A regularity criterion for the density-dependent magnetohydrodynamic equations

Research Article

Received 20 March 2009 Published online 15 December 2009 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/mma.1255MOS subject classification: 35 Q 35; 76 N 10

A regularity criterion for thedensity-dependent magnetohydrodynamicequations

Yong Zhoua∗† and Jishan Fanb

Communicated by X. M. Wang

In this paper, regularity criterion for the 3-D density-dependent magnetohydrodynamic equation is considered. It isproved that the solution keeps smoothness only under an integrable condition on the velocity field in multiplier spaces.Hence, it turns out that the velocity field plays a dominant role in the regularity criteria of the weak solutions to thisnonlinear coupling problem even with density. Copyright © 2009 John Wiley & Sons, Ltd.

Keywords: magnetohydrodynamics; regularity criterion; multiplier spaces; Morrey spaces

1. Introduction

We consider the following density-dependent magnetohydrodynamic (MHD) equations:

div u = 0, (1)

�t�+div (�u) = 0, (2)

�t(�u)+div (�u⊗u)+∇�−�u = curl H×H, (3)

�tH−�H = curl (u×H), (4)

div H = 0, (5)

in (0,∞)×R3 with the initial conditions

(�, u, H)|t=0 = (�0, u0, H0), div u0 =div H0 =0 in R3. (6)

Here the unknown functions �, u, � and H are the density, velocity field, pressure and the magnetic field, respectively.The existence of unique local smooth solution to problem (MHD) (1)–(6) has been proved in [1, 2]. More precisely, it is shown in

[2] that if the initial data �0, u0 and H0 satisfy

∇�0 ∈L2 ∩Lq, 0<m��0�M<∞, u0, H0 ∈H2 (7)

aDepartment of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People’s Republic of ChinabDepartment of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, People’s Republic of China∗Correspondence to: Yong Zhou, Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People’s Republic of China.†E-mail: [email protected]

Contract/grant sponsor: Program for New Century Excellent Talents in Universities; contract/grant number: NCET 07-0299Contract/grant sponsor: ZJNSF; contract/grant number: R6090109Contract/grant sponsor: NSFC; contract/grant number: 10971197

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Y. ZHOU AND J. FAN

for some 3<q�6, then there exists a positive time T∗ ∈ (0,∞] and a unique strong solution (�, u,�, H) to the problem (MHD) (1)–(6),such that

∇� ∈ C([0, T∗]; L2 ∩Lq), �t ∈C([0, T∗]; L2 ∩Lq), 0<m��M<+∞,

(u, H) ∈ C([0, T∗]; H2)∩L2(0, T∗; W2,6),

(ut, Ht) ∈ L∞(0, T∗; L2)∩L2(0, T∗; H1),

� ∈ L∞(0, T∗; H1)∩L2(0, T∗; W1,6),

(8)

When H≡0, the system (1)–(3) is the well-known density-dependent Navier–Stokes equations, which has been received manystudies (see [3--5]). In [4], Kim proved the following regularity condition:

u∈Ls(0, T; Lp,∞(R3)) for any (p, s) with2

s+ 3

p=1, 3<p�∞, (9)

where Lp,∞ is the Lorentz space (weak Lp space).As �≡1, the system reduced to the standard MHD equation. Various regularity criteria have been established for the weak

solutions in terms of the velocity field only [6--11]. Recently, regularity was also guaranteed by adding conditions on the pressure[12] (Fan and Zhou, 2008, submitted.).

The aim of this paper is to establish a weaker regularity criterion for this general model on the velocity only. For this purpose,let us introduce the following multiplier spaces M(Hr , L2). By a multiplier acting from one functional space, S1 into another, S2, wemean a function that defines a bounded linear mapping of S1 into S2 by pointwise multiplication. Thus, with any pair of spacesS1, S2, we associate a third, the space of multipliers M(S1, S2) with the following norm:

‖f‖M(S1 ,S2) = sup‖g‖S1 �1

‖fg‖S2 .

