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On the Regularity for Solutions of the Micropolar Fluid Equations

ELVAORTEGA-TORRES(*) - MARKOROJAS-MEDAR(**)

ABSTRACT- We give sufficient conditions on the kinematics pressure in order to obtain regularity and uniqueness of the weak solutions to the micropolar fluid equations.

1. Introduction.

We consider the regularity of the weak solutions for the equations that describes the motion of a viscous incompressible micropolar fluids in a bounded domain VR3, with smooth boundary@V over a time interval [0;T], 05T5‡ 1. These equations are given by

@u

@t‡u ru n1Du‡ rpˆ2mrrotw‡f;

(1)

divuˆ0; (2)

@w

@t ‡u rw n2Dw n3rdivw‡4mrwˆ2mr rotu‡g;

(3)

wheren1ˆm‡mr; n2ˆca‡cd; n3ˆc0‡cd ca.

The unknowns are the functions u(x;t)2R3;w(x;t)2R3 and p(x;t)2R which denote the linear velocity vector, the angular velocity vector of rotation of particles, and the pressure of the fluid, respectively.

The functionsf(x;t)2R3 andg(x;t)2R3 are given, and denote external

(*) Indirizzo dell'A.: Departamento de MatemaÂticas, Universidad CatoÂlica del Norte, Av. Angamos 0610, Casilla 1280, Antofagasta, Chile.

E-mail: eortega@ucn.cl

(**) Indirizzo dell'A.: Depto. de Ciencias BaÂsicas, Univ. del Bio-Bio, Campus Fernando May, Casilla 447, ChillaÂn, Chile.

E-mail: marko@ueubiobio.cl

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sources of linear and angular momentum, respectively. The positive con- stantsm;mr;c0;caandcd, characterize isotropic properties of the fluid:mis the usual Newtonian viscosity;mr;c0;ca;cd are new viscosities related to the appearance of the field of internal rotationwand satisfyc0‡cd>ca. Together with the equations (1)-(3) we prescribe the following bound- ary and initial conditions

u(x;t)ˆ0; w(x;t)ˆ0 on STˆ@V(0;T);

(4)

u(x;0)ˆu0(x); w(x;0)ˆw0(x) in V:

(5)

For the derivation and physical discussion of the equations (1)-(3) see [7]

and [14]. The existence of diverse notions of solutions for the system (1)-(3) has been much studied, as in [13], [17], [18] and [23]. We observe that the equations (1)-(3) include as a particular case the classical Navier-Stokes equations, which widely has been studied (see, for instance, the classical books [12], [21] and the references there in).

In the theory of the three-dimensional Navier-Stokes equations, the problems of uniqueness of weak solutions is the most difficult ones (see for instance [21], p. 297, [12]). For sufficiently smooth solutions, the unique- ness is proved quite easily, while in the class of weak solutions the problem of uniqueness (or non-uniqueness) remain open to discussion. There exist many works devoted to these questions. A pioneer work in this direction was give by Serrin [19], Serrin proved that a weak solution (u;p) is regular if the velocityusatisfies some suitable extra conditions. After many au- thors extend the result of Serrin, see for instance [11], [22], [8], [1], [17].

The pressure term is source of problems for understanding of the Navier-Stokes equations, since it plays a role seemed to a Lagrange multiplier to enforce the incompressibility constraint. Thus, an other interesting question is to look for conditions involving the pressure and not the velocity, as it has been done in almost all the literature. A first sufficient condition for regularity involving the pressure alone, was given by Kaniel in [10] who proved thatp2L1(0;T;Lq(V)), forq>12=5 is a sufficient condition for the smoothness of weak solutions of the three-dimensional Navier-Stokes equations. This approach was ignored by many years but, recently, many authors have write extensions of the results by Kaniel [10], see for instance [2], [3], [16], [5], [6]. Following the idea of Kaniel it is well known that ifu02V, there exists an unique strong local solution for the Navier-Stokes equations, and if ku0kV is small enough and the external force is zero, the solution is global. Since the weak global solution also exist, then the local strong solution is

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equal to the weak solution on the maximal interval of existence, say [0;T), with 05T5T. Thus, the following question comes to our mind:

what conditions on the pressure implies that the unique local strong solution is in fact global?, in other words, it is possible to change the condition on ku0kV by another that involve only the pressure?. If we have an affirmative answer, we would have the uniqueness of weak solutions. It is important to say that these results is not directly com- parable to the Serrin, since the initial condition is take in different spaces. Moreover, it will be interesting to know if it is possible change the hypotheses of Serrin by another that involve only the pressure, but this is a difficult question. In fact, in an early work Heywwod and Walsh [9] give an interesting result on the behavior of the pressure in tˆ0 when the initial condition u02V. They show that there exist initial conditions u02V such that lim sup

t!0‡ kpkL2(V) ˆ ‡ 1, which implies that lim sup

t!0‡ kpkLq(V)ˆ ‡ 1 for anyq2.

