Global Existence and Uniqueness of Strong Solutions for the Magnetohydrodynamic Equations

Volume 2008, Article ID 735846,14pages doi:10.1155/2008/735846

Research Article

Global Existence and Uniqueness of Strong

Solutions for the Magnetohydrodynamic Equations

Jianwen Zhang

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Correspondence should be addressed to Jianwen Zhang,[email protected]

Received 21 June 2007; Accepted 5 October 2007

Recommended by Colin Rogers

This paper is concerned with an initial boundary value problem in one-dimensional magnetohy-drodynamics. We prove the global existence, uniqueness, and stability of strong solutions for the planar magnetohydrodynamic equations for isentropic compressible fluids in the case that vacuum can be allowed initially.

Copyrightq2008 Jianwen Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

MagnetohydrodynamicsMHDconcerns the motion of a conducting fluid in an electromag-netic field with a very wide range of applications. The dynamic motion of the fluids and the magnetic field strongly interact each other, and thus, both the hydrodynamic and electrody-namic effects have to be considered. The governing equations of the plane magnetohydrody-namic compressible flows have the following formsee, e.g.,1–5:

ρ_t ρu_x0,

ρu_t

ρu2p1

2|b|

2

x

λux

x,

ρw_t ρuw−b_x μwx

x,

bt ub−wx

υbx

x,

ρe_t ρeu_x−κθx

xλu2xμwx2υbx2−pux,

1.1

where ρdenotes the density of the fluid, u ∈ Rthe longitudinal velocity,w w1, w2 ∈ R2

p pρ, θthe pressure, and e eρ, θthe internal energy;λ andμare the bulk and shear viscosity coefficients, υ is the magnetic viscosity, κ is the heat conductivity. Notice that the longitudinal magnetic field is a constant which is taken to be identically one in1.1.

The equations in1.1describe the macroscopic behavior of the magnetohydrodynamic flow. This is a three-dimensional magnetohydrodynamic flow which is uniform in the trans-verse directions. There is a lot of literature on the studies of MHD by many physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathe-matical challenges, see1–14and the references cited therein. We mention that, whenb 0, the system1.1reduces to the one-dimensional compressible Navier-Stokes equations for the flows between two parallel horizontal platessee, e.g.,15.

In this paper, we focus on a simpler case of 1.1, namely, we consider the magneto-hydrodynamic equations for isentropic compressible fluids. Thus, instead of the equations in

1.1, we will study the following equations:

ρ_t ρu_x0, 1.2

ρu_t

ρu2p1

2|b|

2

x

λux

x, 1.3

ρw_t ρuw−bx

μwx

x, 1.4

bt ub−wx

υbx

x, 1.5

where p Rργ with γ ≥ 1 being the adiabatic exponent and R > 0 being the gas constant. We will study the initial boundary value problem of1.2–1.5in a bounded spatial domain

Ω 0,1 without loss of generalitywith the initial-boundary data:

ρ, ρu, ρw,b x,0 ρ₀, m0,n0,b0

x, x∈Ω, 1.6

u,w,b|_x0,10, 1.7

where the initial dataρ₀ ≥ 0, m0, n0, b0 satisfy certain compatibility conditions as usual and

some additional assumptions below, andm0 n0 0 whenever ρ₀ 0. Here the boundary

conditions in1.7mean that the boundary is nonslip and impermeable.

The purpose of the present paper is to study the global existence and uniqueness of strong solutions of problem 1.2–1.7. The important point here is that initial vacuum is allowed; that is, the initial density ρ₀ may vanish in an open subset of the space-domain

Ω 0,1, which evidently makes the existence and regularity questions more difficult than the usual case that the initial densityρ₀ has a positive lower bound. For the latter case, one can show the global existence of unique strong solution of this initial boundary value problem in a similar way as that in3,9,14. The strong solutions of the Navier-Stokes equations for isentropic compressible fluids in the case that initial vacuum is allowed have been studied in

16,17. In this paper, we will use some ideas developed in16,17and extend their results to the problems1.2–1.7. However, because of the additional nonlinear equations and the non-linear terms induced by the magnetic fieldb, our problem becomes a bit more complicated.

