Strong solutions to the incompressible magnetohydrodynamic equations

Research Article

Received 10 August 2009 Published online 28 June 2010 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1338MOS subject classification: 76 W 05; 35 Q 35; 35 D 05; 76 X 05

Strong solutions to the incompressiblemagnetohydrodynamic equations

Qing Chen∗†, Zhong Tan and Yanjin Wang

Communicated by M. Groves

In this paper, we are concerned with strong solutions to the Cauchy problem for the incompressible Magnetohydro-dynamic equations. By the Galerkin method, energy method and the domain expansion technique, we prove the localexistence of unique strong solutions for general initial data, develop a blow-up criterion for local strong solutions andprove the global existence of strong solutions under the smallness assumption of initial data. The initial data are assumedto satisfy a natural compatibility condition and allow vacuum to exist. Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: magnetohydrodynamics; strong solutions; incompressible; vacuum; blow-up

1. Introduction

The incompressible nonhomogeneous Magnetohydrodynamic system (MHD) is the couple of the Navier–Stokes equations andMaxwell equations, see [1], i.e. ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

�t +div(�u)=0

(�u)t +div(�u⊗u)−�u+∇p=curl H×H

Ht −curl (u×H)+curl curl H=0

div u=div H=0

in (0,∞)×R3. (1)

Here �, u, p represent the density, velocity, pressure of the fluid and H is the magnetic field. The nonhomogeneous system canbe seen as a model for the evolution of a multi-phase flow consisting of several immiscible incompressible fluids with differentconstant densities, see [1--3]. For simplifications, we have set the physical constants to be one. We complement the system (1.1)with the Cauchy initial data

(�,�u, H)|t=0 = (�0,�0u0, H0) in R3. (2)

The purpose of this paper is to deal with the strong solutions for the Cauchy problem (1)–(2). This is motivated by our earlierwork [4] for the compressible Magnetohydrodynamic equations. In [4] the authors proved the global existence and convergencerates of smooth solutions for the compressible MHD equations under the condition that the initial data are close to the constantequilibrium state in H3-framework, and by a similar method one can also obtain global strong solutions for the small initial data inthe H2-framework. The essential point in the proof of the global existence of small solutions is that under the assumption the initialdensity is bounded far away from the vacuum. However, if there exists vacuum initially, one cannot expect the global strong orsmooth solutions at all. This is because by considering the particular case H0 =0, it furnishes H=0 and hence the system reducesto the compressible Navier–Stokes equations. It is well known that the smooth or strong solutions will blow up in finite time if theinitial density is compactly supported, no matter how small the initial data are, see Xin [5], Cho and Jin [6], Tan and Wang [7]. Whenthere is vacuum initially, only the local existence of strong solutions can be proved in Fan and Yu [8] under a natural compatibilitycondition. Our interest is that for the incompressible MHD equations whether we can obtain the global existence of unique strong

School of Mathematical Sciences, Xiamen University, Fujian 361005, People’s Republic of China∗Correspondence to: Qing Chen, School of Mathematical Sciences, Xiamen University, Fujian 361005, People’s Republic of China.†E-mail: [email protected]

Contract/grant sponsor: China-NSAF; contract/grant number: 10976026Contract/grant sponsor: NSF; contract/grant number: 10531020

94

Copyright © 2010 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2011, 34 94–107

Q. CHEN, Z. TAN AND Y. WANG

solution when there is vacuum. This is also motivated by the previous results on strong solutions for the incompressible Navier–Stokes equations [9, 10]. We shall prove that when there is vacuum but the initial data satisfy a natural compatibility condition,there exists a unique local strong solution for general large initial data. Then we develop a blow-up criterion for this local solution,and we prove the global existence of strong solutions under the assumption that the initial data are small enough. More precisely,we assume that the initial data satisfy the following regularity and compatibility conditions:

0��0 ∈L32 (R3)∩L∞(R3)∩D1(R3), u0 ∈D1

0(R3)∩D2(R3), H0 ∈H2(R3),

�u0 +curl H0 ×H0 −∇p0 =�120 g for some (p0, g)∈D1(R3)×L2(R3),

div u0 =div H0 =0 in R3.

(3)

Remark 1.1Formally applying the operator div to the magnetic equation (1.1)3, we find that div H is invariable with respect to the time. Thus wehave div H(t)=0 if we prepare it initially, hence Equation (1.1)3 is purely parabolic concerning the magnetic field H. The compatibilitycondition in (3), roughly speaking is equivalent to the L2-integrability of

√�ut at t =0. Indeed, it is natural and plays a crucial role

in deducing the regularity of the time derivatives of u, H.

Notations. For any domain �⊂R3, we denote by Lp(�), Wm,p(�) the usual Lebesgue and Sobolev spaces and Hm(�)=Wm,2(�),with norms |·|Lp , |·|Wm,p , |·|Hm , respectively. Moreover,

Dk(�) = {v ∈L1loc(�)|∇ lv ∈L2(�), |l|=k}, |v|D1 =|∇v|L2 , |v|D2 =|∇2v|L2 ,

D10(�) = Completion ofC∞

0 (�) in |·|D1 , H10(�)=Completion ofC∞

0 (�) in |·|H1 .

Recall that D10(R3)=D1(R3) and D1

0(�)=H10(�) for �⊂⊂R3. For the sake of conciseness, we do not precise in functional space

names when they are concerned with scalar-valued or vector-valued functions. We assume C to be a positive generic constantthroughout this paper that may vary at different places.

