stationary solutions for generalized boussinesq models

This article was downloaded by: [Moskow State Univ Bibliote]On: 07 February 2014, At: 21:40Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Applicable Analysis: An International JournalPublication details, including instructions for authors and subscriptioninformation:http://www.tandfonline.com/loi/gapa20

Stationary solutions for generalized boussinesqmodelsSebastián A. Lorca a & José Luiz Boldrini aa Campinas, 13081-970, SP, BrazilPublished online: 02 May 2007.

To cite this article: Sebastián A. Lorca & José Luiz Boldrini (1995) Stationary solutions for generalizedboussinesq models, Applicable Analysis: An International Journal, 59:1-4, 325-340

To link to this article: http://dx.doi.org/10.1080/00036819508840408

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”)contained in the publications on our platform. However, Taylor & Francis, our agents, and ourlicensors make no representations or warranties whatsoever as to the accuracy, completeness, orsuitability for any purpose of the Content. Any opinions and views expressed in this publication arethe opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis.The accuracy of the Content should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoevercaused arising directly or indirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes. Any substantialor systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, ordistribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use canbe found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/gapa20

http://dx.doi.org/10.1080/00036819508840408

http://www.tandfonline.com/page/terms-and-conditions

Applicublc Amlysis, Vol. 59, pp. 325-340 Reprints available directly from the publisher Photocopying permitted by license only

Q 1995 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands

under license by Gordon and Breach Science Publishen SA F'rinted in Malaysia

Stationary Solutions for Generalized Boussinesq Models Communicated by J. Hale

by

Sebastia'n A. Lorca and Jose' Luiz Boldrini UNICAMP - IMECC

C.P. 6065 13081-970, Campinas, SP, Brazil

AMS: 35Q10,35Q20,76D05

Abstract. We study the existence, regularity and conditioas for uniqueness of solutions of a generalized Boussinesq model for thermally driven convection. The model allows temperature dependent viscosity and thermal conductivity

KEY WORDS: Boussinesq, thermally driven convection, stationary solutions, temperature dependent viscosity temperature

(Received for Publication June 1994)

Mathematical Subject Classification. Primary 35Q10; Secondary 35820, 76D05.

1. Introduction

We study the stationary problem for the equations governing the coupled mass and heat flow of a viscous incompressible fluid in a generalized Boussinesq approximation by assuming that viscosity and heat conductivity are temperature dependent. The equations are

(1.1)

-div(v(T)Vu) + u.Vu - aTg + Vp = 0, div u = 0

-div(k(T)VT) + u.VT = 0 in 0 ,

where R is a bounded domain in R N , N = 2 or 3 throughout the paper. Here u ( x ) E RN denotes the velocity of the fluid at a point x E 0 ; p(x) E R is the

hydrostatic pressure; T(x) E HZ is the temperature; g(x) is the external force by unit of mass; v(.) > 0 and k(.) > 0 are kinematic viscosity and thermal conductivity, respectively; a is a positive constant associated to the coefficient of volume expansion. Without loss of generality, we have taken the reference temperature as zero. For a derivation of the above equations, see for instance Drazin and Reid [6].

The expressions V, A and diu denote the gradient, Laplace and divergence operators, respectively (we also denote the gradient by grad); the ith component of u.Vu is given by

dui d T (u.Vu); = x u j - ; u.VT = x u j - .

j=1 dx j j=1 ax j The boundary conditions are as follows

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014

(1.2) u = 0 and T = To on X l , where To is a given function on 8R (the boundary of 0).

The classical Boussinesq equations correspond to the special case where v and k are positive constants. This case has been much studied (see for instance Morimoto [9], [lo] and Rabinowitz [13] and the references therein).

Equations (1.1) are much less studied, and they correspond to the following physical situation. For certain fluids we cannot disregard the variation of viscosity (and thermal conductivity) with temperature, this being important in the determination of the details of the flow. It is found, for example, that a liquid usually rises in the middle of a polygonal convection cell, while a gas falls. Graham [7] suggested that this is because the viscosity of a typical liquid decreases with temperature whereas that of a typical gas increases. This suggestion was subsequently confirmed by Tippelkirch's experiments (see 113)) on convection of liquid sulphur, for which the viscosity has a minimun near 153" C. He found that the direction of the flow depended on the temperature being above or below 153" C. Palm [lo] was the first to analyse the effect of the variation of viscosity with temperature; other papers on the subject are, for instance, Busse [4] and Palm, Ellingsen, Gjevik [12]. All these papers have the Theoretical Fluid Dynamics flavour. A rigorous mathematical analysis is more difficult in this case than in the case of the classical Boussinesq equations due to the stronger nonlinear coupling between the equations.