By the other hand, the condition given by Kaniel is: kpkLq(V)C0 for each t in the interval of existence of the weak solution, with q>12

5, in particular, this implies that the lim sup

t!0‡ kpkL2(V) is finite and consequently some degree of regularity on the pressure is in fact necessary. Thus, other interesting question in this direction is find the minimal hypotheses on the pressure. The work of Berselli [5] is in that direction. In fact, the Berselli's results, roughly speaking, are theLq- versions of the Kaniel result (q>3).

The purpose of this paper is to extend the proof presented in [5] to the micropolar equations, and thus, we research the same level of knowledge as the one in the case of the classical Navier-Stokes equations. We note that, as remarked in [5], the proof is done by following techniques in- troduced in [4] and developed in [3].

An outline of this paper is as follows. In Section 2, we introduce some notations and functions spaces, and we state our main theorem. In Section 3, we prove some estimates for the proof of our main result. Finally, in Section 4, we prove our main theorem.

2. Statements, notations and main result.

We denote by Lp(V), 1p ‡1 the usual Lebesgue spaces of scalar functions and for vector value functions we define Lp(V)ˆ

ˆ(Lp(V))3 with norm kvkpˆ k jvjkp where j j denotes the Euclidean

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length. As customary, H10(V) denotes the Sobolev space given by the functions vanishing on the boundary and belonging toL2(V) with their first-order distributional derivatives, and the norm is defined by kvkH1

0 k jrvj k2. W hen X is a Banach space, we denote by C(0;T;X) the set of continuous functions on [0;T] with values in X and by Lp(0;T;X) the Banach space of measurable functionsvwith values inX, endowed with the norm

kvkLp(0;T;X)ˆ ZT

0

kv(t)kpXdt 0

@

1 A

1=p

;

LetC10 (V) denote the set of allC1-real vector functions vwith compact support in V, and C10;s(V) denotes the functions v2C10 (V) such that divvˆ0. Then, we introduce the following function spaces

Haˆclosure of C10;s(V) inLa(V);

Gaˆclosure of C10 (V) inLa(V);

Vˆclosure of C10;s(V) inH10(V):

We denote by the sameCvarious positive constants and byC(;;. . .) the constants depending only on the quantities appearing in parentheses.

Also, we denote the classical bilinear and trilinear forms by (ru;rv)ˆX3

i;jˆ1

Z

V

@uj

@xi

@vj

@xidx; (u rv;w)ˆX3

i;jˆ1

Z

V

uj@vi

@xjwidx for allu;v;wwhere the integrals make sense. Moreover, the trilinear form has the properties

(u rv;jvja 2v)ˆ0; 8u2V;8v2H10(V):

(6)

For simplicity of notations, we consider the following quantities (introduced in reference [4])

Naa(u)ˆ kjuja2 1jruj k22; Maa(u)ˆ k jr juja2j k22 (7)

and, in the calculations will be used the inequalities kuka3aC0Maa(u); 8 juja22H10(V);

(8)

jr jvk jrvj; 8u2H10(V);

(9)

(5)

and for allv2H10(V), the following equalities (rotv;w)ˆ(v;rotw); 8w2H10(V);

(10)

rot (jvja 2v)ˆjvja 2rotv‡(a 2)jvja 3r jvj v;

(11)

div (jvja 2v)ˆjvja 2divv‡(a 2)jvja 3v r jvj;

(12)

kjvja 2vk a

a1ˆkjvjkaa 1: (13)

Now, we set out our main result

THEOREM2.1. Let(u;w)be a weak solution of the problem(1)-(5)with initial conditionsu02Ha,w02Gaandf;g2La(0;T;La(V))fora>3. If p the associated pressure satisfies

p2La(0;T;La‡13a(V)) (14)

then the weak solution satisfies

(u;w)2C(0;T;Ha)C(0;T;Ga) (15)

and

(juja2;jwja2)2L2(0;T;H10(V))L2(0;T;H10(V)):

(16)