Definition 1.1We call (�, u, H) a strong solution to the Cauchy problem (1)–(2) on (0, T) if (�, u, H) satisfies the Equations (1) a.e. in (0, T)×R3 forsome pressure function p, with the regularity⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

�∈L∞(0, T; L32 (R3)∩L∞(R3)∩D1(R3)), �t ∈L∞(0, T; L2(R3)),

u∈L∞(0, T; D10(R3)∩D2(R3))∩L2([0, T]; D1

0(R3)∩D2(R3)∩W2,6(R3)),

H∈L∞(0, T; H2(R3))∩L2([0, T]; D1(R3)∩D2(R3)∩W2,6(R3)),

ut , Ht ∈L2(0, T; D10(R3)),

√�ut , Ht ∈L∞(0, T; L2(R3)).

(4)

Remark 1.2Notice that we do not present pressure p in Definition 1.1 since the existence and regularity of it follows immediately from themomentum conversation equation (1.1)2, the incompressible condition div u=0 and the regularity (4) by a classical consideration,see Simon [3].

We define the functional of (�, u, H) associated with the regularity (4)

�(T) = sup0�t�T

{|(∇�,�t)(t)|L2 +|u(t)|D10∩D2 +|H(t)|H2 +|Ht(t)|L2}+esssup

t∈[0,T]|√�ut|L2

+∫ T

0[|(u, H)(t)|2

D10∩D2 +|(u, H)(t)|2W2,6 +|(ut , Ht)(t)|2

D10

] dt, (5)

the ‘instantaneous’ energy functional

E(t)=|(√�u, H,√

�ut , Ht ,∇u,∇H)|2L2 , (6)

and the initial ‘instantaneous’ energy functional

E(0)=|(√�0u0, H0, H0t ,∇u0,∇H0, g)|2L2 ,

the ‘dissipative’ energy functional

F(t)=|(√�ut , Ht ,∇u,∇H,∇2u,∇2H,∇ut ,∇Ht)|2L2 , (7)

and finally the ‘blow-up’ functional

J(t)=|(∇u(t),∇H(t))|4L2 +|(∇u(t),∇H(t))|8L2 . (8)


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Definition 1.2A finite time T∗ is called the finite blow-up time provided that

�(T)<+∞ for all 0�T<T∗ and limT→T∗ �(T)=+∞. (9)

Remark 1.3As |�(t)|

L32 ∩L∞ is invaluable in time, we do not present it in the functional � of Definition 1.2.

Now our main results can be stated as:

Theorem 1.1Let initial data (�0, u0, H0) be given satisfying (3), we have

(i) (Local existence) there exist a small time T∗ and a unique strong solution on (0, T∗),(ii) (Blow-up criterion) T∗ is the finite blow-up time of (�, u, H) if and only if∫ T∗

0J(t) dt =+∞, (10)

(iii) (Global existence) if in addition, E(0)�� with �>0 a sufficiently small constant, then T∗ =∞.

MHD has spanned a very large range of applications [1, 11--13]. Owing to the physical importance and the mathematical challenges,the study on (MHD) has attracted many physicists and mathematicians. There are many mathematical results related to theincompressible MHD system. For the homogeneous viscous incompressible MHDs equations (i.e. �(t, x)≡ �), there have been extensivestudies. Duraut and Lions [14] constructed a class of global weak solutions and showed the smoothness and uniqueness in thetwo-dimensional case provided given initial data are smooth, but when dimension is three the strong solution is only local exceptfor small data, see also Sermange and Teman [15]. Thus there are many related works to study the regularity criteria for weaksolutions to be smooth in three dimensions, see [15--19]. For the nonhomogeneous case, Gerbeau and Le Bris [20], Desjardins andLe Bris [21] studied the global existence of weak solutions of finite energy in the whole space or in the torus. Local existence ofstrong solutions in some Besov spaces was considered by Abidi and Hmidi [22] and they also showed that the solution is globalwhen the initial data are small, see also Abidi and Paicu [23]. Both [22, 23] allowed variable viscosity and conductivity coefficientsbut required an essential assumption that there is no vacuum (more precisely, the initial data are closed to a constant state). Itis to be noted that we extend the results [19, 22, 23] essentially since we allow vacuum to exist. To prove Theorem 1.1 whenvacuum exists is strongly in the spirit of the papers [9, 10] where the authors proved the local or global strong solutions for theincompressible nonhomogeneous Navier–Stokes equations, however, we should derive the new estimates arising from the presenceof the magnetic field and overcome the strong coupling between the equations. More precisely, to close the uniform estimates,we have to collect all the estimates for u, H, ut , Ht , respectively, and it requires very delicate calculations. For relevant results onincompressible Navier–Stokes equations we refer to the books [2, 24--26] and the references therein. Finally, we let readers refer tothe works [27--30] for the global existence of weak solutions to the multidimensional compressible MHD equations.

The remainder of this paper is devoted to prove Theorem 1.1. In Section 2, we employ the Galerkin method to construct theapproximate solutions. After deriving the uniform estimates we establish the local existence of weak solutions for the Cauchyproblem (1)–(2) by the weak convergence method and domain expansion technique. In Section 3, we improve the regularity ofweak solutions to obtain local existence of unique strong solutions, and we derive the finite blow-up criterion to answer whether thelocal strong solution blows up or not. Under the assumption that the initial data are small enough, we obtain the global existenceof strong solutions.

