A global existence result in Sobolev spaces for MHD system in the half-plane

A Global Existence Result in Sobolev Spaces for MHD System in the Half-Plane.

EMANUELA CASELLA(*) - PAOLA TREBESCHI(*)

ABSTRACT- The main result of this paper is a global existence theorem in suitable Sobolev spaces for 2D incompressible MHD system in the half-plane. The existence result derives by the existence of a global classical solution in Hölder spaces, by proving some a-priori estimates in Sobolev spaces and, finally, by applying the Banach-Caccioppoli fixed point theorem. Hence, the uniqueness of the solution follows.

1. Introduction.

Let V be the half-plane R²₁»4 ](x₁,x2)R²: x1D0(, and let G be the boundary of V. In QT»4V3( 0 , T), with TD0, we consider the equations of magneto-hydrodynamics for 2D incompressible ideal fluid

ut1(uQ˜)u1˜p1 1

2˜NBN²2(BQ˜)B40 in QT, (1)

B_t1(uQ˜)B2(BQ˜)u2mDB40 in Q_T, (2)

divu40 in QT, (3)

divB40 in QT, (4)

(*) Indirizzo degli AA.: Dip. di Matematica, Università di Brescia, Facoltà di Ingegneria, Via Valotti 9, 25133 Brescia.

uQn40 on G3( 0 ,T), (5)

BQn40 on G3( 0 ,T), (6)

rotB40 on G3( 0 , T), (7)

u(x, 0 )4u₀(x) in V, (8)

B(x, 0 )4B₀(x) in V. (9)

Here u4u(x,t)4(u¹(x,t),u²(x,t) ), B4B(x,t)4(B¹(x,t), B²(x,t) ) andp4p(x,t) denote the unknown velocity field, the magnetic field and the pressure of the fluid respectively. The functionsu₀4(u₀¹(x),u₀²(x) ) andB04(B₀¹(x),B0²(x) ) denote the given initial data,n the unit outward normal on G and m a real positive constant. Moreover, we use the notation

f_t4 ¯f

¯t , ¯_i4 ¯

¯xi

, ˜4(¯₁,¯₂), uQ˜4u¹¯₁1u²¯₂,

¯²_ij4 ¯²

¯_i¯_j , D4¯₁₁² 1¯₂₂² .

In case the magnetic fieldBis identically equal to zero, i.e. in the case of Euler equations, such a problem for global classical solutions was stu- died by many authors, starting from Lichtenstein [10] and Wolibner [15].

The existence of global solutions in Hölder spaces in bounded domains has been proven by Kato [6]. This result was extended to the exterior do- main case by Kikuchi [8]. On the other hand, the existence of a classical solution for MHD system was shown by Kozono [9] and by Casella, Sec- chi and Trebeschi [5] in the bounded and unbounded case, respectively.

Existence results in Sobolev spaces were proved by several authors.

For the Euler equation we refer to Temam [14], Kato and Lai [7] and Beira˜o Da Veiga [3], [4]. Existence and uniqueness results inW^k-spaces for the equations of magneto-hydrodynamics, when m40 , have been proved by Alexseev [1]. Moreover, in this case, Secchi [12] and Schmidt [11] proved not only existence and uniqueness results, but also the con- tinuous dependence on the data. In this paper we prove a global exis- tence result in suitable Sobolev spaces for MHD system in the half-plane case. To prove this result, firstly, we show a local existence theorem in Sobolev spaces. Then we derive some a-priori estimates, global in time, which come from the all-time existence of classical solution of system (1)-

(9) in Hölder-spaces. We underline that energy-method works well, since the classical solution (u,B) is such that Vu(t)V_LQ(V), VB(t)V_LQ(V), V˜u(t)V_LQ(V),V˜B(t)V_LQ(V)andVBt(t)V_LQ(V)are uniformly bounded in time on the whole interval [ 0 ,T].

We observe that the main result obtained in the present paper is a necessary first step in the analysis of slightly compressible MHD fluids, which will be the object of a forecoming work.

The plan of the paper is the following. In next section we fix some no- tations and we introduce some preliminary results and the main theo- rem. In Section 3 we show some a-priori estimates, and finally in Section 4 we prove the main result.

2. – Notations and results.

For a scalar-valued function f, we set Rotf4(¯₂f,2¯₁f) ,

for a vector-valued function u4(u¹,u²), we use the notation rotu4¯₁u²2¯₂u¹ and divu4˜Qu4¯₁u¹1¯₂u².