REMARK2.2. To the Navier-Stokes equations it is well known that a weak solution in the sense of Leray-Hopf (see[19], [15], [20], [21])is reg- ular if

u2Lr(0;T;Ls(V)); 2 r‡3

s1; s>3:

(17)

REMARK2.3. If we put rˆaand sˆ3a=(a‡1)we get as a sufficient condition for regularity that

p2Lr(0;T;Ls(V)) withs>9=4 and 2 r‡3

sˆ1‡3 r: (18)

Note that the right-hand side in the equality of(18)is a number greater than 1and strictly lower than2. Note that if we could take rˆ3we would get a pressure belonging to L3(0;T;L94(V)). By taking into account that p' juj2, this condition is equivalent tou2L6(0;T;L92(V)), which satisfies(17).

3. Apriori estimates.

With the notations given in Section 2 and in order to prove the main Theorem 2.1, we give the following Lemma:

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LEMMA3.1. Let(u;p;w)be a smoothsolution of the micropolar equa- tions(1)-(5)inV(0;T]. Then the following estimate is satisfied

1 a

d

dt(kukaa‡kwkaa)‡3n1

4 Naa(u)‡n2

2 Naa(w)‡4n1(a 2)

a2 Maa(u)‡C1(a 2) a2 Maa(w)

‡4mrkwkaa (rp;juja 2u)‡C(kfkaa‡ kgkaa)‡C(kukaa‡ kwkaa);

(19)

where C1ˆ4n2 an3>0.

PROOF. We multiply both sides of (1) by juja 2u(as in the proof of Lemma 1.1, [4]) and then doing integration by parts inV, taking into ac- count (6) and (10), it follows that

1 a

d

dtkukaa‡n1Naa(u)‡4n1(a 2) a2 Maa(u)

(rp;juja 2u)‡2mr(w;rot (juja 2u))‡(f;juja 2u):

(20)

In analogous form multiplying (3) byjwja 2w, taking into account (6), (10) and (12), we get

1 a

d

dtkwkaa‡n2Naa(w)‡4n2(a 2)

a2 Maa(w)‡n3kjwja2 1jdivwj k22‡4mrkwkaa 2mr(u;rot (jwja 2w))‡(g;jwja 2w)‡n3(a 2) (divw;jwja 3wr jwj):

(21)

Now, taking into account (11) and (9), we have

(w;rot (juja 2u))ˆ(w;juja 2rotu)‡(a 2)(w;juja 3r juj u)

ˆ(jwj;juja 2jruj)‡(a 2)(jwj;juja 3jr juj uj) (jwj;juja 2jruj)‡(a 2)(jwj;juja 2jruj) (a 1)(jwj;uja 2jruj):

(22)

Since 1

a‡a 2 2a ‡1

2ˆ1, from (22) together the HoÈlder and Young in- equalities, we get

2mr(w;rot (juja 2u))2mr(a 1)kwkakjuja2 1ka2a2kjuja2 1jrujk2 2n1(a 1)kwkakjujkaa22(Naa(u))1=2 4n1(a 1)2kwk2ak jujkaa 2‡n1

4 Naa(u):

(23)

(7)

Again, since2

a‡a 2

a ˆ1, by using the Young's inequality in (23), we obtain 2mr(w;rot (juja 2u))C(a;n1)kwkaa‡C(a)kukaa‡n1

4 Naa(u);

(24)

Similarly as to (23) and (24), we can obtain 2mr(u;rot (jwja 2w))2m2r(a 1)2

n2 kuk2ak jwjkaa 2‡n2

2 Naa(w) C(a;mr)kukaa‡C(a;n2)kwkaa‡n2

2 Naa(w) (25)

Now, considering (13) and since 1

a‡a 1

a ˆ1, by using the Young's in- equality in (23), we have

j(f;juja 2u)j kfkak juj kaa 1Ckfkaa‡Ck juj kaa: (26)

Similarly

j(g;jwja 2w)j Ckgkaa‡Ck jwj kaa: (27)

Now, taking into account thatjr juja2j ˆa

2 juja2 1jr juj ja.e. inV, to the last term in the right-hand side of (21), we get

(divw;jwja 3w r jwj)(jdivwj;jwja 2jr jwk) (jdivwj jwja2 1;jwja2 1jr jwk) kjdivwj jwja2 1k2 kjwja2 1jr jwk k2 2

a k jdivwj jwja2 1k2k jr jwja2j k2: (28)