2. Weak solutions

In this section, we shall prove the local existence of weak solutions for the Cauchy problem (1)–(2) by making use of the domainexpansion technique. We will first prove the local existence of weak solutions for bounded domains via the semi-discrete Galerkinmethod. The uniform estimate for the weak solutions is independent of the size of the underlying domain, hence the case of thewhole space follows by passing to the limit in the domain parameters.

2.1. Bounded domains

We consider the following initial boundary problem⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

�t +div(�u)=0

(�u)t +div(�u⊗u)−�u+∇p=curl H×H

Ht −curl (u×H)+curl curl H=0

div u=div H=0

in (0,∞)×�, (11)

96



where �⊂R3 is a bounded smooth domain and the initial boundary conditions are imposed as

(�,�u, H)|t=0 = (�0,�0u0, H0) in �, (12)

u=0, H ·n = 0, curl H×n=0 on (0, T)×��. (13)

Theorem 2.1Assume that the data �0, u0, H0 satisfy the regularity conditions

0��0 ∈L32 (�)∩L∞(�), u0 ∈H1

0(�)∩H2(�), H0 ∈H2(�) (14)

and the compatibility conditions

div u0 =div H0 =0 in �, H0 ·n=0, curl H0 ×n = 0 on ��, (15)

�u0 +curl H0 ×H0 −∇p0 = �120 g in � (16)

for some (p0, g)∈D1(�)×L2(�), then there exist a time T∗>0 and a weak solution (�, u, H) to the problem (11)–(13) on (0, T∗)such that for any t ∈ (0, T∗),

|�(t)|Lq =|�0|Lq , 32 �q�∞, (17)

sup0�t�T∗

|(√�u, H,∇u,∇H,√

�ut , Ht)|2L2 +∫ T∗

0|(√�ut , Ht ,∇u,∇H,∇2u,∇2H,∇ut ,∇Ht)|2L2 dt�C. (18)

Here the local existence time T∗ and the positive constant C depend only on |�0|L

32 ∩L∞ , |∇u0|L2 , |H0|H2 and |g|L2 , but independent

of the size of the domain �.

2.1.1. Galerkin scheme. We shall employ the semi-discrete Galerkin scheme to construct approximate solutions �m ∈C1([0, T); C2(�)),Hm ∈C1([0, T); Xm), um ∈C1([0, T); Ym) to the initial boundary problem (11)–(13) with the initial data (��

0, u0, H0), where

��0 = J�∗�0 +�, J� is the standard molifier.

It is well known that

0<��0�|�0|L∞ +� and ��

0 →�0 a.e. in � as �→0. (19)

Here we take our basic function space for u

X ={�∈H10(�) | div�=0 in �}

with its finite-dimensional subspaces Xm defined as

Xm =span{�1,. . . ,�m}⊂X ∩C2(�), m=1, 2,. . .

where �m is the mth eigenfunction of the Stokes operator A=−P� in X , with P being the usual projection operator on thedivergence-free vector field. And we take the basic function space for H as

Y ={�∈H1(�) | div�=0 in �, � ·n|�� =0}with its finite-dimensional subspaces Ym defined as

Ym =span{�1,. . . ,�m}⊂Y ∩C2(�), m=1, 2,. . .

where �m is the mth eigenfunction of the operator B in Y , B is the operator defined by the unique solution of a steady-statemagnetic problem (see [1, P. 58]). It has been verified in [1, 26] that {�m}∞m=1, {�m}∞m=1 form the orthogonal bases of the spacesX , Y , respectively.

We are looking for the approximate solutions in the form

�m(t, x)∈C∞([0, T]×�), um(t, x)=m∑

j=1�j(t)�j(x), Hm(t, x)=

m∑j=1

�j(t)�j(x)

solving the following problem

�mt +um ·∇�m = 0, (20)∫�

�mumt ·v dx =

∫�

(−�mum ·∇um +�um +curl Hm ×Hm) ·v dx, v∈Xm, (21)

∫�

Hmt ·w dx =

∫�

[−curl curl Hm +curl (um ×Hm)] ·w dx, w∈Ym (22)


97


equipped with the initial conditions

�m(0)=��0, um(0)=

m∑k=0

(u0,k)L2k, Hm(0)=m∑

j=0(u0,�j)L2�j . (23)

We solve this system (20)–(23) at first locally in time and then derive the uniform bounds for the solutions with respect to the timeto extend the solution globally. Assume that um exists, then the existence of unique solution �m to (20) follows from the classicalcharacteristic method. Indeed, since um is regular, and the trajectory Xm =Xm(s; x, t) of a particle located in x at time t is defined by

d

dsXm =um(Xm(s; x, t), s), Xm(t; x, t)=x, (24)

thus with the aid of um it is divergence-free and vanishes near ��, the solution �m is given by

�m(t, x)=��0(Xm(0; x, t)) ∀(t, x)∈ [0, T]×�. (25)

This yields �m =��0(�um) where � : um →Xm is continuous at least and furnishes the dependence of �m on �j , j =1, . . . , m. Obviously,

we obtain from (2.15) that

�m(t, x)�� and |�m(t)|L

32 ∩L∞ =|��

0|L

32 ∩L∞ . (26)

Thus according to the system (21)–(22), we obtain the following system of 2m nonlinear first-order ordinary differential equations

MZmt =F(Zm), (27)

where the vector Zm = (�1, . . . ,�m,�1, . . . ,�m), the block diagonal matrix M=diag(N, Im) and F denotes the right-hand side ofthe system (21)–(22) by substituting the expression of (26). Here Im is the identity matrix of order m and the entries of N areNi,j = (�m�i ,�j), i, j=1, . . . , m. As �m ∈C∞([0, T]×�) and �m��, according to [31, Lemma 2.5] the matrix N is nonsingular and eachcomponent of its inverse belongs to C∞[0, T]. Hence M is nonsingular and we deduce from (27) that

Zmt =M−1F(Zm)

together with the initial data

Zm(0)= (�01, . . . ,�0

m,�01, . . . ,�0

m) with �0j = (u0,�j), �0

j = (H0,�j), j=1,. . . , m.

