) # P * Rd# * Rd# #* w# P R * Rd5\ P )c(#w# t P(1#"""u""")##g# P ). (33) * /R * Rd*P By Eqs. (30), (31) and (33), one has finally Eq. (20). 䊐 4.2. Study of the Stokes like equations Eq. (14) Let us now give some estimates of the solution of Eq. (14). Proposition 8. (i) ¸et 1(s)R and w3¸Q(0,¹; ¸P())), * w3¸Q(0,¹;¼\ P())). ¹hen the system R Eq. (14) has a unique solution (z,n) such that z3¸Q(0,¹;¼ P())), * z3¸Q(0,¹;¼ P())), R n3¸Q(0,¹;¼ P())), * n3¸Q(0,¹;¸P())). R (ii) If moreover w3C([0,¹];¸P())), then
and for all t, 0(t)¹, #z# ## n# * Rd5 P * Rd*P )c #w# , * Rd*P ##* n# Q #* z# Q R * Rd5\ P R * Rd5 P )c #* w# Q . R * Rd5\ P
(34a)
(34b)
Proof. This is a consequence of the following lemma which is related to similar results for Stokes equations, given for example in [26, Theorems 4.6, p. 128, p. 4.18, p. 136] or in [24, Theorem 6.1, p. 225]. Lemma 9. (i) ¸et w3¼\ P()). ¹hen the system (14) has a unique solution such that z3¼ P()), n3¸P()). It satisfies #z# P ## n# \ P )c #w# \ P . 5 5 5 (ii) ¸et w3¸P()). ¹hen z3¼ P()), n3¼ P()), and 1 #z# P # # n# P )c #w# P . 5 * * a
From Proposition 8, we deduce a possible value for the constant c defined in inequality (16) in Section 2. 4.3. Local existence of a solution of Eq. (2) This proof is based on the Schauder’s fixed-point Theorem that we can find, for example, in [27, Corollary 3.6.2, p. 163]. We define a non empty bounded and convex set by K *"+u3¸(0,¹*;¼ P()))5C([0,¹*];¼ P())): 2 u satisfies Eqs. (12) and (35), where 0(¹*)¹ will be chosen later and #u# * P )2M, * 2 d5 where
z3C([0,¹];¼ P())), n3C([0,¹];¼ P())),
493
(35)
M"c c (# f # P ##u !a*u # P ). * /2 * The constants c , c are given by Eqs. (20) and (34). It follows from Eqs. (20) and (34) that there exists ¹*3(0,¹] such that u | z maps K * into itself. 2 We provide K * with the topology of 2 X *"C([0,¹*];¼ P())). 2 Let us first prove that K * is closed in X *. 2 2 Let u 3K * with u Pu in X *. Let us prove that L 2 L 2 u3K *. Since u 3K *, u is bounded in ¸(0,¹*, 2 L 2 L ¼ P())) then, it converges to z in ¸(0,¹*, ¼ P())) 夹 weak and #z# * P )lim inf#u # * P . * 2 5 L * 2 5 By uniqueness of the limit in D (Q *), we obtain 2 that u3¸(0,¹*,¼ P())). Now, let us remark that u | z maps K * into 2 a relative compact set K * of X *. From the esti2 2 mates Eqs. (20) and (34), we deduce that K * 2 is bounded in ¸(0,¹*;¼ P()))5¼ P(0,¹*; ¼ P())). So using Lemma 4, we deduce that K * is relatively compact in X *. 2 2 Let us now prove that the map uPz is continuous provided with the topology of X * from 2 K * into itself. 2 At first let us prove that u | w is continuous from K * into C([0,¹*];¼\ P())), where K * is 2 2 endowed with the topology of X *. 2
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Let u and u in K *, u Pu in X *. Let w and w , L 2 L 2 L the solutions of Eq. (13), be associated with u and u . Let us denote ¼L"w !w. Then ¼L satisfies L L
fixed point, say u. This fixed point gives a solution (u,n,w) of Eqs. (10)—(12) on [0,¹*]. At most, from Propositions 5 and 8, one has
l * ¼L# ¼L#u ) ¼L#ºL ) w # u ) ¼L L R a
u3C([0,¹*];¼ P())),
# ºL ) w
n3¼ P(0,¹*;¸P()))5C([0,¹*];¼ P())).