As a step in this direction, in this paper we will study the existence and regularity of solutions of (1.1) by using a spectral Galerkin method combined with fixed point arguments; we will need more estimates than the ones required in the classical case in order to handle the nonlinearity in the higher order terms of the equations.

We will show the existence of weak and strong solutions of problem (1.1), (1.2) under certain conditions on the temperature, dependency of the viscosity and thermal conductivity; we allow more general external forces than the usual one (constant gravitational field) because this could be useful in certain geophysical models. Properties of regularity and uniqueness are also studied. Questions concerning stability and bifurcation are left for future work.

We observe that if we take u r 0 in (1.1) and g = (0,0, l ) , the usual approximation for the gravitational acceleration, we are left with grad p = aTy = (0,O, aT). Consequently,

dT 8T curl (O,O, aT) = 0 in R, and so - = - - - 0 in R. Therefore, we see that an arbitrary

dx dy temperature on the boundary will require, in general, motion. That is, in general the solution of (1.1), (1.2) are not trivial.

Finally, we would like to comment that analogous questions can be considered for the corresponding evolution problem. Results along these lines will appear elsewhere. Also, concerning numerical results we would like to mention the interesting paper by C. Bernard;, F. Laval, M. Metivet and B. Pernand-Thomas [3] that treat a related problem.

This paper is organized as follows: in section 2 we describe the notation and the ba-

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014

BOUSSINESQ MODELS 327

sic facts to be used later on: we also state the our main results. In Section 3, we make a technical preparation by proving certain a priori estimates that will be useful for the proofs of the main results. The proofs of Theorem 2.1 (existence of a weak ~olu t ion)~ Theorem 2.2 (existence of strong solution), Theorem 2.3 (regularity) and Theorem 2.4 (uniqueness) are done in Section 4,5, 6 and 7, respectively.

2. Preliminaries and results

In this article the functions are either R or RN valued (N = 2 or 3) and we will not distinguish them in our notations; this being clear from the context. The L2(R)-pr~duct and norm are denoted by ( , ) and I 1, respectively, the LP(R) norm by 1 I,, 1 < p 5 W; the Hm(R) norm is denoted by 1 1 11, and the Wk+(R) norm by I lk,,. Here Hm(R) = Wm*2(R) and Wk-P(R) are the usual Sobolev spaces (see Adams (11 for their properties).

Let D(R) = {v E (C,6"(R))N div v = 0 in R), H = completion of D(R) in L2(R), and V = completion of D(R) in H1(R). We note that, when R is Lipschitz-continuous, the spaces H and V can be characterized as

H = {v E L2(R);div v = 0 in R,v.n = 0 on dR),

V = {v E H,'(R);div v = 0 in R},

where n is the external orthonormal field to dR. Let P be the orthogonal projection af L2(R) onto H obtained by the usual Helmholtz decomposition. We shall denote by vk and ak respectively the eigenfunctions and eigenvalues of the Stokes operator & = -PA: Dom (A) c L2(R) + L2(R), where Dom (A) = V n H2(R) is the domain of i. If R is Lipschitz continuous, it is known that vk are orthogonal in the inner product (., .) and (V-, V.) and are complete in the spaces H and V; when R is of class C2 then the eigenfunctions vk are also complete in V n H2(R) and orthogonal with respect to the inner product (A*, &a) (see, for example, Temam (141.

Similar considerations are true for the Laplace operator A; we will denote by $k and Xk respectively the eigenfunctions and eigenvalues of the operator -A.

Let W'-:*P(~R) be the trace space obtained as the image of W1+'(R) by the boundary value mapping on BR, equipped with the norm iy]l-;,p,m = inf {IV~~,,, v E W19p(R), v = y on 8R). Similarly, when dR is sufficiently smooth, we can define the trace spaces W k - A , ~ p (80) with norm ( (k-;,p,m. When p = 2, we denote Hk-1/2(dR) = Wk-1/212(dR)

and 11 Ilk-l/z,an = I lk-l/2,2,an (see Adams [I]). We assume that we can find a function S defined in R satisfying S = To on dR; then,

we can transform the equations (1.1), (1.2), by introducing the new variable cp = T - S to obtain

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014

- div(v(cp + S)Vu) + u.Vu - acpg - aSg + V p = 0,

(2.1) div u = 0

- div(k(cp + S)Vcp) + u.Vcp - diu(k(cp + S)VS) + u.VS = O in R u = 0 and p = 0 on 80.