Consequently, the standard theory of ordinary differential equations implies that there exist some Tm >0 and a solution Zm(t) onthe time interval (0, Tm). This along with (25) in turn implies the existence of �m in the same time interval.

In order to prove Tm =+∞, we need to derive the uniform estimate for the approximate solutions. For this purpose, by takingv=um in (21), using (20) and the fact that div um =0, and integrating by parts, we obtain

1

2

d

dt

∫�

�m|um|2 dx+∫

�|∇um|2 dx =

∫�

(curl Hm ×Hm) ·um dx. (28)

Similarly taking w=Hm in (17), we have

1

2

d

dt

∫�

|Hm|2 dx+∫

�|∇Hm|2 dx =

∫�

curl (um ×Hm) ·Hm dx

=∫

�(um ×Hm) ·curl Hm dx

= −∫

�(curl Hm ×Hm) ·um dx, (29)

where we have used the elementary vector formula (a×b) ·c=−(c×b) ·a.Thus by combining (28) and (29), we deduce the standard energy identity

1

2

d

dt

∫�

(�m|um|2 +|Hm|2) dx+∫

�|∇(um, Hm)|2 dx =0. (30)

Hence integrating directly in time, it yields

sup0�t�T

{|√

�mum(t)|2L2 +|Hm(t)|2L2}+∫ T

0|∇(um, Hm)(s)|2L2 ds�C. (31)

The bound (31) implies that the obtained approximate solution (�m, um, Hm) exists globally. Indeed, we can prove Tm =+∞ byreductio ad absurdum. In fact, if Tm<+∞, then the solution would tend to infinity in that instant. However, the boundedness ofthe coefficient of Zm follows from (31) together with the bounds of �m (26). Thus we arrive at the contradiction and deduce theglobal existence of um, Hm which in turn implies the global existence of �m through the relations (20).

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2.1.2. Uniform bounds. We shall derive in this subsection some uniform bounds for the approximate solutions we constructedabove. As we always have to estimate |f ·∇g|L2 -like terms, we deal with them with the following lemma:

Lemma 2.2For any f, g∈D1

0(�)∩D2(�), there holds that

|f∇g|L2�C(�)|∇f |2L2 |∇g|L2 +�|∇2g|L2 for any �>0. (32)

ProofIt follows directly from Hölder’s inequality, Sobolev’s inequality and Cauchy’s inequality that

|f∇g|L2�|f |L6 |∇g|L3�C|∇f |L2 |∇g|12L6 |∇g|

12L2�C(�)|∇f |2L2 |∇g|L2 +�|∇2g|L2 . �

We divide the estimates into four steps.

Step 1: Estimate for |(∇um(t),∇Hm(t))|L2

Taking v=umt in (21), by (26) and Cauchy’s inequality, we have∫�

�m|umt |2 dx+ 1

2

d

dt

∫�

|∇um|2 dx = −∫

��m(um ·∇um) ·um

t dx+∫

�(curl Hm ×Hm) ·um

t dx

� 1

2

∫�

�m|umt |2 dx+C

∫�

|um|2|∇um|2dx+∫

�|curl Hm||Hm||um

t |dx. (33)

We estimate the second term in the right-hand side of (33) by Lemma 2.2 as∫�

|um|2|∇um|2 dx�C(�)|∇um|6L2 +�|∇2um|2L2 ,

and estimate the third term as∫�

|curl Hm||Hm||umt |dx � |∇Hm|L2 |Hm|L3 |um

t |L6

� C|∇Hm|L2 |Hm|12L6 |Hm|

12L2 |∇um

t |L2

� C|∇Hm|32L2 |Hm|

12L2 |∇um

t |L2

� C(�)|∇Hm|4L2 |Hm|2L2 +�(|∇umt |2L2 +|∇Hm|2L2 ).

Thus we have

d

dt|∇um|2L2 +|

√�mum

t |2L2�C(�)(|∇um|6L2 +|∇Hm|4L2 |Hm|2L2 )+�(|∇Hm|2L2 +|∇um

t |2L2 +|∇2um|2L2 ). (34)

Taking w=Hmt in (22) and using the fact div Hm =0, we obtain∫

�|Hm

t |2 dx+ 1

2

d

dt

∫�

|∇Hm|2 dx�1

2

∫�

|Hmt |2 dx+ 1

2

∫�

|curl(um ×Hm)|2 dx.

As ∫�

|curl(um ×Hm)|2 dx � C

∫�

(|∇um|2|Hm|2 +|um|2|∇Hm|2) dx

� C(�)(|∇um|2L2 |∇Hm|4L2 +|∇um|4L2 |∇Hm|2L2 )+�(|∇2um|2L2 +|∇2Hm|2L2 )

� C(�)(|∇um|6L2 +|∇Hm|6

L2 )+�(|∇2um|2L2 +|∇2Hm|2L2 ),

thus together with (34), it implies that

d

dt[|∇um|2L2 +|∇Hm|2L2 ]+|

√�mum

t |2L2 +|Hmt |2L2 � C(�)(|∇um|6

L2 +|∇Hm|6L2 +|∇Hm|4L2 |Hm|2L2 )

+�(|∇Hm|2L2 +|∇umt |2L2 +|∇2um|2L2 +|∇2Hm|2L2 ). (35)


99


Step 2: Estimate for |(∇umt (t),∇Hm

t (t))|L2

Differentiating (21) with respect to t, taking v=umt , we can derive

1

2

d

dt

∫�

�m|umt |2 dx+

∫�

|∇umt |2 dx =

∫�

(−�mt um

t −�mt um ·∇um −�mum

t ·∇um −�mum ·∇umt ) ·um

t

+(curlHmt ×Hm) ·um

t +(curlHm ×Hmt ) ·um

t dx.