Theorem 1.1. Assume thatρ₀, m0 ρ₀u0,n0ρ₀w0, andb0satisfy the regularity conditions:

ρ₀∈ H1, ρ₀≥0, u0,w0

∈ H1

0∩ H2, b0∈ H10. 1.8

Assume also that the following compatibility conditions hold for the initial data:

λu0xx−

Rργ₀ 1

2b0

2

x

ρ1₀/2f for somef∈ L2Ω, 1.9

μw0xxb0xρ₀1/2g for someg∈ L2Ω. 1.10

Then there exists a unique global strong solution ρ, u,w,b to the initial boundary value problem

1.2–1.7such that for allT ∈0,∞,

ρ∈ L∞0, T;H1_{, u,w∈ L}∞_{0, T;H}1

0∩ H2

, b∈ L∞0, T;H1₀,

ρ_t,√ρut,√ρwt

∈ L∞_{0, T;L}2_{, u}

t,wt∈ L2

0, T;H1_{, b}

t,bxx∈ L2

0, T;L2_. 1.11

Remark 1.2. The compatibility conditions given by1.9, 1.10play an important role in the proof of uniqueness of strong solutions. Similar conditions were proposed in16–18when the authors studied the global existence and uniqueness of solutions of the Navier-Stokes equa-tions for isentropic compressible fluids. In fact, one also can show the global existence of weak solutions without uniqueness if the compatibility conditions1.9,1.10are not valid.

We will prove the global existence and uniqueness of strong solutions in Sections3and 4, respectively, whileSection 2is devoted to the derivation of some a priori estimates.

We end this section by introducing some notations which will be used throughout the paper. LetWm,pΩdenote the usual Sobolev space, andWm,2Ω HmΩ,W0,pΩ LpΩ. For simplicity, we denote byCthe various generic positive constants depending only on the data andT, and use the following abbreviation:

Lq_{0, T;W}m,p_{≡ L}q_{0, T;W}m,p_{Ω, L}p _{≡ L}p_{Ω, ·}

p ≡ ·Lp_Ω. 1.12

2. A priori estimates

This section is devoted to the derivation of a priori estimates ofρ, u,w,b. We begin with the observation that the total mass is conserved. Moreover, if we multiply1.3,1.4, and1.5by

u,w, andb, respectively, and sum up the resulting equations, we have by using1.2that

1 2ρ

u2w21

2b

2

t

1 2ρu

u2w2ub2−w·b

x

upx

λuuxμw·wxυb·bx

x−

λu2xμwx2υbx2

.

2.1

Lemma 2.1. For any t∈0, T, one has

Ωρx, tdx Ωρ0xdx≤C,

Ω

Gρ 1

2ρ

u2w21

2b

2

x, tdx

t

0 Ω

λu2_xμwx2υbx2

x, sdx ds≤C,

2.2

whereGρis the nonnegative function defined by

Gρ

⎧ ⎪ ⎨ ⎪ ⎩

Rργ

γ−1 if γ >1

Rρlnρ−ρ1 if γ 1.

2.3

The next lemma gives us an upper bound of the densityρx, t, which is crucial for the proof ofTheorem 1.1.

Lemma 2.2. For anyx, t∈QT : Ω×0, T,ρx, t≤Cholds.

Proof. Notice that1.3can be rewritten as

ρu_t

λux−ρu2−p−

_b2

2

x

. 2.4

Set

ψx, t: t

0

λux−ρu2−p−

1 2b

2

x, sds

x

0

m0ζdζ, 2.5

from which and2.4, we find thatψ satisfies

ψ_x ρu, ψ_tλux−ρu2−p−

1 2b

2

, ψ|t0

x

0

m0ζdζ. 2.6

In view ofLemma 2.1and2.6, we have by using Cauchy-Schwarz’s inequality that

_ψxL∞

0,T;L1≤C,

_Ωψx, tdx≤C, 2.7

which imply

ψ_L∞_0,T×Ω ≤ψ_xL∞_0,T;L1

_Ωψx, tdx≤C. 2.8

LettingDt : ∂tu∂x denote the material derivative and choosing F expψ/λ, we obtain after a straightforward calculation that

DtρF:∂tρF u∂xρF −1

λ

p b

2

2

ρF ≤0, 2.9

To be continued, we need the following lemma because of the effect of magnetic fieldb.