Suppose that S E H1(R) (thus To E H'/*(~R)); then we can reformulate (2.1) in weak form as follows: to find u E V and cp E HA(R) satisfying

(~(cp + S)Vu, Vv) + B(u, u,v) - a(cpg,v) - a(Sg,v) = 0, for all v in V (k(p + S)Vcp, Vll) + b(u, cp, ll) + (k(p + S)VS, V$) + b(u, S, $1 = 0, for all 1C, in HA(R)

N 8ui where B(u,v,w) = (u.Vv,w) = JQ C U ~ ( X ) - ( X ) W ; ( X ) ~ X and b ( u , ~ , $ ) = (u.VV,$) =

i,j=l 8s;

Definition. A pair of functions (u, T) E V x H1(R) is called a weak solution of (1.1), (1.2) if there exists a function S in H1(R) such that cp = T - S E H,'(R), S = TO on iXl , and, (u, cp) is a solution of (2.2)

Based on physical assumptions, throughout the paper we will suppose that

( 1 . v(7) > 0; k(7) > 0 for all T E R

We observe that assumption (A.l) allows the cases lim inf v(T) = 0 or T-tm

lim sup v(T) = t o o (the same holds for k(.)). T++s

Our first result concerns the existence of weak solutions.

Theorem 2.1. Let R be a hounded domain in R N ( N = 2 or 3) with Lipschitz continuous boundary; let the functions u, k be continuous, g E L2(R) and To E H ' I ~ ( ~ R ) fl Lw(dO). Then, there exists a weak solution of (1.1), (1.2). In case that inf{v(r), k(r) ; T E IR) > 0 and sup {v(T), k(7); T E R) < ca the result is true under the weaker assumption TO E H112(dR).

We remark that in the proof of this result we will obtain approximate solutions by using the eigenfunctions of the Stokes and Laplace operators. Actually, for this is not necessary to obtain the result in theorem 2.1, and we could use any basis V and H,'(R).

If. we have stronger assumptions, we are able to prove the following. Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014


Theorem 2.2. Let R be a bounded domain in nN ( N = 2 or 3) with C1ll boundary; we suppose that v, k are of class C1, g E L3(R) and To E H ~ / ~ ( ~ R ) . Then, if IITol13/~,an is small enough, there exists a strong solution of (1.1), (1.2), that is, there exists a pair (u ,T) E (V n H2(R) ) x H2(R) such that

We ~ b s e r v e that there exists a unique function p (the pressure) in H1(R) n Lg(R), with Li(R) = { f E LZ(R)/( f , 1) = 0), such that

For this, see Temam [Is]. We also observe that Theorem 2.2 is true if we take a small instead of IITol1312,m small.

The next result is concerned with the regularity of ( 1 1 , T, p).

Theorem 2.3 Let R be a bounded doma.in in RN ( N = 2 or 3) with a Ck+',' boundary; let the functions v, k be of class Ck+l,g E Wkv3(Q) and To E Wk+7/4v4(d~) . Then a strong solution (u ,T) satisfies u E Hk+'(R) and T E Wk+234(R). Moreover, the associated pressure satisfies p E Hk+' ( 0 ) n Lg(R).

The following is a uniqueness result for "smalln weak solutions.

Theorem 2.4. Let R be a bounded domain in nN (N = 2 or 3), with a Ck+'*' boundary and v, k, k' Lipschitz continuous. There exists E > 0 such that , if there exists a weak solution (u ,T) of (1.1), (1.2) satisfying lVul+ llTllz < E , then it is unique.

Finally we state two lemmas for convenience of reference. By Hdder's inequality and Sobolev imbeddings, we have

Lemma 2.5. There exists a constant CB depending on R such that IB(u,v,w)l 5 C~lVul lVvl [Vwl for Vu E H:(R),Vv 'V H1(Q),Vw E HA(R) and Ib(u,p,$)( 5 CBIVUIIV~IIV$I for Vu E Ht(R) ,Vp E H1(R),V$ E Hd(R).

By density arguments and integration by parts (see Temam [14]), we have

Lemma 2.6 (i) B(u, v, W) = -B(u, W, v) for VU E V, and Vv, w E H1(R); b(u, p , $) = -b(u, +,cp) for Vu E V, and V ~ , 9 E H1(R)

(ii) B(u , v, v) = 0 for Vu E V, and Vv E H1(R); b(u, cp,p) = 0 for VU E V, and Vp E H1 (R).