Differentiating (22) with respect to t, and taking w=Hmt , we obtain

1

2

d

dt

∫�

|Hmt |2 dx+

∫�

|∇Hmt |2 dx =

∫�

[curl(umt ×Hm) ·Hm

t +curl(um ×Hmt ) ·Hm

t ] dx

=∫

�[(um

t ×Hm) ·curlHmt +(um ×Hm

t ) ·curlHmt ] dx

=∫

�[−(curlHm

t ×Hm) ·umt +(um ×Hm

t ) ·curlHmt ] dx.

Combining these two equalities, we deduce that

1

2

d

dt

∫�

(�m|umt |2 +|Hm

t |2) dx+∫

�(|∇um

t |2 +|∇Hmt |2) dx

=∫

�[div(�mum)(um

t +um ·∇um) ·umt −�m(um

t ·∇um) ·umt −�m(um ·∇um

t ) ·umt +curlHm ×Hm

t ) ·umt +(um ×Hm

t ) ·curlHmt ] dx

�∫

�(3|�m||um||um

t ||∇umt |+|�m||um||um

t ||∇um|2 +|�m||um|2|umt ||∇2um|+|�m||um|2|∇um||∇um

t |+|�m||umt |2|∇um|

+|∇Hm||Hmt ||um

t |+|um||Hmt ||∇Hm

t |) dx

=7∑

j=1Ij .

We can estimate I1 ∼ I7 as follows:

I1 � C|�m|12L∞|um|L6 |

√�mum

t |L3 |∇umt |L2

� C|�m|12L∞|∇um|L2 |

√�mum

t |12L2 |√

�mumt |

12L6 |∇um

t |L2

� C|�m|34L∞|∇um|L2 |

√�mum

t |12L2 |∇um

t |32L2

� C(�)|∇um|4L2 |√

�mumt |2L2 +�|∇um

t |2L2 ,

I2 � C|�m|L∞|um|L6 |umt |L6 |∇um|L6 |∇um|L2

� C|�m|L∞|∇um|2L2 |∇umt |L2 |∇2um|L2

� C(�)|∇um|4L2 |∇2um|2L2 +�|∇umt |2L2 ,

I3 � C|�m|L∞|um|2L6 |umt |L6 |∇2um|L2

� C|�m|L∞|∇um|2L2 |∇umt |L2 |∇2um|L2

� C(�)|∇um|4L2 |∇2um|2L2 +�|∇umt |2L2 ,

I4 � C|�m|L∞|um|2L6 |∇um|L6 |∇umt |L2

� C|�m|L∞|∇um|2L2 |∇umt |L2 |∇2um|L2

� C(�)|∇um|4L2 |∇2um|2L2 +�|∇umt |2L2 ,

I5 � C|�m|12L∞|

√�mum

t |L3 |umt |L6 |∇um|L2

� C|�m|34L∞|∇um|L2 |

√�mum

t |12L2 |∇um

t |32L2

10

0



� C(�)|∇um|4L2 |√

�mumt |2L2 +�|∇um

t |2L2 ,

I6 � |∇Hm|L2 |Hmt |L3 |um

t |L6

� C|∇Hm|L2 |Hmt |

12L2 |∇Hm

t |12L2 |∇um

t |L2

� C(�)|∇Hm|4L2 |Hmt |2L2 +�|∇um

t |2L2 +�|∇Hmt |2L2 ,

I7 � |um|L6 |Hmt |L3 |∇Hm

t |L2

� C|∇um|L2 |Hmt |

12L2 |∇Hm

t |32L2

� C(�)|∇um|4L2 |Hmt |2L2 +�|∇Hm

t |2L2 .

Thus we have, by taking � small enough,

d

dt[|√

�mumt |2L2 +|Hm

t |2L2 ]+|∇umt |2L2 +|∇Hm

t |2L2�C(|∇um|4L2 +|∇Hm|4L2 )(|√

�mumt |2L2 +|Hm

t |2L2 +|∇2um|2L2 ). (36)

Step 3: Estimate for |(∇2um(t),∇2Hm(t))|L2

For convenience, we rewrite (21) as∫�

−�um ·v dx =∫

�(curlHm ×Hm −�mum

t −�mum ·∇um) ·v dx,

then by taking v=Aum and Cauchy’s inequality, we obtain

|Aum|2L2 � C

(∫�

|Hm|2|∇Hm|2 dx+∫

�|�mum

t |2 dx+∫

�|�mum ·∇um|2 dx

)� C(�)(|

√�mum

t |2L2 +|∇um|6L2 +|∇Hm|6

L2 )+�(|∇2um|2L2 +|∇2Hm|2L2 ).

Similarly, taking w=curl curl Hm in (22), we have

|curl curl Hm|2L2 � C

[∫�

|Hmt |2 dx+

∫�

|curl(um ×Hm)|2 dx

]� C(�)(|Hm

t |2L2 +|∇um|6L2 +|∇Hm|6

L2 )+�(|∇2um|2L2 +|∇2Hm|2L2 ).

