global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic...

Nonlinear Analysis 72 (2010) 4438–4451

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Nonlinear Analysis

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Global existence and convergence rates of smooth solutions for thecompressible magnetohydrodynamic equationsI

Qing Chen ∗, Zhong TanSchool of Mathematical Sciences, Xiamen University, Fujian 361005, China

a r t i c l e i n f o

Article history:Received 1 June 2009Accepted 10 February 2010

MSC:76W0535Q3535D0576X05

Keywords:MagnetohydrodynamicsCompressibleGlobal existenceSmooth solutionsLp-Lq convergence rates

a b s t r a c t

In this paper, we are concerned with the global existence and convergence rates of thesmooth solutions for the compressible magnetohydrodynamic equations in R3. We provethe global existence of the smooth solutions by the standard energy method under thecondition that the initial data are close to the constant equilibrium state in H3-framework.Moreover, if additionally the initial data belong to Lp with 1 ≤ p < 6

5 , the optimalconvergence rates of the solutions in Lq-norm with 2 ≤ q ≤ 6 and its spatial derivatives inL2-norm are obtained.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper,we consider the equations ofMagnetohydrodynamicswhich describes themotion of electrically conductingmedia in the presence of a magnetic field. The interactions between the viscous, isentropic, compressible fluid motion andthe magnetic field are modeled by the magnetohydrodynamic system (MHD) which describes the coupling between thecompressible Navier–Stokes equations and the magnetic equations, i.e.,{

ρt + div(ρu) = 0(ρu)t + div(ρu⊗ u)+∇p = ∆u+∇divu+ curlH× HHt − curl(u× H)+ curlcurlH = 0, divH = 0

(1.1)

in (0,∞) × R3. Here ρ,u,H represent the density, velocity of the fluid and the magnetic field respectively. p = p(ρ) =ργ , γ > 1 is its pressure. Notice that we have normalized some physical constants to be unit but without reducing anyessential difficulties for our analysis. We complement (1.1) with the Cauchy data

(ρ,u,H)(0, x) = (ρ0(x),u0(x),H0(x)) x ∈ R3. (1.2)

Magnetohydrodynamics has spanned a very large range of applications [1–3]. Due to the physical importance andmathematical challenges, the study on (MHD) has attracted many physicists and mathematicians. Many results concerning

I Supported by National Natural Science Foundation of China - NSAF (Grant No. 10976026).∗ Corresponding author.E-mail addresses: [email protected] (Q. Chen), [email protected] (Z. Tan).

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Q. Chen, Z. Tan / Nonlinear Analysis 72 (2010) 4438–4451 4439

the existence and uniqueness of (weak, strong or smooth) solutions in one dimension can be found in [4–7] and thereferences cited therein. Here, we are interested in the global solvability and large-time behavior of the solutions for theCauchy problem (1.1)–(1.2) in the multi-dimensional case. There will arise some significant difficulties caused by its multi-dimensionality and the strong interaction between the magnetic fluid and the hydrodynamic motion. The initial-boundaryvalue problem for the MHD system is studied recently by [8–11] concerning the existence of weak solutions. These worksimply that for γ > N

2 , where N is the dimension, the problem admits at least a global weak solution for any boundedsmooth domain. The constraint on the adiabatic exponent γ > N

2 is somehow optimal and the uniqueness of weak solutionremains open, and this is also well known in the framework of the compressible Navier–Stokes equations [12–14]. Hencewe plan to investigate the more regular solutions for the Cauchy problem (1.1)–(1.2). Indeed, we shall firstly prove theglobal existence and uniqueness of the smooth solutions under the assumption that the initial data are sufficiently closeto the constant equilibrium state. Since the solution obtained is global, we will then investigate the large-time behaviorof such solution. Obviously, the solution will converge to the constant equilibrium state as time goes to infinite, and ourinterest is its convergence rates. Motivated by [15–17], where Duan et al. considered optimal Lp-Lq convergence rates forthe compressible Navier–Stokes equations, we shall prove the optimal Lp-Lq convergence rates of the smooth solutions for(1.1)–(1.2). Our main results are formulated as the following theorem:

Theorem 1.1. Assume divH0 = 0 and the initial data are close enough to the constant state (ρ̄, 0, 0)with ρ̄ > 0, i.e. there existsa constant δ0 such that if

|(ρ0 − ρ̄,u0,H0)|H3 ≤ δ0, (1.3)

then there exists a unique globally smooth solution (ρ,u,H) of the Cauchy problem (1.1)–(1.2) such that for any t ∈ [0,∞),

|(ρ − ρ̄,u,H)(·, t)|2H3 +∫ t

0|∂xρ(·, s)|2H2 + |(∂xu, ∂xH)(·, s)|

2H3ds ≤ C |(ρ0 − ρ̄,u0,H0)|

2H3 . (1.4)

Moreover, if in addition, there is some p ∈ [1, 65 ) such that

|(ρ0 − ρ̄,u0,H0)|Lp < +∞, (1.5)

we have

|∇k(ρ − ρ̄,u,H)(t)|L2 ≤ C(1+ t)

−σ(p,2;1), k = 1, 2, 3 (1.6)

and

|(ρ − ρ̄,u,H)(t)|Lq ≤ C(1+ t)−σ(p,q;0), ∀q ∈ [2, 6], (1.7)

where σ(p, q; k) are defined by

σ(p, q; k) =32

(1p−1q

)+k2. (1.8)

Notations.We denote by Lp, Wm,p the usual Lebesgue and Sobolev spaces onR3 andHm = Wm,2, with norms | · |Lp , | · |Wm,p ,| · |Hm respectively. For the sake of conciseness, we do not precise in functional space names when they are concerned withscalar-valued or vector-valued functions. We denote ∇ = ∂x = (∂1, ∂2, ∂3), where ∂i = ∂xi , and put ∂

lxf = ∇

lf = ∇(∇ l−1f )for l = 1, 2, 3, . . . .We assume C be a positive generic constant throughout this paper that may vary at different places andthe integration domain R3 will be always omitted without any ambiguity.