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014

330 SEBASTIAN A. LORCA AND JOSE LUIZ BOLDRINI

In what follows we will use C as a generic positive constant which depends only on R, through constants appearing in the Poincark inequality and Sobolev inequalities.

3. A Priori Estimates

In this section we will show that problem (1.1), (1.2) satisfies a weak maximum principle. Also we will obtain a priori estimate for weak solutions.

Lemma 3.1. Let {u, T ) be a weak solution of (1.1), (1.2). Then we have

(3.1) ikf To 5 T ( r ) j sup To a.e. in an

Proof. Assume 1 = supan To < co (If 1 = oo we are done). We take $J = T+ in (2.2), where T+ = sup{T - I , 0) to obtain (k(T)VTl VT+) = -b(u, T, T+). An easy computation shows that (k(T)VT+, VT+) = (k(T)VT, VT+) = -b(u, T, T+) = -b(u, T+, T+). There-

fore, by using Lemma 2.6 (ii), we have k(T)(VT+I2 = 0. Thus, k(T)(VT+IZ = 0 a.e in L 0 , and consequently JVT+12 = 0 a.e. in R. This last equality implies that T+ = 0 since T+ E HA; thus, the right hand side of (3.1) follows. The left hand side (3.1) is similarly obtained o

An interesting consequence of the previous lemma is that we can transform problem (1.1), (1.2) into an equivalent one. Suppose that inf {k(t);t E R) = 0 or sup{k(t), t E R) = +oo and supan lTol < too; then, we consider the modified function &, with the same regularity as k and satisfying &(T) = k(7) for all 171 I SUPITOI and

an

Analogous considerations can be done for u(.).

Clearly, a pair (u, T) is a weak solutions of (1.1), (1.2) if and only if it is a weak solution of the following problem

-div(fi(T)Vu) + u.Vu + V p = aTg, div u = 0,

- d i v ( i ( ~ ) ~ ~ ) + u.VT = 0 in $2.

(3.3) T = T o , u = O on 80. Therefore, hereafter we can suppose that the functions u(.) and k(.) satisfy

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014

BOUSSINESQ MODELS 33 1

where uO(TO) = inf{v(t); It( 5 suplTol)/2, vl(To) = 2 sup{v(t); It( I suplTol) with analogous rn a~

definitions for ko(T0) and kl (To).

Remark 3.2 Obviously, if we assume

then, the above modification is unnecessary.

Now, we prove an a priori estimate. Let {u, y ) be a weak solution of (1.1), (1.2). Thus, by taking v = u and $ = y in (2.2), we have

From Lemma 2.6, Holder's inequality and (3.4), we obtain vo(To)JVu12 5 a(yg,u) + cr(Sg, u) I a1g1(1y13 + I S [ ~ ) I U ~ ~ . By Sobolev imbeddings, we find

C (3.8) lVul 5 ~~----(~~IIVYI + Isl IlSlll)

vo(T0)

Similarly, we have, ko(To)lVv12 5 -b(u,S, y ) - (k(y + S)VS, Vy) = b ( u , ~ , S ) - (k (y+ S)VS,Vy) I IulslV~llSls + kl(To)lVSllV~I I c l V ~ l l s l 3 l V ~ l + h(To) lVsl lV~l .

Thus,

(3.9) C kl (To)

IVYI I -IS131VuI + -IISIIi ko(T0) ko(T0)

Substituting (3.9) into (3.S), we obtain

If we assume

(3.10)

then we have

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014


Substituting this in (3.9), we are left with

k1 (To) +-IlSIll

ko(T0)

We summarize these estimates in

4. Exis tence of W e a k Solutions

We start by proving the following result (compare with the one in Morimoto [9]).

Lemma 4.1. Let R be a bounded domain in R N , N = 2 or 3, with Lipschitz continuous boundary. If To is a function in ~ ' l ~ ( d R ) , then for any positive numbers E and 1 < p < 6 if n = 3 or any finite p > 1 if n = 2, there exists an extension S E H1(0) of To such that

P I P < E

Proof By definition of H112(dR), we can obtain an extension Po E H1(fl), of To. For any 6 > 0, we consider dRs = { x E RN;d(x,8R) < 6) and P(.) E C?(RN) such that 0 5 P(.) 5 I,/?(.) 1 in dfls12,P(.) r 0 in RN\dRa (we can obtain such function, by applying a differential version of Urysohn's Lemma).