We denote the norm ofL^p(V), 1GpG Q, byVQV_Lp.H^m(V) denotes the usual Sobolev space of ordermF1 , andVQV_Hmdenotes its norm. For sim- plicity we use the abbreviated notation L^p,H^m. We also use the same symbol for spaces of scalar and vector-valued functions.

Moreover, if X is a normed space, then L^p(0,T;X), with 1GpE1Q, denotes the set of all measurable functionsu(t) with values inXsuch that:

VuV_Lp(X)»4

u

0 T

Vu(t)V_X^pdt

v

where VQV_X is the norm in X.

GivenTD0 arbitrary, the set of all essentially bounded (with respect to the norm ofX) measurable functions oftwith values inXis denoted by L^Q( 0 , T; X). We equip this space with the usual norm

VfV_LQ(X)4 sup

t[ 0 ,T]

Vf(t)V_X.

In particular, the norm of L^Q( 0 ,T;L^p), 1GpE 1Q, is denoted by VQV_LQ(L^p).

Let C^{m( [ 0 , T];}X) denote the set of allX-valued m-times continuou- sly differentiable functions of t, for 0GtGT.

We define X^m(T)»4^m_k4

1

VuV²_Xm»4sup

[ 0 ,T]

!

k40 m21

V¯_t^ku(t)V_H2^m2k.

We denote by B^{(V) (resp.} B^(QT)) the Banach space of all real valued continuous and bounded functions on V (resp. Q_T), with the usual norm.

For 0EaE1 , C^a(V) denotes the usual space of functions in B^(V), uniformly Hölder continuous onVwith exponenta; the norm ofC^a(V) is VQV_LQ1[Q]_a, where

[f]_a»4 sup

xcy,x,yV

Nf(x)2f(y)N Nx2yN^a

.

For 0EaE1 and integerk,C^k1a(V) denotes the space of functions f with D^bfB^{(V) for}_NbN Gk, and D^gfC^a(V) forNgN 4k. The norm is

NfN^k1a4max

NbN Gk

VD^bfV_LQ1max

NgN 4k[D^gf]_a.

WithC^k,j(QT) for integersk,jF0 we mean the set of all functionsffor which every ¯_x^q¯_t^rf exists and is continuous on QT, for 0G NqN Gk, 0GrGj.C^k1a,j1b(QT), for integersk,jF0 and 0Ga,bE1 is the sub- set of C^k,j(QT), consisting of Hölder continuous functions with expo- nents a in x and b in t.

For every function fC^k1a,j1b(QT), we consider the following seminorm:

[f]_a,b»4 sup

xcy,t[ 0 ,T]

Nf(x,t)2f(y,t)N

Nx2yN^a 1 sup

tcs,xV

Nf(x,t)2f(x,s)N Nt2sN^b

, and the norm

NfN^k1a,j1b»4 max

NqN Gk,rGj sup

(x,t)Q_T

N¯^q_x¯^r_tf(x,t)N 1max

NqN 4k[¯^q_x¯^j_tf]_a,b. We shall denote by C and by C_i,iN, some real positive constants which may be different in each occurrence, and byC_Q(t) a real function in L^Q( 0 , T) depending on Vu(t)V_LQ,VB(t)V_LQ,V˜u(t)V_LQ, V˜B(t)V_LQ, VB_t(t)V_LQ and some their suitable powers.

We now set Z»4rotu,j»4rotB,Z₀»4rotu₀ and j0»4rotB₀. By applying rot to both sides of equations (1) and (2), we get

Zt1uQ˜Z2BQ˜j40 , (10)

j_t1uQ˜j2BQ˜Z12¯1u¹DB12¯2B²Du2mDj40 , (11)

whereDu4¯1u²1¯2u¹, andDB4¯1B²1¯2B¹. Finally, letF,f,F^,c be defined as

F4 2uQ˜j1BQ˜Z22¯₁u¹DB22¯₂B²Du, f(s)»4

!

k40 3

V¯_t^kj(s)V_H21, F^(s)»₄

!

k40 3

V¯_t^kF(s)V_H232k, c(s)»4

!

k40 3

V¯_t^kj(s)V_H252k.

We now recall a result (see [5]) which will be fundamental to prove that, under suitable assumptions on initial data, the classical solution of problem (1)-(9) belongs to suitable Sobolev spaces.