Lemma 2.3. The magnetic fieldbsatisfies the following estimates:

sup

0≤t≤T

_btL∞bxtL2

bt_L2_0,T;L2≤C, bxx_L2_0,T;L2≤C. 2.10

Proof. Multiplying1.5bybtand integrating over0, t×Ω, we have

1 4

t

0 Ω

_bt2

x, sdx dsυ

2 _Ωbx

2 x, tdx

≤ υ

2 _Ωb0x

2

xdx

t

0 Ω

u2bx2u2xb

2_w

x2

x, sdx ds

≤C2 t

0

Ωu

2

xx, sdx

Ω

bx2x, sdx

ds,

2.11

where we have used Cauchy-Schwarz’s inequality,Lemma 2.1, and the following inequalities:

max x∈Ω u

2_{·, s≤u}

xs2_L2, max

x∈Ω b·, s

2_≤b

xs2_L2. 2.12

Sinceux_L2_0,T;L2 ≤ Cbecause ofLemma 2.1, we thus obtain the first inequality

indi-cated in this lemma from2.11by applying Gronwall’s lemma and then Sobolev’s inequality. To prove the second part, we multiply1.5bybxxand integrate the resulting equation over0, T×Ωto deduce that

T

0 Ω

bxx2x, tdx dt

≤C

T

0 Ω

_bt2_w

x2u2xb

2

u2bx2

x, tdx dt

≤CCsup t∈0,T

_bt2

L∞ T

0

_uxt2

L2dtC

T

0

_uxt2

L2bxt2_L2dt

≤CCsup t∈0,T

_bt2

L∞bxt2_L2 T

0

_uxt2

L2dt≤C,

2.13

where we have used Cauchy-Schwarz’s inequality, Sobolev’s inequality2.12,Lemma 2.1, and the first part of the lemma. This completes the proof ofLemma 2.3.

Lemma 2.4. The following estimates hold for the velocityu,w:

sup

0≤t≤T

_utL∞uxtL2

√ρut_L2_0,T;L2≤C,

sup

0≤t≤T

_wtL∞wxtL2

√ρwt_L2_0,T;L2≤C.

Proof. Multiplying1.3byutand then integrating overΩ, by Young’s inequality we obtain

λ

2

d

dt _Ωu

2

xdx 1 2 _Ωρu

2

tdx≤ 1 2 _Ωρu

2_u2

xdx

Ωputxdx dt

1 2 _Ω|b|

2_u

xtdx. 2.15

It follows from1.2and1.3that

Ωputxdx

d

dt _Ωpuxdx−

R

2λ d

dt _Ωρ

γ1_u2_dx−Rγ −2

2λ _Ωρ

γ1_u2_u

xdx

1

2λ _Ωp 2_u

xdx− 1

λ _Ωpu

ρuuxb·bx

dx γ −1

Ωpu

2

xdx,

1 2 _Ω|b|

2_u

xtdx 1 2

d dt _Ω|b|

2_u

xdx−

Ωb·btuxdx.