Using these results and with the help of the fact that

|∇2um|2L2�C|Aum|2L2 and |∇2Hm|2L2�C|curl curl Hm|2L2 ,

we get

|∇2um|2L2 +|∇2Hm|2L2�C[|√

�mumt |2L2 +|Hm

t |2L2 +|∇um|6L2 +|∇Hm|6

L2 ]. (37)

Step 4: ConclusionFinally from (30), (35)–(37), we deduce the following inequality which is the key to our results,

d

dt|(√

�mum, Hm,∇um,∇Hm,√

�mumt , Hm

t )|2L2 +|(√

�mumt , Hm

t ,∇um,∇Hm,∇2um,∇2Hm,∇umt ,∇Hm

t )|2L2

�C(|∇um|4L2 +|∇Hm|4L2 +|∇um|8L2 +|∇Hm|8

L2 )×(|∇um|2L2 +|∇Hm|2L2 +|√

�mumt |2L2 +|Hm

t |2L2 +|Hm|2L2 ). (38)

Hence if we define

Em(t) = |(√

�mum, Hm,√

�umt , Hm

t ,∇um,∇Hm)|2L2 ,

Fm(t) = |(√

�mumt , Hm


t )|2L2 ,

Jm(t) = |∇um(t)|4L2 +|∇Hm(t)|4L2 +|∇um(t)|8L2 +|∇Hm(t)|8

L2 .

(39)

We can derive the following three types of energy estimates

dEm(t)

dt+Fm(t)�C(Em(t)3 +Em(t)5), (*)


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1


dEm(t)

dt+Fm(t)�CJm(t)Em(t), (**)

dEm(t)

dt+Fm(t)�C(Em(t)2 +Em(t)4)Fm(t). (***)

2.1.3. Proof of Theorem 2.1 We define

C�,m0 = C(��

0, um(0), Hm(0), p0)

=∫

�(��

0)−1|�um(0)+curl Hm(0)×Hm(0)−∇p0|2 dx, (40)

C�0 = C(��

0, u0, H0, p0)=∫

�(��

0)−1|�u0 +curl H0 ×H0 −∇p0|2 dx, (41)

and

C0 =C(�0, u0, H0, p0)=∫

�(�0)−1|�u0 +curl H0 ×H0 −∇p0|2 dx =|g|2L2 . (42)

By (19) and the fact that

(um(0), Hm(0))→ (u0, H0) in H2(�) as m→+∞, (43)

we can easily derive that

limm→+∞C�,m

0 =C�0 and lim

�→0C�

0 =C0,

hence, there exists �1 >0 sufficiently small such that

C�0�C0 +1 ∀�∈ (0,�1), (44)

and for any fixed �∈ (0,�1), there exists a large number M=M(�) such that

C�,m0 �C�

0 +1�C0 +2 for all m>M(�). (45)

On the other hand, taking v=umt in (21),∫�

�|umt |2 dx =

∫�

(−�mum ·∇um +�um +curl Hm ×Hm) ·umt dx,

we obtain that, by using Cauchy’s inequality and div umt =0,∫

��m|um

t |2 dx�2

∫�

�m|um|2|∇um|2 +(�m)−1|�um −∇p0 +curl Hm ×Hm|2 dx, (46)

and one can obtain similarly from (22) that∫�

|Hmt |2 dx�

∫�

|∇2Hm|2 +|Hm|2|∇um|2 +|um|2|∇Hm|2 dx. (47)

Letting t →0 in (46)–(47), using (43), (45) and recalling the expression of Em(t), we have

Em(0)�C(1+C0). (48)

Then the energy inequality (�) and (48) imply that there exists a small time T∗, independent of �, m and �, such that

sup0�t�T∗

Em(t)�C. (49)

Thus substituting (49) into (�) and integrating in time, we obtain

sup0�t�T∗

|(√

�mum, Hm,∇um,∇Hm,√

�mumt , Hm

t )|2L2 +∫ T∗

0|(√

�mumt , Hm


t )|2L2 dt�C. (50)

For any fixed �∈ (0,�1), we denote by (��,m, u�,m, H�,m) the approximate solution obtained by the Galerkin scheme with the initialdata (��

0, u0, H0). As the uniform bound (50) is independent of �, m and �, by the standard argument, one can easily conclude that

the sequence (��,m, u�,m, H�,m) converges, letting first m→+∞ and later �→0, up to the extraction of subsequences, to some limit(�, u, H) which is a weak solution to the problem (1)–(2) with the original initial data (�0, u0, H0) and satisfies the estimates (17), (18).The proof of Theorem 2.1 is completed. �

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2



2.2. Whole space

Theorem 2.1 implies the local existence of weak solutions for the particular case when �=BR, where BR is the open ball of radiusR centered at 0. As T∗ and C are independent of the size of the domain �, we can prove by the domain expansion technique thelocal of weak solutions for the Cauchy problem (1)–(2).

Theorem 2.3Assume that the data �0, u0, H0 satisfy⎧⎪⎨⎪⎩

0��0 ∈L32 (R3)∩L∞(R3), u0 ∈D1

0(R3)∩D2(R3), H0 ∈H2(R3)

div u0 =div H0 =0, �u0 +curl H0 ×H0 −∇p0 =�120 g in R3,

(51)

then there exist a time T∗ >0 and a weak solution (�, u, H) to the initial value problem (1)–(2) satisfying (17)–(18).