Remark 1.1. Notice that the divergence free of themagnetic fieldH can be justified by the initial assumption that divH0 = 0.Indeed, this can be easily and formally observed by take div to themagnetic equation. Hence themagnetic equation is purelyparabolic with respect to H.

We will prove the global existence of smooth solutions by the standard energy method in spirit of Matsumura andNishida [18,19]. And we do not try to prove the global existence of smooth solutions for the general large initial data,since it is an outstanding unsolved problem even for the pure compressible Navier–Stokes equations. And we remark thatwe can also obtain the global existence of strong solutions for the small initial data in the H2-framework as (1.3), whichcan be proved similarly by the method of proving Theorem 1.1. We point out that the essential point in the proof of theglobal existence of small solutions is that under the initial condition (1.3) the initial density is bounded far away from thevacuum. A natural question to ask is whether one can still obtain the global smooth or strong solutions when there existsvacuum initially and surely the small initial data. However, the answer is negative. We can explain this as: by consideringthe particular case H0 = 0, this furnishes H = 0 and hence the system (1.1) reduces to the compressible Navier–Stokesequations. It is well known in [20,21] that the smooth or strong solutions will blow up in finite time if the initial densityis compactly supported, no matter how small the initial data are. When there is vacuum initially, only the local existence

4440 Q. Chen, Z. Tan / Nonlinear Analysis 72 (2010) 4438–4451

of strong solutions can be proved in [22] under a natural compatibility condition. On the other hand, our decay rates forthe smooth solutions are optimal since (1.6)–(1.7) concerning (ρ − ρ̄,u) and H are the same as the optimal decay ratesfor the compressible Navier–Stokes equations [15] and the heat equation [23] respectively. Related convergence rates ofsolutions for the Navier–Stokes equations can be found in [15–18,24–27]. Although our proofs are in spirit of those for theNavier–Stokes equations [15–19], we should derive the new estimates arising from the presence of the magnetic field andovercome the strong coupling between the equations.The rest of this paper is devoted to prove Theorem 1.1. In Section 2, we firstly do some careful a priori estimates for the

smooth solutions and then the global existence of the smooth solutions is established by combining our a priori estimatesand the local existence result. In Section 3wewill reformulate the problem and use the energymethod to derive a Lyapunov-type energy inequality of all the derivatives controlled by the first order derivatives, then we utilize the decay-in-timeestimates for the linearized system to control the first order derivatives by the higher order derivatives. Hence, the decayrates of the global smooth solutions follow from these two kinds of estimates.

2. Global existence

In this section, we will prove the existence part of Theorem 1.1. In order to state the ideas more clearly, we will derivethe uniform-in-time a priori estimates for smooth solutions at first and these estimates also hold for our H3 local solutionswhich can be proved rigorously by the standardmethod ofMatsumura andNishida [19], using themollifier technique. Basedon these uniform estimates, global existence will be proved at the end of this section.The first bright idea to reducemany complicated computations lies in thatwe just need to do the lowest order and highest

order energy estimates for the solutions. This is motivated by the following observation:

|f |2Hk ≤ C |(f , ∂kx f )|

2L2 , ∀f ∈ H

k. (2.1)

The inequality (2.1) can be easily proved by combing Young’s inequality and Gagliardo–Nirenberg’s inequality

|∂ ixf |Lp ≤ C(p)|f |αLq |∂

kx f |

(1−α)Lr , ∀f ∈ Hk (2.2)

where 1p −i3 =

1qα + (

1r −

k3 )(1− α)with i ≤ k.

2.1. A priori estimates

For this purpose, we assume that (ρ,u,H) is a smooth solution to (1.1)–(1.2) on the time interval (0, T )with ρ > 0. Weshall establish the following theorem:

Theorem 2.1. There exists a constant δ � 1 such that if

sup0≤t≤T

|(ρ − ρ̄,u,H)(·, t)|H3 ≤ δ (2.3)

then for any t ∈ [0, T ], there exists a constant C1 > 1 such that it holds

|(ρ − ρ̄,u,H)(·, t)|2H3 +∫ t

0|∂xρ(·, s)|2H2 + |(∂xu, ∂xH)(·, s)|

2H3ds ≤ C1|(ρ0 − ρ̄,u0,H0)|

2H3 . (2.4)

Proof. Setting % = ρ − ρ̄ and h′(ρ) = 1ρp′(ρ), we rewrite the system (1.1) as

%t + div[(% + ρ̄)u] = 0, (2.5)

ut + u · ∇u+∇[h(% + ρ̄)− h(ρ̄)] =1

% + ρ̄∆u+

1% + ρ̄

∇divu+1

% + ρ̄curlH× H, (2.6)

Ht + curl(u× H)−∆H = 0, divH = 0. (2.7)

We mention here that for convenience, sometimes we still denote % + ρ̄ by ρ.Firstly, we observe that the a priori assumption (2.3) and the Sobolev inequality together with the continuity equation

(2.5) imply

supx∈R3|(%, %t , ∂x%,u, ∂xu,H, ∂xH)(t)| ≤ C |(%,u,H)(·, t)|2H3 ≤ Cδ. (2.8)

In particular

ρ̄

2≤ ρ = % + ρ̄ ≤ 2ρ̄, (2.9)


and there exists a constant C0 such that

0 <1C0≤ h′(ρ) ≤ C0 <∞, |h(k)(ρ)| ≤ C0 for any positive integer k. (2.10)

In what follows, we will always use the smallness assumption of δ and (2.8)–(2.10).We divide the a priori estimates into four steps.