2.16

Thus, inserting2.16into2.15, and integrating over0, t, we see that

Ωu

2

xx, tdx t

0 Ω

ρu2_tdx ds

≤CC

Ω

puxργ1u2|b|2uxx, tdx

C

t

0 Ω

ρu2u2_xργ1u2uxp2uxpub·bxpux2b·btuxdx ds,

2.17

where the terms on the right-hand side can be bounded by using Lemmas2.1–2.3as follows:

Ω

puxργ1u2|b|2uxtdx≤Cδ δ Ωu

2

xtdx, δ >0, 2.18

t

0 Ω

ρu2u2_xργ1u2uxp2uxpu2x

dx ds

≤CC

t

0

max x∈Ω u

2_{·, su}

xs2_L2ds≤CC

t

0

_uxs4

L2ds,

2.19

t

0 Ω

_{pub·bxb·btuxdx ds≤C} t

0 Ω

ρu2bx2bt2u2x

dx dt≤C. 2.20

Therefore, takingδappropriately small, we conclude from2.17–2.20that

Ωu

2

xtdx t

0 Ωρu

2

tdx ds≤CC

t

0

_uxs4

L2ds, 2.21

where, combined with the fact that ux_L2_0,T;L2 ≤ Cdue to Lemma 2.1, we obtain the first

part of Lemma 2.4 by applying Gronwall’s lemma and then Sobolev’s inequality. Similarly, multiplying1.4bywtand integrating the resulting equation overΩ, we get that

μ

2

d

dt _Ωwx

2

dx 1

2 _Ωρwt

2 dx≤ Ω 1 2ρu

2_w

x2bt·wx

dx− d

where we have also used Cauchy-Schwarz’s inequality. Integration of2.22over0, tgives

Ω

wxx, t2dx t

0 Ω

ρwt2dx ds≤C1

2 _Ωwxx, t

2 dx

C

t

0 Ω

bt2wx2

dx ds≤C1

2 _Ωwxx, t

2 dx,

2.23

where Lemmas2.1–2.3and the first conclusion of this lemma have been used. Therefore, from the above inequality we obtain the second part, and soLemma 2.4is proved.

Notice that1.3,1.4can be written as follows:

ρutρuuxGx, ρwtρuwxKx, 2.24

whereG:λux−p− |b|2/2 andK:μwxb. Thus, by Lemmas2.1–2.4, we see that

_G,KL∞

0,T;L2Gx,Kx_L2_0,T;L2≤C, 2.25

which immediately implies

T

0

_uxt2

L∞dt≤C T

0

G2_L∞p2_L∞b4_L∞tdt

≤C

T

0

G2_L2Gx2_L2p

2

L∞b4_L∞tdt≤C,

T

0

_wxt2

L∞dt≤C T

0

K2

L∞b2_L∞

tdt

≤C

T

0

K2

L2Kx_L22b2L∞

tdt≤C.

2.26

Hence, we have the following lemma.

Lemma 2.5. There exists a positive constantC, such that

T

0

_uxt2

L∞Gxt2_L2

dt≤C,

T

0

wxt2_L∞Kxt2_L2

dt≤C, 2.27

whereG:λux−p− |b|2/2andK:μwxb.

To prove the uniqueness of strong solutions, we still need the following estimates.

Lemma 2.6. The pressurepρ Rργ satisfiessup_0≤t≤Tpx·, tL2 ≤C. Furthermore, if the

compati-bility conditions1.9,1.10hold, then

sup

0≤t≤T Ωρ

u2_t wt2

x, tdx

T

0 Ω

u2_txwtx2

Proof. It follows from the continuity equation1.2thatpsatisfies

ptpxuγpux0, 2.29

which, differentiated with respect tox, leads to

pxtpxxu γ 1pxuxγpuxx0. 2.30

Multiplying the above equation bypxand integrating overΩ, we deduce that

d

dtpxt 2

L2 ≤C Ω

_px2

uxppxuxxx, tdx

≤Cuxt_L∞pxt2_L2pxt

2

L2bxt

2

L2Gxt

2

L2

,

2.31

where we have used the inequality

_uxx2

L2 ≤CGx2_L2px2_L2bx2_L2

, 2.32

which follows from the definition ofG. Therefore, applying the previous Lemmas2.1–2.5and Gronwall’s lemma, one has

sup

0≤t≤T

_pxtL2 ≤C, 2.33

which proves the first part of the lemma.