THEOREM 2.1. Let TD0 be arbitrary. Let u₀C^{11u(V), rotu}0 L¹(V), B₀C^21u(V)OH¹(V) for some 0EuE1, such that divu₀4 4divB040 inVand u0Qn4B0Qn40 onG. Then there exists a positive constant C

* such that, ifVB0V_H11NB0N²1uGC

*,then there exists a solu- tion ]u,B,p(C^{1 , 1(Q}T)3C^{2 , 1(Q}T)3C^{1 , 0(Q}T)of system (1)-(9). Such a solution is unique up to an arbitrary function of t which may be added to p.

REMARK 2.2. In [5] Kozono’s result obtained in [9] and Kikuchi’s re- sult, see [8], are extended to the exterior domain case and to the half- plane case, and to the MHD equations, respectively. In [5] the existence of the global classical solution for MHD system in Hölder spaces is proved by applying the Schauder fixed point theorem. The authors fol- lowed the idea of Kato [6], Kikuchi [8], Kozono [9]. The crucial step is the definition of a map, defined on a suitable class, the same already consid- ered by Kozono in [9], which satisfies the conditions of the Schauder fixed point theorem. The uniqueness of the solutions of the studied pro- blem is obtained by following standard tecniques, see Temam [13].

The main result, we are going to prove, is:

THEOREM 2.3. Let TD0 be arbitrary. Let the couple (u₀,B₀) H⁵OL¹,rotu₀L¹, divu₀4divB₀40inVand u₀Qn4B₀Qn40onG.

Assume also that, for some 0EuE1 , VB0V_H11 NB0N²1uGC

*, where C

* is the constant obtained in Theorem 2.1.

Then problem (1)-(9) has a unique solution (u,B,p) such that uX⁵(T), BX⁵(T)OL²( 0 ,T;H⁶), ˜pX⁴(T).

3. Some a-priori estimates.

We devote this section to prove

LEMMA 3.1. The following energy-type estimate 1

2 d

dt

(

)

(

)

(12)

holds in [ 0 ,T].

Note that Theorem 2.1 ensures us that the time functionsVu(t)V_LQ(V), VB(t)V_LQ(V), V˜u(t)V_LQ(V),V˜B(t)V_LQ(V), and VB_t(t)V_LQ(V) are uniformly bounded in time on the whole interval [ 0 ,T]. Consequently, the real functionC_Q(t) (appearing in (12) and in some preliminary lemmata given below) belongs to L^Q( 0 ,T). We shall prove (12) for regular sol- utions.

LEMMA 3.2. The couple (u,B) satisfies the following energy-type estimate

VuV_L2_Q(L2)

1VBV_L2_Q(L2)

12mVjV_L22(Q_T)

GVu₀V_L22

1VB₀V_L22. (13)

PROOF. We multiply equations (1) and (2) by u and B respectively.

By standard calculations and by summing the resulting expressions, we get easily the thesis. r

LEMMA 3.3. The following inequality holds VZV_L2Q(L²)

1VjV_L2Q(L²)

1 m 2

V˜jV_L22(Q_T)

GC(VZ₀V_L22

1Vj0V_L22).

PROOF. We multiply equations (10) and (11) by Z and j respectively.

Since

2

V

(BQ˜)jQZ dx4

V

(BQ˜)ZQjdx,

by summing the resulting expressions, we obtain (14) 1

2 d

dt(VZ(t)V_L²²1Vj(t)V_L22)12

V

(¯₁u1DB1¯₂B2Du)jdx1

1mV˜j(t)V_L22

40 . SinceV˜u(t)V_L2GCVZ(t)V_L2andV˜B(t)V_L4GCVj(t)V_L1 /22V˜j(t)V_L1 /22, we easily obtain that

(15) 2

V

N(¯₁u₁DB1¯₂B₂Du)jNdxG m 2

V˜j(t)V_L22

1CVZ(t)V_L22Vj(t)V_L22. By collecting (14) and (15), we get

1 2

d

dt(VZ(t)V_L22

1Vj(t)V_L22)1 m

2V˜j(t)V_L22

GCVj(t)V_L22(VZ(t)V_L22

1Vj(t)V_L22).

The thesis follows by using Lemma 3.2 and the Gronwall lem-

ma. r

The next step is to estimate theL^Q( 0 ;T; L²)-norm of¯^aZ, wherea is a multi-index such that 1G NaN G4 . We get the following result.

LEMMA 3.4. Let eD0 . Then the following inequality 1

2 d dt

V¯^aZ(t)V_L22

GC_Q(t)V¯^aZ(t)V_L22

1eVj(t)V_H25, holds in [ 0 ,T], where C_Q depends also on e.