Step 1: L2-norms of u,HMultiplying the magnetic equation (2.7) byH and integrating overR3 (by parts), with the help of Hölder’s inequality and

(2.2), (2.8), we obtain

12ddt|H|2L2 + |∂xH|

2L2 = −

∫(u× H) · curlHdx

≤ |u|L3 |H|L6 |∂xH|L2 ≤ C |u|H1 |∂xH|2L2 ≤ Cδ|∂xH|

2L2 . (2.11)

Multiplying the momentum equation (2.6) by u and integrating over R3, we have

12ddt|u|2L2 −

∫1

% + ρ̄(∆u+∇divu) · udx

=

∫ (1

% + ρ̄curlH× H

)· u− (u · ∇u) · u+ [h(% + ρ̄)− h(ρ̄)]divudx. (2.12)

We estimate the second term on the left-hand side of (2.12) as

−

∫1

% + ρ̄(∆u+∇divu) · udx =

∫1

% + ρ̄(|∇u|2 + |divu|2)−

1(% + ρ̄)2

[(∇% ⊗ u) : ∇u+ divu(∇% · u)]dx

≥ C |∇u|2L2 − Cδ|(∂x%, ∂xu)|2L2 , (2.13)

where a ⊗ b = (aibj)3×3, for vectors a, b ∈ R3 and A : B = aijbij, for matrixes A, B ∈ R3×3. Also we estimate the terms onthe right-hand side of (2.12) as∫ (

1% + ρ̄

curlH× H)· udx ≤ C |curlH|L2 |H|L6 |u|L3 ≤ Cδ|∂xH|

2L2 (2.14)

−

∫(u · ∇u) · udx ≤ C |u|L3 |∂xu|L2 |u|L6 ≤ Cδ|∂xu|

2L2 (2.15)∫

[h(% + ρ̄)− h(ρ̄)]divudx = −∫[h(% + ρ̄)− h(ρ̄)]

%t + u · ∇%% + ρ̄

dx

= −ddt

∫H(%)dx−

∫h′(ρ̄ + θ%)% + ρ̄

%u · ∇%dx

≤ −ddt

∫H(%)dx+ Cδ|∂x%|2L2 (2.16)

where θ ∈ (0, 1) and H(%) is defined as

H(%) =∫ %

0

h(s+ ρ̄)− h(ρ̄)s+ ρ̄

ds. (2.17)

Hence since δ is small, we deduce from (2.11)–(2.16) that

12ddt

[|(u,H)|2L2 +

∫2H(%)dx

]+ C |(∂xu, ∂xH)|2L2 ≤ Cδ|∂x%|

2L2 . (2.18)

Step 2: L2-Norms of ∂3x u, ∂3xH

Applying the differential operator ∂ijk to Eq. (2.7), multiplying the resulting equation by ∂ijkH, and integrating it over R3,we obtain

12ddt

∫|∂ijkH|2dx+

∫|∇∂ijkH|2dx = −

∫∂ijk(u× H) · curl∂ijkHdx

≤ Cδ|(∂2x u, ∂3x u, ∂

2xH, ∂

3xH, ∂

4xH)|

2L2 ,


hence we get

ddt|∂3xH|

2L2 + |∂

4xH|

2L2 ≤ Cδ|(∂

2x u, ∂

3x u, ∂

2xH, ∂

3xH)|

2L2 . (2.19)

Applying ∂ijk to Eq. (2.6), multiplying the resulting equation by ∂ijku, and integrating it over R3, we have

12ddt

∫|∂ijku|2dx+

∫1

% + ρ̄[|∂ijk∇u|2 + (∂ijkdivu)2]dx =

∫∂ijk

(1

% + ρ̄curlH× H

)· ∂ijku− ∂ijk(u · ∇u) · ∂ijku

+ ∂ijk[h(% + ρ̄)− h(ρ̄)]∂ijkdivu+ ∂ijk(

1% + ρ̄

∆u)· ∂ijku

+1

% + ρ̄|∂ijk∇u|2 + ∂ijk

(1

% + ρ̄∇divu

)· ∂ijku+

1% + ρ̄

(∂ijkdivu)2dx. (2.20)

The estimates of the right-hand side of (2.20) are lengthy and we should estimate it term by term,∫∂ijk

(1

% + ρ̄curlH× H

)· ∂ijkudx = −

∫∂ij

(1

% + ρ̄curlH× H

)· ∂ijkkudx

≤ Cδ|(∂2x %, ∂4x u, ∂xH, ∂

2xH, ∂

3xH)|

2L2 (2.21)

−

∫∂ijk(u · ∇u) · ∂ijkudx =

∫∂ij(u · ∇u) · ∂ijkkudx

≤ Cδ|(∂2x u, ∂3x u, ∂

4x u)|

2L2 (2.22)∫

∂ijk[h(% + ρ̄)− h(ρ̄)]∂ijkdivudx = −∫∂ijk[h(% + ρ̄)− h(ρ̄)]∂ijk

(%t +∇% · u% + ρ̄

)dx

= −

∫[h′′′(ρ)∂i%∂j%∂k% + h′′(ρ)∂ik%∂j% + h′′(ρ)∂i%∂jk% + h′′(ρ)∂ij%∂k% + h′(ρ)∂ijk%]

×

[−6ρ4(%t +∇% · u)∂i%∂j%∂k% +

2ρ3(∂k%t∂i%∂j% + %t∂ik%∂j% + %t∂i%∂jk%

+ ∂k∇% · u∂i%∂j% +∇% · ∂ku∂i%∂j% +∇% · u∂ik%∂j% +∇% · u∂i%∂jk% + ∂j%t∂i%∂k%+ %t∂ij%∂k% + ∂j∇% · u∂i%∂k% +∇% · ∂ju∂i%∂k% +∇% · u∂ij%∂k% + ∂i%t∂j%∂k%

+ ∂i∇% · u∂j%∂k% +∇% · ∂iu∂j%∂k%)−1ρ2(∂jk%t∂i% + ∂j%t∂ik% + ∂k%t∂ij%

+ %t∂ijk% + ∂jk∇% · u∂i% + ∂j∇% · ∂ku∂i% + ∂j∇% · u∂ik% + ∂k∇% · ∂ju∂i%+∇% · ∂jku∂i% +∇% · ∂ju∂ik% + ∂k∇% · u∂ij% +∇% · ∂ku∂ij% +∇% · u∂ijk%+ ∂ik%t∂j% + ∂i%t∂jk% + ∂ik∇% · u∂j% + ∂i∇% · ∂ku∂j% + ∂i∇% · u∂jk%+ ∂k∇% · ∂iu∂j% +∇% · ∂iku∂j% +∇% · ∂iu∂jk% + ∂ij%t∂k% + ∂ij∇% · u∂k%