Define S(x) = ,B(x)po(x). Then S is a required extension, because S E H1(fl) and

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014


Since by Sobolev S* l~o (x ) I~dx < -t right hand side of

embedding, H1(R) c LP(R), with p satisfying the stated conditions, .oo, and, therefore, we can choose 6 > 0 sufficiently small so that the the above inequality is less than E . 0

Now, we are ready to prove Theorem 2.1.

Proof of Theorem 2.1. According to Lemma 4.1, with p = 3, we can choose an extension S of To such that S E H1 and satisfies (3.10).

n

As nth aproximate solution of equation (2.1) we choose functions un(x) = Cc , ,kvk (~ ) k = l

and cpn(x) = xdn,k+k(x) satisfying the equations k= 1

(4.1) (u(cpn + S)Vun, V d ) + B(un, un, d) - cr(cpng, d) - cr(Sg, d) = 0,

(4.2) (k(cpn + S)Vcpn, Vllr') + b(un, pn , llrf ) + (k(cpn + S)VS, VV) + b(un, S, + j ) = 0,

for 1 5 j 5 n. First we assume the existence of (un,cpn) for any n f N. Note that solutions (un,cpn)

must satisfy estimate (3.11). In fact, the identity (3.6) for un is obtained by multiplying (4.1) by G,, and summing over j from 1 to n. Similarly, we have estimate (3.7) for 9".

Estimates (3.11) follow from equations (3.6), (3.7) as in Section 3. Therefore, the sequence (un,cpn) is bounded in V x Hi.

Since V (respectively HA) is compactly imbedded in H (respectively L2(R)) we can choose subsequences, wich we again denote by (un, cpn), and elements u E V, cp E Hi such that un 4 u weakly in V and strongly in H and also cpn + cp weakly in HA, strongly in L2 and almost everywhere in R. Furthermore, we can suppose that Vun + Vu, weakly in L2 and Vcpn + Vcp, weakly in L2.

This is enough to take the limit as n goes to w in (4.1), (4.2) and obtain

In fact, in taking this limit, there is no difficulty with the nonlinear term. It is easy to see that B(un, un, V ) --+ B(u, U, v ) Vv E V and that b(un, cpn, $) 4 b(u, cp, @), V$ E H1(R) (see, for example, Temam [13]). Also, we observe that

( 4 9 " + s p u n , vd) = p u n , u(cpn + S)VV") -, (Vu, v(cp + s)vV") = ( ~ ( $ 0 + S)VU, v d )

because v(cpn + S ) V ~ + v(cp + S ) V ~ strongly in L2(R) due to Lebesgue Dominated Convergence Theorem. Similarly, (k(cpn + S)Vcpn, V@) 4 (k(cp + S)Vv, V?).

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014

Since the system {vk} (respectively { + k } ) is complete in V (respectively Hi(R)), (4.3) implies that (u, cp) satisfies (2.1). Therefore, (u,cp f S ) is a required weak solution.

In order to prove the solvability of the system (4.1), (4.2) for any n E N, we follow Heywood [8] in applying Brouwer's Fixed Point Theorem.

Let Vn be the subspace of V spanned by {vl, . . . , vn), and let Mn be the subspace spanned by {$I,. . . , $ J ~ } . For every (w, J ) E Vn x Mn we consider the unique solution L(w, J) = (u, cp) E V, x Mn of the linearized equations

for 1 5 j 5 n. This is a system of 2n linear equations for the coefficients in the expansions u = ELl ckvk , (P = dk+k. Equations (4.4), (4.5) have a unique solution because the associated homogeneous system (S = 0) has an unique solution. In fact, if (u,(P) is a solution of the homogeneous system, proceeding as before, we multiply (4.4) by cj, (4.5) by dj, and sum over j from 1 to n, to obtain vo(To)lVu12 = 0, ko(To)lVyIZ = 0. Hence u = O,(P = 0. The continuity of L follows by applying arguments that are similar to the ones used to take the limit in (4.1), (4.2).

We also have the following estimates

which are shown in exactly the same way as it was done for a solution (un, cpn) of (4.1), (4.2). Substituting (4.7) into (4.6) and proceeding as we did to obtain (3.11) we find

CYC2 CYC lVul 5

vo(To)ko(To) I g ' 's'3'vw' + vo(To)ko(To) I91 IlSlll(kl(T0) + Ico(T0)).