Then, by using the Young's inequality, from (28) we have (divw;jwja 3wr jwj)1

ak jwja2 1jdivwj k22‡1

aMaa(w);

and since,1

35a 2

a 51 fora>3, then

(29) n3(a 2) (divw;jwja 3wr jwj)n3k jwja2 1jdivwj k22‡n3(a 2) a Maa(w):

Setting (24) and(26) in (20), as well as (25), (27) and (29) in (21), we obtain 1

a d

dtkukaa‡3n1

4 Naa(u)‡4n1(a 2) a2 Maa(u)

(rp;juja 2u)‡Ckwkaa‡Ckukaa‡Ckfkaa: (30)

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and 1 a

d

dtkwkaa‡n2

2 Naa(w)‡(4n2 an3)(a 2)

a2 Maa(w)‡4mrkwkaa Ckukaa‡Ckwkaa‡Ckgkaa;

(31)

thus adding inequalities (30) and (31) is followed (19).

LEMMA3.2. Let(u;p;w)be a smoothsolution of the micropolar equa- tions(1)-(3)inV(0;T]. Then the following estimate is satisfied

(rp;juja 2u)C2kpka3a

a‡1‡n1(a 2)

a2 Maa(u)‡n1

4 Naa(u);

(32)

where C2is a positive constant dependent onn1;a.

PROOF. Taking into account thatu2Haand (12), we have (rp;juja 2u)ˆ(p;div (juja 2u))ˆ(a 2) (p;juja 3u rjuj)

(a 2) (jpj;juja 2jruj) (a 2) (jpj juja2 1;juja2 1jruj):

(33)

Since a‡1

3a ‡a 2 6a ‡1

2ˆ1, by applying HoÈlder's inequality in (33), we have

(rp;juja 2u)(a 2)kpk3a

a‡1kjuja2 1k6a

a 2kjuja2 1jrujk2 (a 2)kpk3a

a‡1kuka3a22kjuja2 1jrujk2 (34)

Then, by using the Young's inequality in (34), we get (rp;juja 2u)(a 2)2

n1 kpk23a

a‡1kuka3a2‡n1 4Naa(u):

(35) Now,2

a‡a 2

a ˆ1, then by using the Young's inequality and (8), we have (a 2)2

n1 kpk23a

a‡1kuka3a2C2kpka3a

a‡1‡n1(a 2) a2 ( 1

C0)kuka3a C2kpka3a

a‡1‡n1(a 2) a2 Maa(u);

(36)

whereC2ˆa 2 n1

a

(C0a)a222n1

a .

Thus, inequalities (35) and (36) imply (32).

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4. Proof of Theorem 2.1

The proof of Theorem 2.1 is based on Lemma 3.1 and Lemma 3.2 with the aid of smoothing properties.

From Lemma 3.1 and Lemma 3.2, we have that the smooth solution (u;p;w) of the micropolar equations (1)-(3) satisfies

d

dt(kukaa‡ kwkaa)‡an1

2 Naa(u)‡an2

2 Naa(w)‡3n1(a 2) a Maa(u)

‡C1 (a 2)

a Maa(w)C2kpkaa‡13a‡C(kfkaa‡kgkaa)‡C(kukaa‡kwkaa):

(37)

Moreover, by observing the notation given in (7) and since a 2>1;

a 2 a >1

3, from (37) we get d

dt(kukaa‡kwkaa)‡n1Naa(u)‡n2Naa(w)‡n1kr juja2k22‡C1

3 kr jwja2k22 C2kpka3a

a‡1‡C(kfkaa‡ kgkaa)‡C(kukaa‡ kwkaa):

(38)

By integrating (38) from 0 tot2(0;T), we have ku(t)kaa‡kw(t)kaa‡n1

Zt

0

Naa(u(t))dt‡n2 Zt

0

Naa(w(t))dt

‡n1

Zt

0

kr ju(t)ja2k22dt‡C1 3

Zt

0

kr jw(t)ja2k22dt

ku0kaa‡kw0kaa‡C Zt

0

kp(t)ka3a

a‡1dt‡C Zt

0

(kf(t)kaa‡kg(t)kaa)dt

‡C Zt

0

(ku(t)kaa‡ kw(t)kaa)dt:

(39)

Then, taking into account the hypotheses onf;gandp(see (14)), inequality (39) implies

ku(t)kaa‡kw(t)kaa‡n1

Zt

0

Naa(u(t))dt‡n2

Zt

0

Naa(w(t))dt‡n1

Zt

0

kr ju(t)ja2k2dt

‡C1

3 Zt

0

kr jw(t)ja2k2dtC ‡C Zt

0

(ku(t)kaa‡ kw(t)kaa)dt:

(40)

(10)

and using the Gronwall's inequality in (40), we obtain ku(t)kaa‡ kw(t)kaa‡n1

Zt

0

kr ju(t)ja2k2dt‡C1

3 Zt

0

kr jw(t)ja2k2dt C(1‡t) exp (C t):ˆK(t)K(T)

(41)

then follows that (u;w)2L1(0;t;Ha)L1(0;t;Ga) and consequently by the hypothesisa>3 (see [8]) we have that (u;w)2C(0;t;Ha)C(0;t;Ga).