+ ∂i∇% · ∂ju∂k% + ∂j∇% · ∂iu∂k% +∇% · ∂iju∂k%)+1ρ(∂ijk%t + ∂ijk∇% · u

+ ∂ij∇% · ∂ku+ ∂ik∇% · ∂ju+ ∂i∇% · ∂jku+ ∂jk∇% · ∂iu+ ∂j∇% · ∂iku+ ∂k∇% · ∂iju+∇% · ∂ijku)]dx

≤ Cδ|(∂x%, ∂x%t , ∂2x %, ∂2x %t , ∂

3x %, ∂

2x u)|

2L2 −

∫[h′′(ρ)∂ik%∂j% + h′′(ρ)∂i%∂jk%

+ h′′(ρ)∂ij%∂k% + h′(ρ)∂ijk%] ×[−1ρ2(∂j%t∂ik% + ∂k%t∂ij% + ∂j∇% · u∂ik%

+ ∂k∇% · u∂ij% + ∂i%t∂jk% + ∂i∇% · u∂jk%)+1ρ(∂i∇% · ∂jku+ ∂j∇% · ∂iku

+ ∂k∇% · ∂iju)]dx−

∫[h′′′(ρ)∂i%∂j%∂k% + h′′(ρ)∂ik%∂j% + h′′(ρ)∂i%∂jk%

+ h′′(ρ)∂ij%∂k% + h′(ρ)∂ijk%] ×[1ρ(∂ijk%t + ∂ijk∇% · u)

]dx. (2.23)


The estimate of (2.23) is tedious. Observing from (2.2) and (2.8), we have that∫h′′(ρ)ρ2

∂i%∂jk%∂j%t∂ik%dx ≤ Cδ|∂2x %|2L4 |∂x%t |L2 ≤ Cδ|∂x%|L∞ |∂

3x %|L2 |∂x%t |L2

≤ Cδ|(∂x%t , ∂3x %)|2L2 .

Similarly

−

∫[h′′(ρ)∂ik%∂j% + h′′(ρ)∂i%∂jk% + h′′(ρ)∂ij%∂k% + h′(ρ)∂ijk%]

[−1ρ2(∂j%t∂ik% + ∂k%t∂ij% + ∂j∇% · u∂ik%

+ ∂k∇% · u∂ij% + ∂i%t∂jk% + ∂i∇% · u∂jk%)+1ρ(∂i∇% · ∂jku+ ∂j∇% · ∂iku+ ∂k∇% · ∂iju)

]dx

≤ C |(∂x%t , ∂2x %, ∂2x %t , ∂

3x %, ∂

2x u, ∂

3x u)|

2L2 . (2.24)

On the other hand, since

−

∫1ρh′′′(ρ)∂i%∂j%∂k%∂ijk%tdx =

∫∂i

[1ρh′′′(ρ)∂i%∂j%∂k%

]∂jk%tdx

≤ Cδ|(∂x%, ∂2x %, ∂2x %t)|

2L2 ,

and

−

∫h′′(ρ)ρ

∂ik%∂j%∂ijk%tdx =∫∂i

[h′′(ρ)ρ

∂ik%∂j%

]∂jk%tdx

≤ Cδ|(∂2x %, ∂2x %t , ∂

3x %)|

2L2 +

∫h′′(ρ)ρ

∂ik%∂ij%∂jk%tdx

≤ Cδ|(∂2x %, ∂2x %t , ∂

3x %)|

2L2 ,

we obtain

−

∫[h′′′(ρ)∂i%∂j%∂k% + h′′(ρ)∂ik%∂j% + h′′(ρ)∂i%∂jk% + h′′(ρ)∂ij%∂k%]

[1ρ(∂ijk%t + ∂ijk∇% · u)

]dx

≤ Cδ|(∂2x %, ∂2x %t , ∂

3x %)|

2L2 . (2.25)

Finally,

−

∫h′(ρ)ρ

∂ijk%(∂ijk%t + ∂ijk∇% · u)dx = −12ddt

∫h′(ρ)ρ

(∂ijk%)2dx+

12

∫ [h′(ρ)ρ

]t(∂ijk%)

2dx

+

∫ [h′′(ρ)2ρ

(∂ijk%)2∇% · u−

h′(ρ)2ρ2

(∂ijk%)2∇% · u+

h′(ρ)2ρ

(∂ijk%)2divu

]dx

≤ −12ddt

∫h′(ρ)ρ

(∂ijk%)2dx+ Cδ|∂3x %|

2L2 ,

together with (2.24)–(2.25), thus (2.23) can be replaced by∫∂ijk[h(% + ρ̄)− h(ρ̄)]∂ijkdivudx ≤ Cδ|(∂x%, ∂x%t , ∂2x %, ∂

2x %t , ∂

3x %, ∂

2x u, ∂

3x u)|

2L2 −

12ddt

∫h′(ρ)ρ

(∂ijk%)2dx. (2.26)

Since ∫∂ijk

(1

% + ρ̄∆u)· ∂ijku+

1% + ρ̄

|∂ijk∇u|2dx

=

∫ [−

6(% + ρ̄)4

∂i%∂j%∂k%∆u+2

(% + ρ̄)3(∂ik%∂j%∆u+ ∂i%∂jk%∆u+ ∂i%∂j%∂k∆u

+ ∂ij%∂k%∆u+ ∂i%∂k%∂j∆u+ ∂j%∂k%∂i∆u)−1

(% + ρ̄)2(∂ijk%∆u+ ∂ij%∂k∆u

+ ∂ik%∂j∆u+ ∂i%∂jk∆u+ ∂jk%∂i∆u+ ∂j%∂ik∆u+ ∂k%∂ij∆u)]· ∂ijku−

1(% + ρ̄)2

(∇% ⊗ ∂ijku) : ∇∂ijkudx

≤ Cδ|(∂3x %, ∂2x u, ∂

3x u, ∂

4x u)|

2L2 + C |∂

3x %|L2 |∂

2x u|L3 |∂

3x u|L6 + C |∂

2x %|L3 |∂

3x u|L6 |∂

3x u|L2

≤ Cδ|(∂3x %, ∂2x u, ∂

3x u, ∂

4x u)|

2L2 , (2.27)


and similarly, we can obtain∫∂ijk

(1

% + ρ̄∇divu

)· ∂ijku+

1% + ρ̄

(∂ijkdivu)2dx

=

∫ [−

6(% + ρ̄)4

∂i%∂j%∂k%∇divu+2

(% + ρ̄)3(∂ik%∂j%∇divu+ ∂i%∂jk%∇divu

+ ∂i%∂j%∂k∇divu+ ∂ij%∂k%∇divu+ ∂i%∂k%∂j∇divu+ ∂j%∂k%∂i∇divu)