PROOF. By applying ¯^a to both sides of equation (10), we get

¯^aZt1(uQ˜)¯^aZ4 2[¯^a,uQ˜]Z1¯^a( (BQ˜)j), (16)

where [Q,Q] denotes the commutator operator. We now multiply (16) by

¯^aZ and we estimate term by term. We use the Hölder and Young in- equalities and some suitable interpolation inequalities (obtained by the

well-known Gagliardo-Nirenberg one). More precisely, VD²uV_L4GCV˜uV_L1 /2QVZV_H1 /22, (17)

VD²BV_L4GCV˜BV_L1 /2QVjV_H1 /22, (18)

VD³uV_L4GCV˜uV_L1 /2QVZV_H1 /24, (19)

VD³BV_L4GCV˜BV_L1 /2QVjV_H1 /24, (20)

VD⁴uV_L4GCV˜uV_L1 /6QVZV_H5 /64, (21)

VD⁴BV_L4GCV˜BV_L1 /6QVjV_H5 /64, (22)

VD²BtV_L4GCVBtV_L1 /2QVjtV_H1 /23. (23)

By using (17)-(23), we easily obtain the thesis. r LEMMA 3.5. The following estimate

1 2

d

dtf(t)1C₁c(t)GC₂(F^(t)₁^{f(t) )} holds in [ 0 ,T].

PROOF. We write (11) in the form ¯_tj2mDj4F. For each integer k41 ,R, 4 , we take (k21 ) time derivatives and we obtain the follow- ing problems

. /´

¯_t^kj2mD¯^k_t²¹j4¯^k_t²¹F in V,

¯_t^k21j40 on ¯V. (24)

For each fixedk, we multiply the first equation of (24) by¯_t^k21j and by 2D¯tk21

j. By using the Hölder inequality one has (25) 1

2 d

dtV¯_t^k21j(t)V_H²¹1m

2

(

1VD¯tk21j(t)V_L22

)

GC

(

1V¯_t^k21j(t)V_L22

)

. /´

2mD¯^k21_t j4 2¯_t^kj1¯^k21_t F in V,

¯_t^k21j40 on ¯V. (26)

By well-known results on the regularity of the solutions of problems

(26), we get

V¯_t^k21j(t)V_Hm12GC(V¯_t^kj(t)V_Hm1V¯_t^k21j(t)V_Hm1V¯_t^k21F(t)V_Hm).

(27)

We now sum (25) fork41 ,R, 4 , and we add to both sides of the re- sulting expression the term

m

2

g

^!

h

By observing that Vj_ttt(t)V_H2 is equivalent to Vj_ttt(t)V_H11VDj_tttV_L2, we get

1 2

d

dtf(t)1C₁c(t)GC₂

g

^!

h

We use inequality (27) firstly fork41 , 2 , 3 and m442k, and again in the casesk41 , 2 and m41 . By summing the resulting expressions we obtain the thesis. r

We now estimate each term appearing in F(t). The result we are go- ing to show is

LEMMA 3.6. The following inequality

F^(t)_G^C_Q^(t)(f(t)₁^VZ(t)VH2⁴) holds in [ 0 ,T].

PROOF. We split the proof of the previous statement in several steps.

As first step we write explicity VF(t)V_H3. By using the Hölder and Gagliardo-Nirenberg inequalities, we easily obtain

VF(t)V_H23

GC_Q(t)(Vj(t)V_H24

1VZ(t)V_H24).

By using (27) in the following cases (k,m)4( 1 , 2 ), (k,m)4( 2 , 0 ), and finally (k,m)4( 1 , 0 ), one has

Vj(t)V_H24

GC

g

^!

h

By straightfull calculations, we get VF(t)V_H22

GC_Q(t)(Vj(t)V_H21

1Vjt(t)V_H21

1VZ(t)V_H24), VFt(t)V_L22

GC_Q(t)(Vj(t)V_H21

1Vjt(t)V_H21

1VZ(t)V_H24).

Hence

VF(t)V_H23

GC_Q(t)(f(t)1VZ(t)V_H24).