We are now in a position to prove the second part. We first derive the estimate for the longitudinal velocityu. To this end, we firstly rewrite1.3as

ρutρuux−λuxx

pb

2

2

x

0. 2.34

Differentiation of2.34with respect totgives

ρuttρuuxt−λuxxt

p1

2b

2

xt

−ρ_tutuux

−ρutux, 2.35

which, multiplied byutand integrated by parts overΩ, yields

1 2

d

dt _Ωρu

2

t dxλ Ωu

2

xtdx− Ω

p1

2b

2

t

uxtdx−

Ω

ρuu2_t uuxut

xρu2tux

dx.

2.36

On the other hand, by virtue of1.2we have

−

Ωptutxdx Ωpxuutxdx

γ

2

d

dt _Ωpu

2

xdx−

γ

2 _Ωptu

2

xdx

γ

2

d

dt _Ωpu

2

xdx

Ωpxuutxdx

γ

2 _Ω

−puu2x

x γ −1pu

3

x

dx,

from which and2.36we see that

d

dt _Ω

1 2ρu

2

t

γ

2pu

2

x

dxλ

Ωu

2

txdx

≤

Ω

2ρ|u|ututxρ|u|utux2ρ|u|2utuxxρ|u|2uxutx

ρut2uxpx|u|utxγp|u|uxuxx

γγ −1

2 pux

3_|b|b

tutx

dx

≡9

j1 Ij.

2.38

Using the previous lemmas and Young’s inequality, we can estimate each term on the right-hand side of2.38as follows with a small positive constant:

I1≤2ρ1_L/∞2u_L∞ρut_L2utx_L2 ≤utx

2

L2 C−1ρut

2

L2,

I2≤ ρ1_L/∞2uL∞ρutL2u_xL2u_xL∞ ≤Cu_xL2_∞Cρu_t2_L2,

I3≤ ρ1_L/∞2u2_L∞ρut_L2uxx_L2 ≤Cuxx2_L2Cρut2_L2,

I4≤ ρL∞u2_L∞uxL2u_txL2 ≤u_tx2_L2 C−1,

I5≤ uxL∞ρut2_L2,

I6≤ uL∞px_L2utx_L2 ≤utx2_L2C−1,

I7≤γp_L∞u_L∞ux_L2uxx_L2 ≤CCuxx

2

L2,

I8≤Cp_L∞ux_L∞ux2_L2 ≤CCux2_L∞,

I9≤ b_L∞bt_L2utx_L2 ≤utx

2

L2C−1bt

2

L2.

2.39

Putting the above estimates into2.38and takingsufficiently small, we arrive at

d

dt _Ω

ρu2_t pu2_xdx

Ωu

2

txdx

≤C1ρut

2

L2uxx

2

L2 ux2_L∞bt2_L2ux_L∞

ρut

2

L2

,

2.40

so that, using the relation betweenGxanduxxagain, one infers from2.40that

d

dt _Ω

ρu2_t pu2_xdx

Ωu

2

txdx

≤C1ρut2_L2Gx

2

L2ux

2

L∞bt2_L2

Cux_L∞

ρut

2

L2,

where the first term on the right-hand side of2.41is integrable on0, Tdue to the previous lemmas. Thus, integrating2.41overτ, t⊂0, T, we deduce from1.3that

Ωρu

2

tx, tdx t

τ Ω

u2_txdx ds

≤

Ωρu

2

tx, τdxC

1 t

0

_uxsL∞

ρuts

2

L2ds

Ωρ −1

λuxx−

p 1

2b

2

x

−ρuux

2

x, τdx

C

1 t

0

_uxsL∞

ρuts

2

L2ds

.

2.42

Lettingτ →0 and using the compatibility condition1.9, we easily obtain from2.42that

Ωρu

2

tx, tdx t

0 Ω

u2_txdx ds≤C

1 t

0

_uxsL∞

ρuts

2

L2ds

, 2.43

which, together withux_L1_0,T;L∞≤Cand Gronwall’s lemma, immediately yields

sup

0≤t≤T

ρutt

2

L2utx

2

L2_0,T;L2≤C. 2.44

In a same manner as that in the derivation of2.44, we can show the analogous esti-mate for the transverse velocitywby using the previous lemmas,2.44, and the compatibility condition1.10as well. Thus, we complete the proof ofLemma 2.6.