−1

(% + ρ̄)2(∂ijk%∇divu+ ∂ij%∂k∇divu+ ∂ik%∂j∇divu+ ∂i%∂jk∇divu

+ ∂jk%∂i∇divu+ ∂j%∂ik∇divu+ ∂k%∂ij∇divu)]· ∂ijku−

1(% + ρ̄)2

(∇% · ∂ijku)∂ijkdivudx

≤ Cδ|(∂3x %, ∂2x u, ∂

3x u, ∂

4x u)|

2L2 . (2.28)

Thus (2.19)–(2.22) and (2.26)–(2.28) yield

12ddt

[|(∂3x u, ∂

3xH)|

2+

∫h′(ρ)ρ

(∂ijk%)2dx]+ C |(∂4x u, ∂

4xH)|

2L2 ≤ C |(∂x%, ∂x%t , ∂

2x %, ∂

2x %t , ∂

3x %, ∂

2x u, ∂

3x u, ∂

2xH, ∂

3xH)|

2L2

Since we can directly obtain from Eq. (2.5) that

|∂ ix%t | ≤ Ci+1∑k=1

(|∂kx%| + |∂kxu|), i = 1, 2

this together with the inequality (2.1), we finally arrive at

12ddt

[|(∂3x u, ∂

3xH)|

2+

∫h′(ρ)ρ

(∂ijk%)2dx]+ C |(∂4x u, ∂

4xH)|

2L2 ≤ C |(∂x%, ∂

3x %, ∂xu, ∂xH)|

2L2 . (2.29)

Hence (2.18) together with (2.29) yields

12ddt

[|(u, ∂3x u,H, ∂

3xH)|

2+

∫2H(%)+

h′(ρ)ρ

(∂ijk%)2dx]+ C |(∂xu, ∂4x u, ∂xH, ∂

4xH)|

2L2 ≤ C |(∂x%, ∂

3x %)|

2L2 . (2.30)

Step 3: L2-Norms of ∂x%, ∂3x %We firstly estimate for ∂x%. For this, we calculate as∫ [

12|∇%|2 +

(% + ρ̄)2

2∇% · u

]t=

∫∇% · ∇%t + (% + ρ̄)%t∇% · u+

(% + ρ̄)2

2∇%t · u+

(% + ρ̄)2

2∇% · utdx.

(2.31)

We estimate the first term of the right-hand side of (2.31) as follows:∫∇% · ∇%t = −

∫∇% · ∇div[(% + ρ̄)u]dx

= −

∫∇% · (∇2% · u)+∇% · (∇% · ∇u)+ |∇%|2divu+ (% + ρ̄)∇% · ∇divudx

≤ Cδ|∂x%|2L2 −∫(% + ρ̄)∇% · ∇divudx, (2.32)

where we have used the fact

∇% · (∇2% · u) = ∂i%∂ij%uj = ∇|∇%|2

2· u.

Also we estimate the middle two terms as∫(% + ρ̄)%t∇% · udx ≤ Cδ|(%t , ∂x%)|2L2 ≤ Cδ|(∂x%, ∂xu)|

2L2 , (2.33)


and ∫(% + ρ̄)2

2∇%t · udx = −

∫(% + ρ̄)2

2u · ∇div[(% + ρ̄)u]dx

= −

∫(% + ρ̄)2

2u · ∇(∇% · u)+

(% + ρ̄)2

2u · [divu∇% + (% + ρ̄)∇divu]dx

≤ Cδ|(∂x%, ∂xu)|2L2 +∫(% + ρ̄)3

2(divu)2dx. (2.34)

To estimate the last term, we utilize Eq. (2.6) to find∫(% + ρ̄)2

2∇% · ut =

∫(% + ρ̄)2

2∇% ·

{1

% + ρ̄curlH× H− u · ∇u−∇[h(% + ρ̄)− h(ρ̄)]

+1

% + ρ̄∆u+

1% + ρ̄

∇divu}dx

≤ Cδ|(∂x%, ∂xu, ∂xH)|2L2 −∫(% + ρ̄)2

2h′(ρ)|∂x%|2dx+

∫% + ρ̄

2∇% · (∆u+∇divu)dx. (2.35)

Since ∫(% + ρ̄)∇% · (∆u−∇divu)dx =

∫(% + ρ̄)∂i%∂jjui − (% + ρ̄)∂i%∂jiujdx

= −

∫∂j%∂i%∂jui + (% + ρ̄)∂ij%∂jui − ∂i%∂j%∂iuj − (% + ρ̄)∂ij%∂iujdx

= 0,

together with (2.31)–(2.35), and making use of (2.9) and (2.10), we obtain∫ [12|∇%|2 +

(% + ρ̄)2

2∇% · u

]t+ C

∫|∂x%|

2dx ≤∫(% + ρ̄)3

2(divu)2dx+ Cδ|(∂xu, ∂xH)|2L2 . (2.36)

Now we turn to estimate for ∂3x %. As in (2.31), by direct calculation, we have∫ [12(∂ijk%)

2+(% + ρ̄)2

2∂ijk%∂ijuk

]tdx =

∫−∂ijk%∂ijk[div(% + ρ̄)u] + (% + ρ̄)%t∂ijk%∂ijuk

−(% + ρ̄)2

2∂ijk[div(% + ρ̄)u]∂ijuk +

(% + ρ̄)2

2∂ijk%∂ijuktdx. (2.37)

We estimate the right-hand side of (2.37) as follows:∫−∂ijk%∂ijk[div(% + ρ̄)u]dx = −