M(Hr , L2) has been used in the study of the uniqueness of weak solutions for the Navier–Stokes equations in [13] where it is pointedout that

Lp ⊂Lp,∞ ⊂Mp,q ⊂M(Hr , L2)⊂M3/2,2 for3

r>q>2. (10)

Here Mp,q stands for the homogeneous Morrey space.In this paper, we would like to using the multiplier space Xr :=M(Br

2,1, L2), which has been proved in [14] that

M(Hr , L2)⊂M3/r,2 =Xr :=M(Br2,1, L2), 0<r<1. (11)

Our main result of this paper reads:

Theorem 1.1Assume that the initial data satisfy (7). Let (�, u,�, H) be a strong solution of the problem (MHD) (1)–(6) satisfying the regularity (8). If

u∈L2/ (1−r)(0, T; Xr(R3)) with any r ∈ (0, 1), (12)

then the solution (�, u,�, H) can be extended beyond T . In other words, if T∗ is the maximal time of existence, then

limT→T∗ ‖u‖L2/ (1−r)(0,T;Xr ) =+∞

for some r ∈ (0, 1).

Remark 1.1Our theorem extends the result in [4] for the density-dependent Navier–Stokes equations much, even extends some result for thestandard MHD equations [7, 9, 11].

2. Proof of Theorem 1.1

The proof is based on the establishment of a priori estimates under the condition (12).First, thanks to the maximum principle, we have

0<m��M<∞. (13)

Multiplying (3) by u and using (1), (2), we see that

1

2

d

dt

∫�u2 dx+

∫|∇u|2 dx =

∫u(curl H×H) dx. (14)


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Y. ZHOU AND J. FAN

Multiplying (4) by H, we get

1

2

d

dt

∫H2 dx+

∫|∇H|2 dx =

∫(u×H) ·curl H dx. (15)

Summing up (14) and (15) we have

1

2

∫�u2 +H2 dx+

∫ T

0

∫|∇u|2 +|∇H|2 dx dt�1

2

∫�0u2

0 +H20 dx. (16)

The following is a key estimate for H.

Lemma 2.1

‖H‖L∞(0,T;L6)�C. (17)

ProofMultiplying (4) by |H|4H, setting w :=|H|3 and using (11), we have

d

dt

∫w2 dx+C

∫|∇w|2 dx � C

∫|u|·|w|·|∇w|dx

� C‖uw‖L2 ·‖∇w‖L2�C‖u‖Xr‖w‖Br

2,1‖∇w‖L2

� C‖u‖Xr‖w‖1−r

L2 ‖∇w‖1+rL2

� �‖∇w‖2L2 +C‖u‖2/ (1−r)

Xr‖w‖2

L2 ,

where we used the following inequality due to Machihara–Ozawa [15]:

‖w‖Br2,1

�C‖w‖1−rL2 ‖∇w‖r

L2 for 0<r<1. (18)

Now Gronwall’s inequality yields (17). �

Lemma 2.2

‖(u, H)‖L∞(0,T;H1)∩L2(0,T;H2) � C. (19)

‖ut‖L2(0,T;L2) � C. (20)

ProofMultiplying (3) by ut , integrating by parts and using (1)–(2) and (13), we easily get

1

2

d

dt

∫|∇u|2 dx+C

∫|ut|2 dx �

∣∣∣∣∫

�u ·∇u ·ut dx

∣∣∣∣+∣∣∣∣∫

(curl H×H) ·ut

∣∣∣∣ dx

� C‖u ·∇u‖L2‖ut‖L2 +C‖H‖L6 ·‖curl H‖L3 ·‖ut‖L2

� C‖u‖Xr‖∇u‖Br

2,1‖ut‖L2 +C‖∇H‖L3‖ut‖L2

� C‖u‖Xr‖∇u‖1−r

L2 ‖�u‖rL2‖ut‖L2 +C‖∇H‖1/2

L2 ‖�H‖1/2L2 ‖ut‖L2 , (21)

where (18) and the following Gagliardo–Nirenberg inequality were used.:

‖w‖L3�C‖w‖1/2L2 ‖∇w‖1/2

L2 . (22)

On the other hand, since (u,�) is a solution of the Stokes system:

−�u+∇�= f :=curl H×H−�ut −�u ·∇u. (23)

We get from the classical regularity theory that [16]:

‖u‖H2 +‖�‖H1 � C‖f‖L2

� C‖H‖L6‖curl H‖L3 +C‖ut‖L2 +C‖u ·∇u‖L2

� C‖∇H‖L3 +C‖ut‖L2 +C‖u‖Xr‖∇u‖Br

2,1

� C‖∇H‖1/2L2 ‖�H‖1/2

L2 +C‖ut‖L2 +C‖u‖Xr‖∇u‖1−r

L2 ·‖�u‖rL2 .