ProofAs C∞

0,(R3) is dense in the space {v ∈H2(R3)| div v =0}, without loss of generality, we can choose HR0 ∈C∞

0,(BR) such that

HR0 →H0 in H2(R3) as R→+∞. (52)

For any fixed R>0, let (uR0 , pR

0)∈ (D10(BR)∩D2(BR))×D1(BR) be the unique solution to the boundary value problem

�uR0 +curl HR

0 ×HR0 −∇pR

0 =�120 g, div uR

0 =0 in BR, uR0 =0 on �BR (53)

and we extend uR0 to R3 by defining 0 outside BR, then we have

uR0 →u0 in D1

0(R3) as R→∞. (54)

To show this, note first that

−�(uR0 −u0)+∇(pR

0 −p0)=curl HR0 ×HR

0 −curl H0 ×H0 in BR. (55)

Thus multiplying this by uR0 and integrating over R3, we obtain∫

R3|∇uR

0|2 dx =∫

R3∇u0 ·∇uR

0 dx+∫

R3(curl HR

0 ×HR0 −curl H0 ×H0) ·uR

0 dx. (56)

Using (52), Hölder’s inequality and Cauchy’s inequality, we obtain

|uR0|

D1,20

�C0. (57)

Hence we can easily deduce that there exists a sequence (Rj), Rj →∞, such that

uRj0 →u0 weakly in D1

0(R3) as j→+∞.

Indeed one can use the uniqueness of solutions for the Stokes equations to improve the above convergence, see [9],

uR0 →u0 weakly in D1

0(R3) as R→+∞. (58)

Then the strong convergence (54) follows from (58) and (56).Now let (�R, uR, HR) be a weak solution of the problem (11)–(13) with the initial data (�0, uR

0 , HR0) constructed by Theorem 2.1 for

�=BR. From (52), (54) and �0 ∈L32 (R3), we obtain for all R>0,

|√�0uR0|L2 +|∇uR

0|L2 +|HR0|H2�C0, (59)

and from (53),

C(�0, uR0 , pR

0 , HR0)=

∫�

�−10 |�uR

0 +curl HR0 ×HR

0 −∇pR0|2 dx =|g|2L2 . (60)

Hence (59)–(60) imply that (�R, uR, HR) also satisfies the uniform estimate (17)–(18) with T∗ and C being independent of R. Thus, ifwe extend (�R, uR, HR) to R3 by defining 0 outside BR, then there exists a sequence (Rj), Rj →∞, such that (�Rj , uRj , HRj ) convergesto a limit (�, u, H) which is a weak solution of the problem (1)–(2) satisfying (17)–(18). This completes the proof of Theorem 2.3. �


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3


3. Proof of Theorem 1.1

3.1. Local existence

For more regularity initial data satisfying (3), we can improve the regularity of weak solutions constructed in Theorem 2.3 to obtainthe local existence of strong solutions to the problem (1)–(2).

For this, let (�0, u0, H0) be the initial data in Theorem 1.1, then there exists a weak solution (�, u, H) on (0, T∗) to the system (1)–(2)by Theorem 2.3. By the regularities (17)–(18), we observe that u is a strong solution of the Stokes equations

−�u+∇p=−�ut −�u ·∇u+curl H×H and div u=0 in R3, (61)

for some pressure p. Using the classical regularity results, see [26], we have

|∇2u|L6 � C|−�ut −�u ·∇u+curl H×H|L6

� C(|�|L∞|ut|L6 +|�|L∞|u|L∞|∇u|L6 +|curl H|L6 |H|L∞ ). (62)

This together with (17)–(18) implies ∫ T∗

0|u|2W2,6 dt�C. (63)

Similarly, since H is a strong solution of the elliptic equations

−�H=−Ht +curl (u×H), (64)

using the classical elliptic regularity results, see [32], we can obtain∫ T∗

0|H|2W2,6 dt�C. (65)

To prove that (�, u, H) is a strong solution, it remains to improve the regularity of the density. As � is just a weak solution to theproblem

�t =−∇� ·u, (66)

where u is divergence-free and satisfies the regularity in (18), we only can do some a priori estimates by assuming that � is smoothfor the moment. Taking ∇ to equation (66), multiplying by ∇� and integrating over R3, we obtain

1

2

d

dt

∫R3

|∇�|2 dx = −∫

R3(∇2�·∇u+∇�·∇u) ·∇�dx

� −∫

R3∇(

|∇�|22

) ·u dx+∫

R3|∇u||∇�|2 dx

� |∇u|L∞|∇�|2L2 .

Hence using Gronwall’s inequality, Sobolev’s inequality and (63), we derive

|�(t)|D1�C exp

{∫ t

0C|∇u|L∞ ds

}�C exp

{C

∫ T∗

0|∇u|W1,6 ds

}�C ∀t ∈ (0, T∗). (67)

Owing to (67),the standard argument can be applied to prove that � is indeed a strong solution to (66),and then the regularityproperty of �t follows from the equation (66) and (67), (18). Hence we prove the local existence of strong solutions to theproblem (1)–(2).

The uniqueness of strong solutions is stated as follows.

Theorem 3.1 (Uniqueness)Let (�, u, H), (�, u, H) be two strong solutions to the Cauchy problem (1)–(2) on (0, T) with the same initial data (�0, u0, H0) satisfying (3),then �= �, u= u, H=H.