Since (2.10) holds, we have

1 IVul< llVwl + a

vo(To) ko(To) Isl IlSlll l(kl(T0) + ko(T0)).

If we assume [Vwl 5 Fl(IISIII) (see (3.11)), then (4.7) and (4.8) imply that (u,cp) satisfies (3.11), that is,

Thus, (4.4), (4.5) define a continuous mapping L from the closed and convex set M = {(w,J) E Vn x Mn/IVwl 5 Fl(IIS((l) and IVEl 5 F2(11SIII)) into itself. Using Brower's Fixed Point Theorem, we conclude that the map L has at least one fixed point, which is a solution of (4.1), (4.2): Thus, the proof of Theorem 2.1 is complete.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014


5. Existence of Strong Solutions

In this section we will prove Theorem 2.2; for this we folllows Temam[l4] in using the equivalence of the norm given by the Stokes Operator (respectively Laplacian operator) and the V n H2(R) norm (respectively H;(R) n H2(R) norm) in smooth domain. The main difficult here is to estimate the nonlinearity in the higher order terms in the velocity equation; for this we will need an estimate for the associated pressure in the Stokes' Problem.

Proof of Theorem 2.2. We choose the extension S of To such that ,S is the solution of the problem: -AS =

0 in R, S = To on 80. We know that S is a function in H2(R) and satisfies

According to the proof of Theorem 2.1, we have a, sequence (un ,yn) satisfying (4.1), (4.2) provided (3.10) holds. Since ISI3 5 C((SIIl 5 CJITolJ31z,m, we conclude the existence of this sequence for IITol13/2,m sufficiently small. We need only to show that we can take this sequence bounded in H2(R).

For this, we multiply equation (4.4) by d c i , and then sum over j from 1 to n, to obtain (div(v([ + S)Vu), Au) - B(w, u, AU) + a(Sg, Au) + a(vg, Au) = 0. By using the identity div(v(( + S)Vv) = v([ + S)Av+ vl([ + S)V(< + S)Vv, where V([ + S)Vv denotes the vector field which i th component is given by [V([ + S)VvIi = (V(< + S) , V V ; ) ~ ~ , and where (., denotes the canonical inner product in Rn, we find

(5.2) - ( v ( t + S)AU, h u ) = - B(w, u, Au) + a(vg, AU) + c r ( ~ g , Au)

+ (.I([ + S)V([ + S)Vu, Au).

Since -Au # AU, we need the following decomposition AU + grad q = -Au. It is well known that (see Teman [13])

(5.3) I l Y I l l 5 ~ I W Now, we can rewrite (5.2) as (v(( + S)AU, A U ) = -B(w, U, A U ) + a(yg, AU) t

~ S S , ~ u ) + (vl(S + S)V(< + S)VU, A U ) + (v(< + s ) v ~ , Au) By Holder's inequality, Sobolev imbedding and (3.4), we have

where v:(To) = 2sup{Iv1(t)l; It1 < supJTol). We observe that an D

ownl

oade

d by

[M

osko

w S

tate

Uni

v B

iblio

te]

at 2

1:40

07

Febr

uary

201

4


because du E Vn. Therefore,

Combining estimates (5.4), (5.5) and (5.1), we have

Similarly, the following estimate holds -

(5.7) C

5 -(16wl+ k:(To)(IA<I + IIToIIi,m))(IA~I + 11~011~,m3 ko (To) k1 (To) + -IIToll&m, ko(T0)

and

(5.9)

c ki (To) 1 + 2 k i ( ~ o ) ( l + m ) ] l l ~ ~ l / t , m < - ko (To) 2

C 4aC 1 - [ - I S I . + ~ W ~ ) vo (TO) vo (TO)

Observe that this is possible because lim k(6) > 0 and lim v(6) > 0. 6-O+ 6-O+

Then, estimates (5.6), (5.7) imply

Consequently, if IITol13/l,m is small enough, L maps the closed and convex set { ( w , () E Vn x Mn; IAWI 5 P and lAtl 5 s} into itself. Then we can choose a sequence (un,cpn) bounded in H2(Q) satisfying (4.1), (4.2), and Theorem 2.2 follows.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014


6. Regularity

We first state some lemmas that will be necessary to prove Theorem 2.3. d'h

Lema 6.1 Let h be any function of class Ck such that sup{(-(t)(,t E R} 5 C, i = dt'

0,. . . , k. Then, there exists constants C(k) and Cl(k) such that for all T E Wk+'s4(R) there hold

k k

(i) Ih(T)Ik,m 5 C(k)CITI:+l,4 ; (ii) I ~ ( T ) I L . ~ 5 ci(k)xITIi ,4 t=o t=o

Proof. We only prove (i); the other inequality can be proved in the same way. We proceed by induction on k.