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Y. ZHOU AND J. FAN

Therefore,

‖u‖H2 +‖�‖H1�C‖∇H‖1/2L2 ‖�H‖1/2

L2 +C‖ut‖L2 +C‖u‖1/ (1−r)Xr

‖∇u‖L2 . (24)

Substituting the above estimates into (21), we have

d

dt

∫|∇u|2 dx+C

∫|ut|2 dx � C‖u‖Xr

‖∇u‖1−rL2 (‖∇H‖1/2

L2 ‖�H‖1/2L2 +C‖ut‖L2+C‖u‖1/ (1−r)

Xr‖∇u‖L2 )r‖ut‖L2+C‖∇H‖1/2

L2 ‖�H‖1/2L2 ‖ut‖L2

� �‖ut‖2L2 +C‖u‖2/ (1−r)

Xr‖∇u‖2

L2 +C‖∇H‖2L2 +�‖�H‖2

L2 , (25)

for any �>0 by Young’s inequality.Multiplying (4) by −�H and integrating by parts, using (18), (22), (24) and

−curl (u×H)=u ·∇H−H ·∇u, (26)

we find that

1

2

d

dt

∫|∇H|2 dx+

∫|�H|2 dx � (‖u ·∇H‖L2 +‖H‖L6‖∇u‖L3 )‖�H‖L2

� C(‖u‖Xr‖∇H‖Br

2,1+‖∇u‖L3 )‖�H‖L2

� C‖u‖Xr‖∇H‖1−r

L2 ‖�H‖1+rL2 +C‖∇u‖1/2

L2 ‖�u‖1/2L2 ‖�H‖L2

� C‖u‖Xr‖∇H‖1−r

L2 ‖�H‖1+rL2 +C‖∇u‖1/2

L2 (‖∇H‖1/2L2 ‖�H‖1/2

L2 +‖ut‖L2 +‖u‖1/ (1−r)Xr

‖∇u‖L2 )1/2‖�H‖L2

� �‖�H‖2L2 +C‖u‖2/ (1−r)

Xr‖∇H‖2

L2 +C‖∇u‖2L2 +C‖∇H‖2

L2 +�‖ut‖2L2 +C‖∇u‖4

L2 , (27)

for any �>0 by Young’s inequality.Combining (25) and (27), taking � small enough and using the Gronwall inequality, (19) and (20) can be obtained. �

Lemma 2.3

‖(ut, Ht)‖L∞(0,T;L2)∩L2(0,T;H1)�C. (28)

ProofWe differentiate (3) with respect to time t to obtain

�utt +�u ·∇ut −�ut +∇�t =curl Ht ×H+curl H×Ht −�t(ut +u ·∇u)−�ut ·∇u.

Multiplying the above equation by ut , integrating by parts, and using (1), (2), Lemmas 2.1–2.2 and Hölder’s inequality, we obtain

1

2

d

dt

∫�u2

t dx+∫

|∇ut|2 dx

�‖H‖L6 ·‖curl Ht‖L2 ·‖ut‖L3 +‖curl H‖L2 ·‖Ht‖L6 ·‖ut‖L3 +∣∣∣∣∫

�u ·∇[(ut +u ·∇u)ut] dx

∣∣∣∣+‖�‖L∞ ·‖ut‖L6 ·‖∇u‖L2 ·‖ut‖L3

�C‖∇Ht‖L2‖ut‖L3 +‖�‖L∞ ·‖u‖L6 ·‖ut‖L3 ·‖∇ut‖L2 +‖�‖Ł∞ ·‖u‖2L6 ·‖∇u‖L6 ·‖∇ut‖L2 +‖�‖L∞ ·‖u‖L6 ·‖ut‖L6 ·‖∇u‖L6 ·‖∇u‖L2