∫∂ijk%[∂ijk∇% · u+ ∂ij∇% · ∂ku+ ∂ik∇% · ∂ju+ ∂i∇% · ∂jku+ ∂jk∇% · ∂iu

+ ∂j∇% · ∂iku+ ∂k∇% · ∂iju+∇% · ∂ijku+ ∂ik%∂jdivu+ ∂i%∂jkdivu+ ∂jk%∂idivu+ ∂j%∂ikdivu+ ∂k%∂ijdivu+ (% + ρ̄)∂ijkdivu]dx

≤ Cδ|(∂3x %, ∂3x u)|

2+

∫∇(∂ijk%)

2

2· udx−

∫(% + ρ̄)∂ijk%∂ijkdivudx

≤ Cδ|(∂3x %, ∂3x u)|

2L2 −

∫(% + ρ̄)∂ijk%∂ijkdivu, (2.38)∫

(% + ρ̄)%t∂ijk%∂ijukdx ≤ Cδ|(∂3x %, ∂2x u)|

2L2 , (2.39)

−

∫(% + ρ̄)2

2∂ijk[div(% + ρ̄)u]∂ijukdx = −

∫(% + ρ̄)2

2∂ijuk[∂ijk(∇% · u)+ ∂ik%∂jdivu+ ∂i%∂jkdivu+ ∂jk%∂idivu

+ ∂j%∂ikdivu+ ∂k%∂ijdivu+ (% + ρ̄)∂ijkdivu]dx

≤ Cδ|(∂2x %, ∂2x u, ∂

3x u)|

2L2 +

∫∂i

[(% + ρ̄)2

2∂ijuk

]∂jk(∇% · u)dx+

∫(% + ρ̄)3

2(∂ijdivu)2dx

≤ Cδ|(∂2x %, ∂3x %, ∂

2x u, ∂

3x u)|

2L2 +

∫(% + ρ̄)3

2(∂ijdivu)2dx (2.40)


and ∫(% + ρ̄)2

2∂ijk%∂ijuktdx

=

∫(% + ρ̄)2

2∂ijk%∂ij

{1

% + ρ̄(curlH× H)k − ul∂luk − ∂k[h(% + ρ̄)− h(ρ̄)] +

1% + ρ̄

∂lluk +1

% + ρ̄∂lkul

}dx

≤ Cδ|(∂x%, ∂2x %, ∂3x %, ∂

2x u, ∂

3x u, ∂

2xH, ∂

3xH)|

2L2 −

∫(% + ρ̄)2h′(ρ)

2(∂ijk%)

2dx

+

∫% + ρ̄

2∂ijk%(∂ijlluk + ∂ijlkul)dx. (2.41)

Since

−

∫(% + ρ̄)∂ijk%∂ijkdivu+

∫% + ρ̄

2∂ijk%(∂ijlluk + ∂ijlkul)dx = −

12

∫(% + ρ̄)∂ijk%(∂ijlkul − ∂ijlluk)dx

=12

∫∂l%∂ijk%∂ijkul + (% + ρ̄)∂ijkl%∂ijkul − ∂l%∂ijk%∂ijluk − (% + ρ̄)∂ijkl%∂ijlukdx

=12

∫∂l%∂ijk%∂ijkul − ∂l%∂ijk%∂ijlukdx

≤ Cδ|(∂3x %, ∂3x u)|

2L2 ,

together with (2.37)–(2.41), with the help of (2.9)–(2.10) and the inequality (2.1), we have∫ [12(∂ijk%)

2+(% + ρ̄)2

2∂ijk%∂ijuk

]tdx+ C

∫(∂ijk%)

2dx

≤ Cδ|(∂x%, ∂xu, ∂4x u, ∂xH, ∂4xH)|

2L2 +

∫(% + ρ̄)3

2(∂ijdivu)2dx. (2.42)

Step 4: ConclusionConsequently, multiplying (2.36) and (2.42) by two appropriate small constants α and β respectively, together with

(2.30), we can deduce

12ddt

{|(u, ∂3x u,H, ∂

3xH)|

2+

∫2H(%)+

h′(ρ)ρ|∂3x %|

2dx

+α

[12|∇%|2 +

(% + ρ̄)2

2∇% · u

]+ β

[12(∂ijk%)

2+(% + ρ̄)2

2∂ijk%∂ijuk

]}+ C(α, β)|(∂x%, ∂3x %, ∂xu, ∂

4x u, ∂xH, ∂

4xH)|

2L2 ≤ 0. (2.43)

Integrating (2.43) directly in time, by (2.1), (2.9), (2.10) and (2.17), with the help of the smallness of α and β , we can finishthe proof of Theorem 2.1. �

2.2. Proof of global existence

Wewill finish the proof of existence part of Theorem 1.1 in this subsection. First we state out the local existence withoutproof, since it can be proved in a standard way or can be found in [28,29]:

Theorem 2.2. Under the assumptions of Theorem 1.1, then there exists a positive constant T such that the initial value problem(1.1)–(1.2) has a unique solution (ρ,u,H) which is continuous in [0, T ] × R3 together with its derivatives of first order in t andof second order in x, and there exists a constant C2 > 1 such that the following inequality is satisfied:

|(ρ − ρ̄,u,H)(·, t)|2H3 +∫ t

0|∂xρ(·, s)|2H2 + |(∂xu, ∂xH)(·, s)|

2H3ds ≤ C2|(ρ0 − ρ̄,u0,H0)|

2H3 , (2.44)

for any t ∈ [0, T ].