If k = 0, the result is trivial. So, suppose the result is true for any j E N such that 0 I j < k, and take p = (PI,. . . ,Pn),Pi E N , = P1 + . . . + Pn = k. Then, if i E (1,. . . , n} is such that pi > 0, we have dP(h(T)) = df l (dZi (h(~) ) = aPl (h ' (~) .d , ,~) = x c(y,6)8(h1(T))a6(a,,T), where = /3 - e, and ei is the i th vector in the canonical r+6=01 basis of R n , ~ ( y , 6) are positive constants and y = (yl, . . . ,yn), 6 = (S1,. . . , ~5,),1;.,6~ E BV.

Thus, by the inductive hypothesis and Sobolev imbeddings IdO(h(T))Iw < Ir l

. . . Now, we observe that 171 5 k - 1,161 5 k - 1 and so Jyl + 1 5 k and 161 + 2 5 k + 1.

k

Thus, IdP(h(T))Iw 5 C(k)zlT(i+, , , and the Lemma is proved. 0

(=I

Lemma 6.2: If h satisfies the conditions of Lemma 6.1, then for all T in Wk+'s4(R) I;

and all f in Wkvp(R) there holds (h(T) f In,, 5 Cl(k) (~(TI :+ , , , ) ( f (k,p. e=1

Proof. Let P = (P1 ,..., &,),Pi E N , = P1 + ... + Pn = k. AS before @(h(T)f) = ~ ( ~ , 6 ) 8 ( h ( T ) ) $ f ; therefore, by Lemma 6.1, we have

7+6=P IY I

l@(h(T)f ) I p I C c(y,6)l$(h(T))LI@f I, I C c(y,6)C(~,6)CITIk~+l,4l@f lp 2 7+6=P 7+6=P (=I

k

Cl(k)(CITli+l,4)l f )t,p. This proves the Lemma. t= 1

Lemma 6.3: Let (u,T) be a strong solution of (1.1), (1.2). Assume To E w ' / ~ * ~ ( ~ R ) , then T E W214(R). D

ownl

oade

d by

[M

osko

w S

tate

Uni

v B

iblio

te]

at 2

1:40

07

Febr

uary

201

4

338 SEBASTIAN A. LORCA AND J O S ~ LUIZ BOLDRINI

Proof. According to Section 3, we can suppose that (3.4) holds. We observe that T E H2(R) satisfies

Since To E W7/414(dR) C W5/3~3(dR) and T E H2(R), u E H1(R) (because (u,T) is a kl(T) 1 strong solution) then -IVT12 + -u.VT E L3(R). Thus we can apply the well- k(T) k(T)

know LP-regularity of ~ap lace operator obtaining T in W273(R). By using Sobolev imbedding we have that V T E L9(R), for all 1 5 q < co. Consequently, W ) 1 -IVTIZ + -u.VT E L4(R), and by applying the LP-regularity once again, we find W ) k(T) T E W2s4(R).

Proof of Theorem 2.3. We proceed inductively on k. If k = 0 the result follows by Lemma 6.3. Now we suppose the result is true for k - 1 (that is, T E Wk+'14(R)). Note that if P = (A , , . . , A ) , Pi E N , IP1 = Pi + . . . + Pn = k, then dP(uVT) = x c(j)@u$(VT) +

r+b=O udP(VT), where c(j) are positive constants, 7 = . . ,yn),S = (S1,. . . ,6,),7;,15; E N . Thus, l d P ( u V ~ ) l ~ 5 ~( j ) l@u(~ ld~(VT) I , + IU~, [@(VT)~~, and Sobolev imbed-

7+6=O dings together with the inductive hypothesis imply I u V T I ~ , ~ 5 Cn,k~~~k+1,2\T\k+1,4 +

1 Cn,k11~1121Tlk+1,4 < m. By using Lemma 6.2, we conclude -uVT E w ~ . ~ ( R ) .

YIT) Similarly as before, there holds lblVT121, < x c ( j ) 1 8 v ~ l , l d ~ v ~ 1 4 + l@VT141VTIm 5

7+6=P Cn,ktTIZk+l,4 + C~,k(Tlk+1,41T)2,4 < 00.