+‖�‖L∞ ·‖u‖2L6 ·‖ut‖L6 ·‖�u‖L2 +C‖ut‖L3‖∇ut‖L2

�C‖∇Ht‖L2‖ut‖L3 +C‖ut‖L3‖∇ut‖L2 +C‖�u‖L2‖∇ut‖L2

�C‖∇Ht‖L2 ·‖ut‖1/2L2 ‖∇ut‖1/2

L2 +C‖ut‖1/2L2 ·‖∇ut‖3/2

L2 +C‖�u‖L2‖∇ut‖L2

��‖∇ut‖2L2 +�‖∇Ht‖2

L2 +C‖�u‖2L2 +C‖√�ut‖2

L2 , (29)

due to Young’s inequality.Taking �t to (4), we have the following equation:

Htt −�Ht =curl (ut ×H)+curl (u×Ht).


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Y. ZHOU AND J. FAN

Multiplying the above equation by Ht , integrating by parts, using Lemmas 2.1–2.2 and Hölder’s inequality, we get

1

2

d

dt

∫H2

t dx+∫

|∇Ht|2 dx =∫

(ut ×H+u×Ht)curl Ht dx

� (‖ut‖L3‖H‖L6 +‖u‖L6‖Ht‖L3 )‖curl Ht‖L2

� C(‖ut‖L3 +‖Ht‖L3 )‖∇Ht‖L2

� C(‖ut‖1/2L2 ‖∇ut‖1/2

L2 +‖Ht‖1/2L2 ‖∇Ht‖1/2

L2 )‖∇Ht‖L2

� �‖∇Ht‖2L2 +�‖∇ut‖2

L2 +C‖√�ut‖2L2 +C‖Ht‖2

L2 . (30)

Taking � be small enough, (28) can be obtained by (29), (30) and Gronwall’s inequality. �

Lemma 2.4

‖(u, H)‖L∞(0,T;H2)∩L2(0,T;W2,6) � C. (31)

‖�‖L∞(0,T;H1)∩L2(0,T;W1,6) � C. (32)

ProofSince H is a solution of the elliptic system

−�H=−Ht +H ·∇u−u ·∇H. (33)

By the classical elliptic regularity theory, we deduce that

‖H‖H2 � C‖Ht +u ·∇H−H ·∇u‖L2

� C‖Ht‖L2 +C‖u‖L6‖∇H‖L3 +C‖H‖L∞‖∇u‖L2

� C+C‖∇H‖L3 +C‖H‖L∞

� C+C‖∇H‖1/2L2 ·‖�H‖1/2

L2

� 12 ‖H‖H2 +C+C‖∇H‖L2 ,

which tells us

‖H‖L∞(0,T;H2)�C.

Similarly, from (23) we can get

‖u‖L∞(0,T;H2) +‖�‖L∞(0,T;H1)�C.

From (33) and (28) we easily deduce that

‖H‖L2(0,T;W2,6)�C.

Similarly, from (23) and (28), we have

‖u‖L2(0,T;W2,6) +‖�‖L2(0,T;W1,6)�C.

The proof is complete. �

Lemma 2.5

‖∇�‖L∞(0,T;L2∩Lq) � C, (34)

‖�t‖L∞(0,T;L2∩Lq) � C. (35)

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Y. ZHOU AND J. FAN

ProofDifferentiating (2) with respect to xj , multiplying it by |�j�|q−2�j�, and then integrating by parts, we get

d

dt

∫|∇�|q dx � C‖∇u‖L∞

∫|∇�|q dx

� C‖u‖W2,6

∫|∇�|q dx

This proves (34) by (31) and Gronwall’s inequality.Equation (35) follows easily from (31), (34) and the following equation:

�t =−u ·∇�.

�

The proof for Theorem 1.1 completes.

3. Final remarks

One of the difficulties here is to deal with the equation of density (conservation of mass). So far, no regularity criterion in termsof the gradient of the velocity field or the pressure has been established. We hope we can deal with these problems in the nearfuture.

Acknowledgements

This work is partially supported by the Program for New Century Excellent Talents in Universities (Grant No. NCET 07-0299),ZJNSF (Grant No. R6090109) and NSFC (Grant No. 10971197).

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A regularity criterion for the density-dependent magnetohydrodynamic equations

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