Remark 2.7. From the a priori estimates established above, one sees that the compatibility con-ditions are used to obtain the second part ofLemma 2.6 only. However, this is crucial in the proof of the uniqueness of strong solutions.

3. Global existence of strong solutions

In this section, we prove the global existence of strong solutions to the problem1.2–1.7by applying the a priori estimates given in the previous section. As usual, we first mollify the initial data to get the existence of smooth approximate solutions. For this purpose, we choose the smooth approximate functionsρ

0andb0such that

ρ₀∈C2_{Ω, 0< ≤ρ}

0≤ρ0L∞1, ρ₀−→ρ₀ in H1,

b

0∈C2Ω, b0≤b0L∞1, b₀−→b0 in H1₀.

3.1

Letu₀,w₀∈ C1₀Ω∩C3_{Ω, satisfying u}

0,w0 → u0,w0in H10∩ H2, be the unique

solution to the boundary value problems

λu_0xx

Rρ₀γ 1

2b

0 2

x

ρ₀1/2f inΩ, u₀|x0,10,

μw_0xx−b_0xρ₀1/2g inΩ,w₀|x0,10,

respectively, where

f,g∈C2₀Ω, f,g−→f, g in L2. 3.3

Thus, with the regularized initial dataρ₀, u₀,w₀,b₀satisfying the compatibility condi-tions as above, we can follow the similar arguments as in3,9,14 because ofρ₀≥to show that the problems1.2–1.7admit a global strong solutionρ_{, u,w,b, which satisfies}

0< C≤ρ ≤C withCdepending on. 3.4

Applying the a priori estimates obtained in the previous section, we conclude that the approximate solutionρ, u,w,bsatisfies

sup

0≤t≤T

pt_H1u_x,w_x,b_x

t_L2ρu_t,

ρw

t

t_L2

T

0

_u

x,wx

t2_L∞uxx,wxx , utx,wtx,bt,bxx

t2_L2

dt≤C,

3.5

whereCdepends on the norms of initial data given inTheorem 1.1, but not on. With the help of1.2–1.5and3.5, it is easy to see that

_ρ

xL∞_0,T;L2≤C, ρ_tL∞_0,T;L2≤C, 3.6 _u

xxL∞_0,T;L2≤C

ρ_u

t, ux, px,bx_L∞0,T;L2≤C, 3.7 w

xx_L∞_0,T;L2≤C

ρw

t,wx,bx_L∞

0,T;L2≤C. 3.8

By the uniform in bounds given in 3.5–3.8 we conclude that there exists a sub-sequence of ρ_{, u,w,b which converges to a strong solution ρ, u,w,b to the original} problem and satisfies3.5–3.8as well. This completes the proof ofTheorem 1.1except the uniqueness assertion because of the presence of vacuum, which will be proved in the next section.

4. Uniqueness and stability of strong solutions

In this section, we will prove the following stability theorem, which consequently implies the uniqueness of strong solutions. Our proof is inspired by the uniqueness results due to Choe-Kim16,17and Desjardins19for the isentropic compressible Navier-Stokes equations.

Theorem 4.1. Let ρ, u,w,b andρ, u,w,bbe global solutions to problems 1.2–1.7with ini-tial data ρ₀, u0,w0,b0 andρ₀, u0,w0,b0, respectively. If ρ, u,w,band ρ, u,w,b satisfy the