kl(T) Therefore, we have -IVTI2 E Wk*4(R). Now, by applying LP-regularity for problem

k(T1 \ ,

(6.1), we see, that T E Wk+274(R). As in the proof of Lemma 6.3 we have that u is solution of the following Stokes Problem

where p satisfies (2.2). As above, we show by induction that the right hand side terms in the first equation of (6.2), are in Hk(R). By Cattabriga's Theorem (see [5] and [2]) applied

P to (6.2), we find that u E Hk+'(R), - E Hk+'(R). 4 T )

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014


Applying Lemma 6.2 we conclude that p E H"+'(R). This completes the proof. a

7. Uniqueness

In this section we will prove Theorem 2.4. Let (ul , TI), (u2, T2) be a weak solution of (1.1), (1.2) such that Tl and T2 are in H2(R). Put w = ul - u2,t = Tl - T2. Then w E V, t E H,'(R)n H2(fl) and satisfy for Vv E V, V11, E L2(fl) (v(Tl)Vw, Vv) + B(w, ul, v) + B(u2, w, v) - a(&, v) + ((v(T1) - v(T2))Vu2, Vv) = 0, -(div(k(T~)VO, 11,) + b(w, Ti, $1 + b(u2, t , +) - (div((k(Ti) - k(T2))VT2), $) = 0.

We take v = w and 11, = -A< in these last equalities, thus obtaining

where C1 is a constant such that IV(t) - kl(s)l + (k(t) - k(s)l + Iv(t) - u(s)l 5 Cllt - S I for all t , s in R. This is shown in exactly the same way as in the case of the nth aproximate solutions (un, cpn) in Section 4 and 5.

We note that /(I, 5 CIA[(. Thus, (7.2) implies

Substituting (7.3) into (7.1), we obtain

Thus, if C '* IVulI c 1, we have IVwl = !At[ = 0.

~o(To)ko(To) IIT1ll2(4sl + cllvu' l) + QTJ

Since w E V and E H,'(R) n H2(Q), we see w = 0,( = 0 in R. Therefore, ul = u2,

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

21:

40 0

7 Fe

brua

ry 2

014

References

[l] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2] C. Amrouche, V. Girault, On the existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Japan Acad. 67, 1991, pp. 171-75.

[3] C. Bernardi, F.Lava1, B. Metivet, 8. Pernaud-Thomas, Finite element approximation of viscous flow with varying density, SIAM Jour. Numer. Anal. 29, 1992, pp. 1203-1243.

[4] F. H. Busse, The stability of finite amplitude cellular convection and its relation to an extremun principle. J. Fluid Mech. 30, 1967, pp. 625-49.

[5] I. Cattabriga, Su un problema a1 contorno relativo a1 sistema di equazioni di Stokes. Rend. Sem. Univ. Padova, 31, 1961, pp. 308-40.

[6] P.G. Drazin, W.H. Reid, Hidrodynamic Stability, Cambridge University Press, 1981.

[7] A. Graham, Shear patterns in a unstable layer of air. Phil. Trans. Royal Soc. A 232, 1933, pp. 285-96.

[S] J.G. Heywood, The Navier-Stokes equations: on the existence, regulariy and decay of solutions. Indiana Univ. Math. J . 29, 1980, pp. 639-81.

[9] H. Morimoto, On the existence of weak solutions of equation of natural convection. J. Fac. Sci. Univ. Tokyo 36, 1989, pp. 87-102.

[lo] H. Morimoto, On the existence and Uniqueness of the stationary solution to the equations of natural convection. Tokyo J. Math. 14, 1991, pp. 220-26.

[ll] E. Palm, On the tendency towa.rds hexagonal cells in steady convection, J. Fluid Mech. 8, 1960, pp. 183-92.

[12] E. Palm, T. Ellingsen, B. Gjevik, On the ocurrence of cellular motion in Bknard convection, J . Fluid Mech. 30, 1967, pp. 651-61.

[13] P.H. Rabinowitz, Existence and nonuniqueness of rectangular solutions of the Binard problem, Arch. Rational Mech. Anal. 29, 1968, pp. 32-57.

[14] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.

[15] H. von Tippelkirch, ~ b e r I<onvektionszeller insbesondere im fliissigen Schwefel, Beitrige Phys. Atmos. 20, 1956, pp. 37-54. D

ownl

oade

d by

[M

osko

w S

tate

Uni

v B

iblio

te]

at 2

1:40

07

Febr

uary

201

4

stationary solutions for generalized boussinesq models

Documents