L
Let us now define P3¸P(0,¹*;¸P())) by Eq. (18). By Lemma 4, (u,P) is the solution of Eq. (2). This gives the local existence result of Theorem 1.
l " ºL#(a#b)(N (u )!N (u)) L a #c(N (u )!N (u)), L
4.4. Uniqueness of a solution of Eq. (2)
¼L(0)"0, where ºL"u !u. L We have proved (cf. Eqs. (31) and (33)) that ¼L is bounded in ¸(0,¹*;¸P())) and that * ¼L is R bounded in ¸P(0,¹*;¼\ P())). Then there exists ¼K which converges to ¼ in ¸(0,¹*;¼\ P())) strongly where ¼3¸(0,¹*;¸P())), and * ¼3 R ¸P(0,¹*;¼\ P())). The functions N , N defined by Eq. (3) are con tinuous from K * endowed with the topology of 2 C([0,¹*];¼ P())) into C([0,¹*];¼\ ())), thus since ºLP0 in X *, ¼ satisfies 2
Considering the difference of two solutions of Eq. (2) denoted by (uG,PG) for i"1,2, we obtain the following inequality on º"u!u in D (0,¹*):
a * "º"# * " º"#l " º" R 2 R
1 2
"!a
#a
(u ) )º ) *º! (º ) )u ) º (º ) )u ) *º# (a#b)(N (u,º)
l * ¼# ¼#u ) ¼# u ) ¼"0, R a
(36a)
#N (º,u)) ) º#
¼(0)"0.
(36b)
#N (u,º,u)#N (u,u,º)) ) º,
Since ¼3¸(0,¹*;¸P())) and * ¼3¸P(0,¹*; R ¼\ P())), by uniqueness of a solution ¼ of Eq. (36) in this space (cf. Proposition 5), we obtain ¼"0. Therefore, the whole sequence ¼L goes to 0. This means that
(38)
where N (u,v)"( u#R u) ) *v#2 ) ( u ) (R v)), N (u,v,w) " ) (tr(( u#R u) ) ( v#R v))( w#R w)).
w Pw in C([0,¹*];¼\ P())). L On the other hand, by Proposition 8, the map w | z is continuous from C([0,¹*];¼\ P())) into X *. 2
c(N (º,u,u)
(37)
Therefore, from the continuity of u | w and w | z, we deduce that u | z is continuous from the bounded convex set K * into K *, and by 2 2 Schauder’s fixed point Theorem, it possesses a
Therefore, since uG3C([0,¹*];¼ P())), ) uG"0 and uG" "0, Eq. (38) gives
d dt
("º"#a" º")#l " º")R " º",
where R is a real number. This gives
d dt
R ("º"#a" º")) ("º"#a" º"), a
D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
with º(0)"0. By the Gronwall lemma, we obtain º"0, which gives P"0. 䊐
5. Existence of a global solution
495
Since #N (u)# P )c (a,),r)#w#P , * * #N (u)# P )c (a,),r)#w#P , * * we obtain
In this Section, we prove Theorem 2 which states the existence of a global solution of Eq. (2) for small enough data. Proof of part (i). Let w be a solution of Eqs. (10)—(12) on [0,¹*], ¹*'0 given by Theorem 1. It satisfies the inequality (26) (this inequality was proved only for regular data satisfying Eq. (25); it extends to any data by a density argument). Instead of Eq. (27), let us use the following estimate:
"g" "w"P\)#g# P #w#P\ , *P *
to bound the right-hand side of Eq. (28). Together with Eq. (26), it gives d lr #w#P P # #w#P P * * dt a )r#g# P #w#P\ #r# u# #w#P P . (39) * *P * * Moreover, Lemma 9 gives # u# )Q #u# P )Q c #w# P , * 5 * where Q is the imbedding constant of ¼ P()) in ¼ ()). Therefore, with expression (30) of g, d lr #w#P P # #w#P P * * dt a
#c (a,b,),r) #w#P> #c (a,c,),r) #w#P> . *P *P (40) This estimate does not allow us to obtain the global existence of a solution for all a'0 (see [12]). Therefore we use the supplementary estimates on u and w. First, from the estimate (40), we deduce the following estimate on w: d 2l #w#P # #w#P * * a dt
l )2 # f # P # #u# P #w# P * * a * #c (a,b,),r) #w#P #c (a,c,),r) #w#P . * * (41) Now, mutiplying Eq. (2) by u and integrating, we obtain d l (#u# #a# u# )# # u# * * * dt 2
#c (a,c,),r)#w#P . *
#"c" #N (u)# P ) #w#P\ * *P
l #r # f # P # #u# P #w#P\ * *P a * #r Q c #w#P> . *P
l )r (# f # P # #u# P ) #w#P\ * *P a *
)c (l,),r) # f #P #c (a,b,),r)#w#P * *
)r("a#b" #N (u)# P *
d lr #w#P P # #w#P P * * dt a
(42)
Since, for all e'0 we have, using Young’s inequality and Lemma 9 #u# P )c (e,a,),r) # u# #e #w# P , * * * we obtain after some calculations and for a suitable choice of e, using Eqs. (40) and (41) the following
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D. Bresch, J. Lemoine / International Journal of Non-Linear Mechanics 34 (1999) 485–498
estimates on w:
such that
l d #w#P P # #w#P P * * 2a dt
c F' on [0,k ]. 2
)c (a,l,),r) # f #P P #c (a,l,),r)# u# * * ##w#P\ #c #w#P> #c #w#P> . *P *P *P
(43)
Let us check that if y(0))k /2 and f satisfies c k c # f #P #c # f # ( , * d*P * d*P 2 (47)
and then, for all t3(0,¹*],
l d #w#P # #w#P * * 2a dt
y(t)(k .