The global existence of smooth solutions will be proved by a continued argument, combing the local existence theorem andthe a priori estimates theorem. Suppose

E0 = |(ρ0 − ρ̄,u0,H0)|H3 < min(δ/√C2, δ/

√C1C2), (2.45)


where δ is defines in Theorem 2.1. Since the initial data satisfies E0 < δ/√C2, then by Theorem 2.2, there exists a positive

constant T1 > 0, such that the smooth solution of (1.1)–(1.2) exists on [0, T1] and has the following estimate:

|(ρ − ρ̄,u,H)(·, t)|2H3 +∫ t

0|∂xρ(·, s)|2H2 + |(∂xu, ∂xH)(·, s)|

2H3ds ≤ C2E

20 ,

for 0 ≤ t ≤ T1. It implies

E1 = sup0≤t≤T1

|(ρ − ρ̄,u,H)(·, t)|H3 ≤√C2E0 < δ. (2.46)

Therefore the solution satisfies the a priori estimate (2.3), by Theorem 2.1 and (2.45) we have

E1 ≤√C1E0 < δ/

√C2. (2.47)

Thus by Theorem2.2 the initial problem (1.1) for t ≥ T1, with the initial data (ρ,u,H)(x, T1)has again a unique local solution(ρ,u,H) satisfying:

|(ρ − ρ̄,u,H)(·, t)|2H3 +∫|∂xρ(·, s)|2H2 + |(∂xu, ∂xH)(·, s)|

2H3ds ≤ C2|(ρ − ρ̄,u,H)(·, T1)|

2H3 ,

for T1 < t < 2T1. This together with (2.45) and (2.47), it yields

supT1≤t≤T2

|(ρ − ρ̄,u,H)(·, t)|H3 ≤√C2E1 ≤

√C1C2E0 < δ. (2.48)

Then by (2.46), (2.48) and Theorem 2.1, we have

E2 = sup0≤t≤2T1

|(ρ − ρ̄,u,H)(·, t)|2H3(R3) ≤√C1E0 ≤ δ/

√C2.

Thus we can continue the same process for 0 ≤ t ≤ nT1, n = 3, 4, 5, . . . and finally get the global solution and the estimate(1.4). The proof of uniqueness of solutions is easy, then we omit it.

3. Convergence rate of the solution

In this section we shall prove the decay rates of the solution to finish the proof of Theorem 1.1. In Section 3.1, we willreformulate the problem and list some elementary conclusions on the decay-in-time estimates for the linearized systemand a useful inequality. In Section 3.2, we shall firstly obtain the energy inequality for the derivatives of the orders from thefirst to the third, and then we show a decay-in-time estimate for the first order derivatives, where the error is related to thederivatives of the higher order. Finally, by combining these two estimates we get the optimal decay rates.

3.1. Reformulated system

We will reformulate the problem (1.1)–(1.2) as follows. Set

γ =√P ′(ρ̄), µ =

1ρ̄.

Introducing new variables by

% = ρ − ρ̄, v =1µγ

u, H = H, (3.1)

the initial value problem (1.1)–(1.2) is reformulated as%t + γ divv = S1,vt + γ∇% − µ∆v− µ∇divv = S2,Ht −∆H = S3,(%, v,H)(x, 0) = (%0, v0,H0)(x)→ (0, 0, 0) as |x| → ∞,

(3.2)

where

S1 = −µγ div(%v), (3.3)

S2 =1

µγρcurlH× H+

(1ρ−1ρ̄

)∆v+

(1ρ−1ρ̄

)∇divv− µγ v · ∇v−

1µγ

[P ′(ρ)ρ−P ′(ρ̄)ρ̄

]∇%, (3.4)

S3 = −µγ curl(v× H). (3.5)

We shall consider the convergence rates of the solution (%, v,H) to the steady state (0, 0, 0). For later use, the result onthe global existence of solutions to (3.2) is restated as follows:


Proposition 3.1. Under the assumption of divH0 = 0 and (1.3), there exists a unique globally smooth solution (%, v,H) of theCauchy problem (3.1) satisfying for any t ∈ [0,∞),

|(%, v,H)(·, t)|2H3 +∫ t

0|∂x%(·, s)|2H2 + |(∂xv, ∂xH)(·, s)|

2H3ds ≤ C |(%0, v0,H0)|

2H3 . (3.6)

Moreover, (ρ,u,H) which satisfies (3.1) uniquely solves the initial problem (1.1)–(1.2) for all time.

To use the Lp-Lq estimates of the linear problem for the nonlinear problem (3.2)1–(3.2)2, we rewrite the solution of(3.2)1–(3.2)2 as

U(t) = E(t)U0 +∫ t

0E(t − s)F(U(s),H(s))ds, (3.7)

where we use the notations

U = [%, v]T , U0 = [%0, v0]T , F = [S1, S2]T , (3.8)

and E(t) is the solution semigroup defined by E(t) = e−tA, t ≥ 0, with A being a matrix-valued differential operator givenby

A =(0 γ divγ∇ −µ∆− µ∇div

).

The semigroup E(t) has the following properties on the decay in time, which can be found in [25,26].

Lemma 3.2. Let k ≥ 0 is an integer and 1 ≤ p ≤ 2 ≤ q <∞. Then for any t ≥ 0, it holds that

|∇kE(t)U0|Lq ≤ C(1+ t)−σ(p,q;k)|U0|Lp∩Hk , (3.9)

here σ(p, q; k) is defined by (1.8) and | · |Lp∩Hk = | · |Lp + | · |Hk .

To treat the magnetic field, we notice that the solution of the heat equation (3.2)3 has the formula as:

H(x, t) =∫

1

(4π t)32e−|x−y|24t H0(y)dy+

∫ t

0

∫1

[4π(t − s)]32e−|x−y|24(t−s) S3(y, s)dyds. (3.10)

We state the large-time behavior of solutions to the heat equation as the following lemma which can be obtained by directcalculation or one can refer to [23].

Lemma 3.3. For the solution H of the heat equation (3.2)3 with the Cauchy data H(x, 0) = H0, there exists a constant C suchthat

|∂kxH|Lq ≤ C(1+ t)−σ(p,q;k)

|H0|Lp + C∫ t

0(1+ t − s)−σ(p,q;k)|S3(·, s)|Lpds, k = 0, 1 (3.11)

for any t ≥ 0, 1 ≤ p, q ≤ +∞, as well as σ is defined by (1.8).

We finish this subsection by listing an elementary but useful inequality [17]:

Lemma 3.4. If r1 > 1 and r2 ∈ [0, r1], then it holds that∫ t

0(1+ t − τ)−r1(1+ τ)−r2dτ ≤ C(r1, r2)(1+ t)−r2 .