regularity given inTheorem 1.1, then for anyt∈0, T,

ρ−ρ, p−p,ρu−u,ρw−w,b−bt2

L2

t

0

exp

t

S

Fτdτu−u,w−w,b−b_xs2_L2ds

≤ exp

t

0

Fsds

ρ₀−ρ₀, p0−p₀,ρ₀u0−u0,ρ₀w0−w0

,b0−b0 2

L2

4.1

Proof. From the continuity equation it follows that

ρ−ρ_t ρ−ρ_xuρ_xu−u ρ−ρuxρu−ux0. 4.2

Multiplying this by 2ρ−ρand then integrating overΩ, we have

d

dt _Ω|ρ−ρ| 2_dx

≤

Ω|ρ−ρ|

2_u

x2ρ_x|ρ−ρ||u−u|2|ρ−ρ|2ux2ρ|ρ−ρ|u−uxdx

≤ux_L∞2ux_L∞ρ−ρ2_L22ρ_xL2ρ_L∞ρ−ρ_L2u−u_xL2,

4.3

where we have used the inequality u−u_L∞ ≤u−uxL2. Thus, by Cauchy-Schwarz’s

inequality, one has

d

dtρ−ρ

2

L2 ≤u−u_x

2

L2C−1Atρ−ρ

2

L2, 4.4

whereAt:uxL∞uxL∞ρ_xL22ρ2_L∞

∈ L1_{0, T.}

By virtue of the equations satisfied byuandu, we easily deduce that

ρu−u_tρuu−u_x−λu−u_xx

−ρ−ρutuux

−ρu−uux−p−px− 1 2

|b|2_{− |b|}2

x,

4.5

which, multiplied byu−uand integrated by parts overΩ, gives

1 2

d

dt _Ωρ|u−u|

2_dxλ

Ω

_u−ux2 dx

≤ut_L2u_L∞ux_L2

ρ−ρ_L2u−u_L∞ρu−u

2

L2uxL∞

p−p_L2u−u_xL2

1 2

b_L∞b_L∞b−b_L2u−u_xL2,

4.6

so that we have by using Sobolev’s inequality and Cauchy-Schwarz’s inequality that

d dt

ρu−u2

L2 u−ux

2

L2

≤Btρ−ρ2_L2

ρu−u2

L2p−p

2

L2b−b2_L2

,

4.7

whereBt:Cut_L22 u2_L∞ux2_L2u_xL∞b2_L∞b2_L∞∈ L10, T.

Proceeding the similar argument as that in the derivation of4.7, we also have

d dt

ρw−w2

L2w−wx

2

L2

≤Ctρ−ρ2_L2

ρu−u2

L2

ρw−w2

L2b−b

2

L2

,

whereCt:Cwt2_L2 u

2

L∞wx2_L2wx_L∞

∈ L1_{0, T.}

Furthermore, it follows from the equations for the magnetic fieldsbandbthat

b−btub−bx u−ubx u−uxbuxb−b−w−wxυb−bxx, 4.9

which, multiplied by 2b−band integrated by parts overΩ, gives

d

dtb−b

2

L2 2υb−b_x

2

L2

≤ux_L∞2ux_L∞

b−b2

L22b_xL2b−bL2u−u_L∞

2b_L∞b−b_L2u−u_xL22b−bL2w−w_xL2

≤u−u_x2_L2 w−w_x

2

L2

C−1Dtb−b2_L2,

4.10

whereDt:1uxL∞uxL∞bx_L22b2_L∞

∈ L1_{0, T.}

Finally, from the continuity equations forρandρ, it is easy to see that

p−p_t p−p_xup_xu−u γp−puxγpu−ux0, 4.11

and hence we obtain in a manner similar to the derivation of4.4that

d

dtp−p

2

L2 ≤Cu_xL∞ux_L∞p−p2_L2

Cp_xL2p−p_L2u−uL∞CpL∞p−pL2u−u_xL2

≤u−ux

2

L2C−1Etp−p

2

L2,

4.12

whereEt:uxL∞uxL∞p_x2_L2p2_L∞∈ L10, T.

Summing up 4.4–4.12 and choosing appropriately small, we obtain Theorem 4.1 withFt:CAtBtCtDtEt∈L10, Tby applying Gronwall’s lemma.

Acknowledgment

This work is partly supported by NSFCGrant no. 10501037 and no. 10601008.

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