2a ) # f #P #c (a,l,),r)# u# * * l #c
#w#P #c #w#P . * *
For this, we consider t**0 such that (44)
Now multiplying Eq. (42) by c (a,l,),r)" (4/l)(c #c ) and summing to the inequalities (43) and (44), we obtain, if we denote y(t)"#w#P P ##w#P * * #c (#u# #a# u# ) * * since there exists c (a,)) such that c (#u# #a# u# ))(c #c )# u# * * * ((#u# #a# u# ) and # u# are two * * * equivalent norms on H())) y (t)#F(y)y4c # f #P P #c (a,l,),r)# f #P , * * (45) where F(y)"c (a,l,),r)!(c #c #c )y !(c #c #c )y !yP\
(48)
(46)
and c (a,l,),r)"inf (l/2a, c ). One has F(0)" c '0, therefore, since F is continuous, strictly decreasing on 1>, there exists k (a,l,b,c,),r)'0
y(t*)"k and y(t)(k for all t3[0,t*). Therefore F(y(t))y(t)*(c /2)y(t) for all t3[0,t*]. Then Eq. (45) gives on [0,t*] c y (t)# y(t))c # f #P P #c # f #P * * 2
(49)
and therefore y (t*)(0 which is not possible. So y(t)(k for all t3[0,t*]. Now we prove the global existence of a solution of Eq. (2) by induction. Choosing f satisfying Eq. (47), y(0) satisfying y(0))k /2 and M"c c (# f # #k P) * d*P in Eq. (35) there exists ¹*(1 (independent on y(0) provided that y(0))k ) such that Eqs. (10)—(12) has a solution (u,n,w) on ]0,¹*[. Moreover, y(t)" #w#P P ##w#P #c (#u# #a# u# ) * * * * satisfies Eq. (48) on [0,¹*] and therefore #w(¹*)# P (k P. Suppose now that we have * a solution on ]0,n ¹*[ and that #w(n ¹*)# P ( * k P. Then using the local existence results with w(n ¹*) as initial value, we obtain a solution (u,n,w) on ]n ¹*,(n#1) ¹*[. By uniqueness argument, we obtain a solution on ]0,(n#1) ¹*[ and therefore, from Eq. (48), y((n#1)¹*)(k (which implies that w((n#1) ¹*)(k ; Therefore (u,n,w) is a solu tion of Eqs. (10)—(12) on (0,R).
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This solution satisfies w3C ([0,R);¸P())), @ u3C ([0,R);¼ P())) and n3C ([0,R);¼ P())). @ @ Then Eq. (13) gives * w3¸(0,R;¼\ P())). Since R * u!a** u# * n"* w, we deduce from PropR R R R osition 8 that * u3¸(0,R;¼ P())), * n3¸(0,R;¸P())). R R Lemma 3 gives us the global existence of (u, P) solution of the system (2) with the desired regularity. Proof of part (ii). Let us suppose now that f3¸P(0,R;¸P()))5¸(0,R;¸P())),
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and * n3¸P(0,R;¸P()))5¸(0,R;¸P())). R Lemma 3 gives us the global existence of the (u,P) solution of the system (2) with the desired regularity. 䊐
Acknowledgements The authors would like to thank Professors Che´rif Amrouche and Jacques Simon for useful suggestions.
and satisfies #c # f # (k /2. c # f #P P * d*P * d*P Choosing M"c c (# f # P #k P), * d*P we can prove, as in the previous part, the existence of (u,n,w) on [0,R) such that w3C ([0,R);¸P())), @ u3C ([0,R);¼ P())) and n3C ([0,R);¼ P())). @ @ Since for all t3(0,R) the inequality (49) is satisfied,
R c y(t)# y(t) 2 )c # f #P P #c # f # #y(0). * Rd*P * Rd*P Since f3¸P(0,R;¸P()))5¸(0,R;¸P())), we will have y(t)(R. This means that w3¸P(0,R;¸P()))5¸(0,R; ¸P())). We deduce from Proposition 8 that u3¸P(0,R;¼ P()))5¸(0,R;¼ P())) and n3¸P(0,R;¼ P()))5¸(0,R;¼ P())). The Eq. (13) gives * w3¸P(0,R;¼\ P()))5¸(0,R;¼\ P())), R and therefore by Proposition 8 * u3¸P(0,R;¼ P()))5¸(0,R;¼ P())) R
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