In order to estimate VF_t(t)V_H2,VF_tt(t)V_H1 and VF_ttt(t)V_L2, we follow the same lines as in the previous step, and we consider the following interpo- lation inequalities

VD²BV_L8GCV˜BV_L3 /4QVjV_H1 /44, (28)

VD²uV_L8GCV˜uV_L3 /4QVZV_H1 /44, (29)

VD³BV_L8GCVBV_L5 /16QVjV_H11 /164 , (30)

VD³uV_L8GCVuV_L5 /16Q VZV_H11 /164 , (31)

VD⁴BV_L4GCV˜BV_L3 /8QVjV_H5 /85, (32)

VD⁴BV_L8 /3GCV˜BV_L1 /4QVjV_H3 /44, (33)

VD⁵BV_L8 /3GCV˜BV_L3 /16Q VjV_H13 /165 , (34)

VD³B_tV_L8 /3GCVB_tV_L1 /4QVjtV_H3 /43. (35)

By virtue of the Hölder inequality, of (17)-(23) and of (28)-(35) we get

VF_t(t)V_H22

1VF_ttt(t)V_L22

GC_Q(t)(f(t)1VZ(t)V_H24), VF_tt(t)V_H21

GC_Q(t)

(

)

By (35), by recalling that BtL^Q(V), and by using again (27), we get

VF_tt(t)V_H21

GC_Q(t)

(

1Vj_t(t)V²_H3

)

GC_Q(t)(f(t)1VZ(t)V_H24

1Vjt(t)V²_H11 1Vjtt(t)V²_H11VF_t(t)V²_H1)_G

GC_Q(t)

(

)

Hence, the claim follows. r

By collecting Lemmata 3.3-3.6 we obtain inequality (12).

4. – Proof of Theorem 2.3.

The first topic which we treat is a local existence result in Sobolev spaces for system (1)-(9). Obtained that the classical solution of (1)-(9) belongs locally to H⁵(V), from the a-priori estimates (12) and (13) we can extend such a solution on the whole time interval [ 0 ,T].

PROOF. In order to show the local existence of a solution in Sobolev spaces, we apply the Banach-Caccioppoli theorem. In particular, let 0EtA

GT be sufficiently small and let

S»4 ](u,B)L^Q( 0 ,tA;H⁵(V) ) : V(u,B)V_LQ( 0 ,tA;H⁵)G2A(, whereAis a real positive constant such thatADC₀(Vu₀V_H25

1VB₀V_H25) for a suitable constant C₀, which will be fixed later.

Given the couple (u,B) in S and satisfying (3)-(9), let L:SKL(S)

be the map defined by

U»4(u,B)KUA»4(uA,BA) ,

where UA»4(uA,BA) is the solution of the following linear system uA_t1(uQ˜)uA 2(BQ˜)BA

41

2˜NBN² in QT, (36)

BA

t1(uQ˜)BA

2(BQ˜)uA 2mDBA

40 in Q_T, (37)

divuA 40 in Q_T, (38)

divBA

40 in Q_T, (39)

uAQn40 on G3( 0 , T), (40)

BAQn40 on G3( 0 , T), (41)

rotBA

40 on G3( 0 ,T), (42)

uA(x, 0 )4u₀(x) in V, (43)

BA(x, 0 )4B0(x) in V. (44)

We now show that the mapLsatisfies all the assumptions of the Banach- Caccioppoli theorem. We now apply to both sides of equation (36)-(37) rot and¯^a, wherea is a multi-index with NaN G4 . We multiply the re- sulting expressions by the test functions¯^aZAand¯^ajA, whereZA»4rotuA andjA»4rotBA. By suitable integrations by parts, and by using the Höld- er and Young inequalities, an application of the Gronwall lemma yields that

max

t[ 0 ,tA]

VUA(t)V_H25

G

{

0 tA

VU(t)V_H45

dt

}

where VU( 0 )V_H25

4Vu0V_H25

1VB0V_H25. Since U belongs to S, we get max

t[ 0 ,tA]

VUA(t)V_H25

G(A1C tA( 2A)⁴)e^{C tA( 112A)2}.

Consequently, iftA is small enough, Lmaps the setSinto itself. We now show that L is a contraction with respect to L^Q( 0 ,tA; L²)-norm. Let UA

1»4(uA₁,BA

1) and UA

2»4(uA₂,BA

2) be solutions of system (36)-(44). We now consider the difference between equations (36), written fori41 , 2, and equations (37), again written for i41 , 2 . We use as test functions uA₁2uA₂ and BA

12BA

2 respectively. By standard arguments, we get VUA

12UA

2V_L2Q( 0 ,tA;L²)

GC tAe^4A2tAVU12U2V_L2Q( 0 ,tA;L²).

IftA is sufficiently small,Lis a contraction and the unique fixed point of the map L is a solution of system (1)-(9). The thesis follows by the uniqueness of the classical solution and by using (12) and (13). r

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Manoscritto pervenuto in redazione il 17 aprile 2002.