3.2. Convergence rates

Now we will show the energy inequality as follows:

Lemma 3.5. Under the assumption of Proposition 3.1, let (%, v,H) be the solution to the initial problem (3.2), and (ρ,u,H)satisfies (3.1), then there are two positive constants C and D2 such that if δ0 > 0 in (1.3) is small enough, it holds that

dM(t)dt+ D2M(t) ≤ C |(∂x%, ∂xv, ∂xH)|2L2 , (3.12)

where the energy functional M(t) defined by (3.15) is equivalent to |(∂x%, ∂xv, ∂xH)(t)|2H2 , that is, there exists a positive constantC3 > 0 such that

1C3|(∂x%, ∂xv, ∂xH)(t)|2H2 ≤ M(t) ≤ C3|(∂x%, ∂xv, ∂xH)(t)|

2H2 . (3.13)


Proof. Taking ∂x to (2.6) and (2.7), multiplying the resulting equations by ∂xu and ∂xH respectively, and integrating themover R3, we have

ddt

[|∂xu|2L2 +

∫h′(ρ)ρ|∂x%|

2dx]+ |∂2x u|

2L2 ≤ Cδ0|(∂x%, ∂xu, ∂xH)|

2L2 ,

andddt|∂xH|2L2 + |∂

2xH|

2L2 ≤ Cδ0|∂xu|

2L2 ,

here we omit some complicated calculations since it is analogous to get the resulting inequalities as in Section 2.1. Then,together with (2.29) and (2.42), by using (2.1), it yields

ddt

[|(∂3x %, ∂xu, ∂

3x u, ∂xH, ∂

3xH)|

2L2 +

∫h′(ρ)ρ

(|∂x%|

2+ |∂3x %|

2)+ ρ2∂ijk%∂ijukdx

]+ C |(∂3x %, ∂

2x u, ∂

4x u, ∂

2xH, ∂

4xH)|

2L2

≤ Cδ0|(∂x%, ∂xH)|2L2 + C |∂xu|2L2 . (3.14)

Define the temporal energy functional

M(t) = |(∂3x %, ∂xu, ∂3x u, ∂xH, ∂

3xH)|

2L2 +

∫h′(ρ)ρ

(|∂x%|2+ |∂3x %|

2)+ ρ2∂ijk%∂ijukdx, (3.15)

where we notice thatM(t) is equivalent to |(∂x%, ∂xv, ∂xH)|2H2 . By choosing a sufficiently large constant D1 > 0, and addingD1|(∂x%, ∂xu, ∂xH)|2L2 to both sides of (3.14), we have got (3.12) by (3.1). �

Next we shall estimate the decay rate of the first order derivatives.

Lemma 3.6. Under the assumption of Proposition 3.1, let (%, v,H) be the solution to the initial problem (3.2). Then we have

|(∂x%, ∂xv, ∂xH)|L2 ≤ CK0(1+ t)−σ(p,2;1)

+ Cδ0

∫ t

0(1+ t − s)−σ(p,2;1)|(∂x%, ∂xv, ∂xH)(·, s)|H2ds, (3.16)

where K0 = |(%0, v0,H0)|Lp∩H1 is finite by (1.3) and (1.5). Here 1 ≤ p <65 and σ is defined by (1.8).

Proof. First, it is noticed from (3.3)–(3.5) that

S1 ∼ ∂i%vi + %∂ivi;S2 ∼ (curlH× H)i + %∂jjvi + %∂ijvj + vj∂jvi + %∂i%;S3 ∼ [curl(v× H)]i,

where we use the fact that P′(ρ)

ρ−P ′(ρ̄)ρ̄∼ %, 1

ρ−1ρ̄∼ % and (2.9).

From the integral formula (3.7) and Lemma 3.2, we have

|(∂x%, ∂xv)|L2 ≤ CK0(1+ t)−σ(p,2;1)

+ C∫ t

0(1+ t − s)−σ(p,2;1)|(S1, S2)(·, s)|Lp∩H1ds,

together with (3.11), then we obtain

|(∂x%, ∂xv, ∂xH)|L2 ≤ CK0(1+ t)−σ(p,2;1)

+ C∫ t

0(1+ t − s)−σ(p,2;1)[|(S1, S2)(·, s)|Lp∩H1 + |S3(·, s)|Lp ]ds, (3.17)

where 1 ≤ p < 65 . Hence, to derive (3.16), we need to control |(S1, S2)|Lp∩H1 and |S3|Lp by the L

2-norm of the derivatives of%, v,H.Thus, we estimate them. Since

|(∂i%vi, %∂ivi)|Lp ≤ |(%, v)|L2p2−p|(∂x%, ∂xv)|L2

≤ C |(%, v)|H1 |(∂x%, ∂xv)|L2≤ Cδ0|(∂x%, ∂xv)|L2 ,

|(∂i%vi, %∂ivi)|L2 ≤ Cδ0|(∂x%, ∂xv)|L2 ,

and

|∂x(∂i%vi, %∂ivi)|L2 ≤ Cδ0|(∂x%, ∂2x %, ∂xv, ∂

2x v)|L2 ,


together with (3.11) and by using (3.20)–(3.22) and (3.24), it yields

|(%, v,H)|Lq ≤ CK0(1+ t)−σ(p,q;0) + C∫ t

0(1+ t − s)−σ(p,q;0)[|(S1, S2)(·, s)|Lp∩L2 + |S3(·, s)|Lp ]ds

≤ CK0(1+ t)−σ(p,q;0) + Cδ0

∫ t

0(1+ t − s)−σ(p,q;0)|(∂x%, ∂xv, ∂xH)|H2ds

≤ CK0(1+ t)−σ(p,q;0) + Cδ0

∫ t

0(1+ t − s)−σ(p,q;0)(1+ s)−σ(p,2;1)dsN(t)

≤ CK0(1+ t)−σ(p,q;0) + Cδ0[M(0)+ K 20 ]12

∫ t

0(1+ t − s)−σ(p,q;0)(1+ s)−σ(p,2;1)ds

≤ C(1+ t)−σ(p,q;0) (3.26)

where 2 ≤ q ≤ 6. (3.25)–(3.26) implies (1.6)–(1.7) with the help of (3.1). �

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