ProofTaking the difference of the two momentum equations,

�ut +�u ·∇u−�u+∇(

p+ |H|22

)= div(H⊗H),

�ut + �u ·∇u−�u+∇(

p+ |H|22

)= div(H⊗H),

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4



where we have used the vector formula

curl v×v=div(v⊗v)−∇(

|v|22

)for div v=0,

hence, we obtain

�(u−u)t +�u ·∇(u−u)−�(u−u)+∇(

p− p+ |H|22

− |H|22

)=div[(H−H)⊗H+H⊗(H−H)]−(�− �)(ut +u ·∇u)−�(u−u) ·∇u.

Then multiplying this identity by (u−u), integrating over R3, using Höler’s, Sobolev’s and Cauchy’s inequalities, we have

1

2

d

dt

∫R3

�|u−u|2 dx+∫

R3|∇(u−u)|2 dx

�∫

R3|H−H|(|H|+|H|)|∇(u−u)|+|�− �||ut +u ·∇u||u−u|+�|u−u|2|∇u|dx

�C(|H|L6 +|H|L6 )|H−H|L3 |∇(u−u)|L2 +|�− �|L

32|ut +u ·∇u|L6 |u−u|L6 +|∇u|L∞|√�(u−u)|2L2

�C(�)(|∇H|4L2 +|∇H|4L2 )|H−H|2L2 +|ut +u ·∇u|2L6 |�− �|2L

32

)+�(|∇(u−u)|2L2 +|∇(H−H)|2L2 )+|∇u|L∞|√�(u−u)|2L2 . (68)

Similarly, taking the difference of the two magnetic equations, we have

(H−H)t +curl curl (H−H)=curl ((u−u)×H)+curl (u×(H−H))

Multiplying this identity by (H−H), integrating over R3, we get

1

2

d

dt|H−H|2L2 +|∇(H−H)|2L2 � (|∇(u−u)|L2 |H|L6 +|u−u|L6 |∇H|L2 +|∇u|L2 |H−H|L6 +|u|L6 |∇(H−H)|L2 )|H−H|L3

� �(|∇(u−u)|2L2 +|∇(H−H)|2L2 )+C(�)(|∇u|4L2 +|∇H|4L2 )|H−H|2L2 . (69)

Now using the identity

(�− �)t +∇(�− �) ·u=−∇� ·(u−u)

we can deduce that

d

dt|�− �|

32

L32�3

2

∫|u−u||∇�||�− �| 1

2 dx�3

2|u−u|L6 |∇�|L2 |�− �|

12

L32

.

Thus

d

dt|�− �|2

L32

= 4

3|�− �|

12

L32

d

dt|�− �|

32

L32

� C|∇(u−u)|L2 |∇�|L2 |�− �|L

32

� �|∇(u−u)|2L2 +C(�)|∇�|2L2 |�− �|2L

32

. (70)

Consequently, it follows from (68) to (70) that, by taking � small enough,

d

dt(|�− �|2

L32

+|√�(u−u)|2L2 +|H−H|2L2 )+|∇(u−u)|2L2 +|∇(H−H)|2L2�A(t)(|�− �|2L

32

+|√�(u−u)|2L2 +|H−H|2L2 ), (71)

for some nonnegative function A(t)∈L1(0, T), thanks to the regularities of the solutions and T is finite.Therefore an application of Gronwall’s inequality completes the proof of Theorem 3.1. �

Remark 3.1The proof of Theorem 3.1 is somehow formal and the rigorous proof can be used by the regularizing method of [2]. Revisiting theproof, one can also obtain the uniqueness for less regular solutions or the weak–strong uniqueness theorem.


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3.2. Blow-up criterion

Suppose that T∗ is the finite blow-up time of the strong solution (�, u, H). Exactly as in the derivation of (��) in Section 2 and (63),(65), (67),one can obtain

�(T)�C exp(CJ(T)) for 0�T <T∗. (72)

The regularity of (�, u, H) is not enough to justify the derivation of (72). One can be proved rigorously by the standard method,using the mollifier technique.

Hence if T∗ is the finite blow-up time, by the Definition 1.2, limT→T∗ �(T)=∞, then by (3.12) we have (1.10). And the converseis obvious. This implies the assertion (ii) of Theorem 1.1.

3.3. Global existence

Suppose that (�, u, H) blows up at some finite time T∗ with T∗ <∞. We can get the analogous derivative form as in (��) in Section 2with Em(t) and Fm(t) replaced by E(t) and F(t), respectively. Integrating it from 0 to t, we have

E(t)+∫ t

0F(s) ds�E(0)+C1

∫ t

0[E(s)2 +E(s)4]F(s) ds, 0< t <T∗, (73)

for some constant C1 >1 and here E(t), F(t) are defined in (6)–(7).We choose � such that

0<�<1

4C1<

1

2. (74)

ClaimE(t)<2�, 0�t <T∗.

As E(0)<�, suppose that there is a first time T <T∗ such that

E(t)<2� for t <T and E(T)=2�. (75)

Hence we deduce from (73) and the choice of � in (74) that

E(t)+∫ t

0F(s) ds��+2C1�

∫ t

0F(s) ds��+ 1

2

∫ t

0F(s) ds ∀0�t <T.

This implies E(t)�� for all t <T , which contradicts (75). Then the claim is proved.Consequently the claim implies that ∫ T∗

0J(t) dt�

∫ T∗

0[E(s)2 +E(s)4] ds�CT∗ <+∞.

This contradicts the blow-up criterion (10). Thus we prove the global existence of strong solutions. This proves the assertion (iii) inTheorem 1.1 and the proof of Theorem 1.1 is completed. �

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Strong solutions to the incompressible magnetohydrodynamic equations