Now, if we denote bytthe lowest upper bound of the values in [0;T] for which (41) is satisfied in [0;t], we have thatt>0 and (u;w)2C(0;t;Ha) C(0;t;Ga). Then, takingu(t);w(t) as initial data, ift5Tone show from (39) and (41) that exists a positive constanthsuch that

ku(t‡h)kaa‡ kw(t‡h)kaa‡n1

Zt‡h

t

kr ju(t)ja2k2dt

‡C1

3 Zt‡h

t

kr jw(t)ja2k2dtK(h)K(T);

(42)

thus (u;w)2C(0;t‡h;Ha)C(0;t‡h;Ga), therefore necessarily it must betˆT.

REMARK4.1. Fora>3we have that9 45 3a

a‡153, then L1(0;T;Lq(V))La(0;T;La‡13a(V)); 9

45q53;

(43)

where qˆ 3a

a‡1. Th us, from(14)and(43),we conclude that the same result of the Theorem2.1is obtained if we consider

p2L1(0;T;Lq(V)); 9

45q53:

Moreover, since12 5 >9

4the condition on the pressure given by Kaniel in [10]is improved.

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[1] H. BEIRAÄO DAVEIGA,A new regularity class for the Navier-Stokes equations inRn, Chin. Ann. of Math.,16B, 4 (1995), pp. 1-6.

[2] H. BEIRAÄO DAVEIGA,Concerning the regularity of the solutions to the Navier- Stokes equations via the truncation method Part II, In Equations aux deÂriveÂes partielles et applications Gauthier-Villars EÂd. Sci. MeÂd. Elsevier (Paris 1998), pp. 127-138.

[3] H. BEIRAÄO DAVEIGA,A Sufficient condition on the pressure for the regularity

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of weak Solutions to the Navier-Stokes equations, J. Math. Fluid Mech.,2 (2000), pp. 99-106.

[4] H. BEIRAÄO DAVEIGA,Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J.,36 (1987), pp. 149-166.

[5] L. BERSELLI,Sufficient conditions for the regularity of the solutions of the Navier-Stokes Equations, Math. Meth. Appl. Sci.,22(1999), pp. 1079-1085.

[6] D. CHAE- J. LEE,Regularity criterion in terms of pressure for the Navier-Stokes equations, Nonlinear Anal.,46, no. 5, Ser. A: Theory Methods (2001),pp. 727-735.

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[9] J. G. HEYWOOD- O. D. WALSH,A counter-example concerning the pressure in the Navier-Stokes, as t!0‡, Pacific J. Math.,164(1994), pp. 351-359.

[10] S. KANIEL, A sufficient conditions for smoothness of solutions of Navier- Stokes equations, Israel J. Math.,6(1968), pp. 354-358.

[11] S. KANIEL- M. SHINBROT,Smoothness of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal.,24(1967), pp. 302-324.

[12] O. A. LADYZHENSKAYA, The mathematical theory of viscous incompressible flow, Second edition, Gordon and Breach (New York 1969).

[13] G. LUKASZEWICZ,On the existence, uniqueness and asymptotic properties of solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL, Mem.

Math.,107(vol. XIII) (1989), pp. 105-120.

[14] G. LUKASZEWICZ, Micropolar fluids: theory and applications, BirkhaÈuser (Berlin 1998).

[15] K. MASUDA,Weak solutions of Navier-Stokes equations, Tohoku Math. J.,36 (1984), pp. 623-646.

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[17] E. ORTEGA-TORRES- M. A. ROJAS-MEDAR,On the uniqueness and regularity of the weak solutions for magneto-micropolar equations, Rev. Mat. Apl.,17 (1996), pp. 75-90.

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existence of weak solution, Rev. Mat. Univ. Complutense de Madrid., Vol.11, 2 (1998), pp. 443-460.

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Manoscritto pervenuto in redazione il 6 marzo 2008.

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