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Decay rate of strong solution for the compressible magnetohydrodynamic equations with large initial data
Journal Pre-proof Decay rate of strong solution for the compressible magnetohydrodynamic equations with large initial data
Jincheng Gao, Zhengzhen Wei, Zheng-an Yao
PII: DOI: Reference:
S0893-9659(19)30424-0 https://doi.org/10.1016/j.aml.2019.106100 AML 106100
To appear in:
Applied Mathematics Letters
Received date : 3 September 2019 Revised date : 14 October 2019 Accepted date : 14 October 2019 Please cite this article as: J.C. Gao, Z.Z. Wei and Z.A. Yao, Decay rate of strong solution for the compressible magnetohydrodynamic equations with large initial data, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106100. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
Journal Pre-proof *Manuscript Click here to view linked References
Decay Rate of Strong Solution for the Compressible Magnetohydrodynamic Equations with Large Initial Data Jincheng Gao†
Zhengzhen Wei‡
Zheng-an Yao]
School of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, P. R. China
Abstract
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In this paper, we study the decay rate of strong solution for the compressible magnetohydrodynamics(MHD) equations with large initial data in three-dimensional whole space. First of all, it is shown that both the first and the second 5 order spatial derivatives of magnetic field converge to zero at the same L2 −rate (1 + t)− 4 . Furthermore, we also prove that the lower bounds of decay rate for the magnetic field itself and first order spatial derivative at the L2 −rate are 5 3 (1 + t)− 4 and (1 + t)− 4 respectively. Keywords: Compressible MHD equations; Decay rate; Large initial data.
1
Introduction
In this paper, we investigate the lower and upper bounds of decay rate for the global solution of compressible magnetohydrodynamic(MHD) equations, which describe the motion of electrically conducting media in the presence of a magnetic field. The compressible MHD equations, which describe the coupling between the compressible Navier-Stokes equations and the magnetic equations, can be written as ∂t ρ + div(ρu) = 0, (1.1) ∂t (ρu) + div(ρu ⊗ u) − µ∆u − (µ + λ)∇ div u + ∇P = (∇ × B) × B, ∂t B − ∇ × (u × B) = −∇ × (ν∇ × B), div B = 0,
ur
where (x, t) ∈ R3 × R+ . The unknown functions ρ, u = (u1 , u2 , u3 ), B = (B1 , B2 , B3 ) and P represent the density, velocity, magnetic field and pressure respectively. The pressure P is given by smooth function P = P (ρ) = ργ with the adiabatic exponent γ ≥ 1. The constants µ and λ are the viscosity coefficients, ν is the magnetic diffusion coefficient, which satisfy the following conditions: µ > 0, 2µ + 3λ > 0, ν > 0. Although the electric field E does not appear in (1.1), it can be written in terms of the magnetic field and velocity as follows: E = ν∇ × B − u × B. To complete the system (1.1), initial data is given by (ρ, u, B)(x, t)|t=0 = (ρ0 (x), u0 (x), B0 (x)).
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Furthermore, we also assume that the initial data satisfies the conditions divB0 (x) = 0,
lim (ρ0 , u0 , B0 )(x) = (1, 0, 0).
|x|→∞
Due to their physical importance and mathematical challenges, there are large literature on compressible MHD system (1.1). For the case of dimension one, there are many mathematical results about the non-isentropic compressible MHD equations, for uniqueness and continuous dependence on initial data of weak solution [10], vanishing shear viscosity limit [6], global well-posedness with large initial data on whole space [14], free boundary [1] and bounded domain [2]. For the two-dimensional case, Kawashima [13] obtained the global existence of smooth solution to the general electromagnetic fluid equations when the initial data is small perturbation of some given constant state. For the dimension three, Ducomet and Feireisl [5] obtained the existence of global weak solution on a bounded spatial domain supplemented with conservative boundary conditions. Hu and Wang [11, 12] and Fan and Yu [7] obtained the global weak solution to the nonlinear Email: † [email protected], ‡ [email protected], ] [email protected]
Journal Pre-proof J.C.Gao, Z.Z.Wei, Z.A.Yao
compressible MHD equations with general initial data respectively. Motivated by the work for the compressible NavierStokes equations [16, 17], the recent work [3, 15, 18] not only established the global existence of small solution under the H 3 −framework, but also obtained the decay rate if the initial data belongs to Lq (q ∈ [1, 6/5)) additionally. Later, these work were improved by [8, 9] in the sense that the higher order spatial derivatives of solution obey optimal decay rate. The decay rate of small solution for the compressible MHD equations was also studied [20] if the initial data belongs to some negative Sobolev space rather than Lq (q ∈ [1, 6/5)). Very recently, Chen, Huang and Xu [4] investigated the global-in-time stability of large solution to the compressible MHD equations (1.1), and established the upper decay rate: 3
k(ρ − 1)(t)kH 1 + ku(t)kH 1 + kB(t)kH 1 6 C(1 + t)− 4 .
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Here they required the condition supt∈R+ kB(t, ·)kL∞ + supt∈R+ kρ(t, ·)kC α 6 M for small 0 < α < 1, and the initial data (ρ0 − 1, u0 , B0 ) belongs to L1 (R3 ) ∩ H 2 (R3 ). Based on the recent work [4], we are interested in the decay rate of magnetic field for the compressible MHD equations (1.1) with large initial data. And hence, we hope to address the following three problems: 3 (i)The first order spatial derivative of magnetic field converging to zero has faster decay rate than (1 + t)− 4 or not? (ii)The second order spatial derivative of magnetic field will converge to zero or not? (iii)Can we provide some lower bounds of decay rate for the magnetic field itself and higher order spatial derivatives? The main purpose of this paper is to provide an affirmative answer along this direction. Notation: In this paper, we use H s (s ∈ R3 ) to denote the usual Sobolev space with norm k · kH s and Lp (R3 ) to denote the usual Lp space with norm k · kLp . F(f ) := fb represents the usual Fourier transform of the function f . The constant C denotes the generic positive constant independent of time, and may change from line to line. First of all, we recall the following results obtained in [4], which will be used in this paper frequently. Theorem 1.1 (see [4]). Let (ρ, u, B) be a global and smooth solution of (1.1), if (ρ0 − 1, u0 , B0 ) ∈ L1 (R3 ) ∩ H 2 (R3 ), 0 6 ρ 6 ρ¯, and supt∈R+ kB(t, ·)kL∞ + supt∈R+ kρ(t, ·)kC α 6 M , then the global solution (ρ, u, B) has the uniform-in-time bounds and time decay as follows Z t k(ρ − 1)(t)k2H 2 + ku(t)k2H 2 + kB(t)k2H 2 + (k∇ρ(τ )k2H 1 + k∇u(τ )k2H 2 + k∇B(τ )k2H 2 )dτ 6 C1 , (1.2) 0
3
k(ρ − 1)(t)kH 1 + ku(t)kH 1 + kB(t)kH 1 6 C2 (1 + t)− 4 ,
(1.3)
where the constants C1 and C2 are independent of time.
In this paper, we hope to establish optimal decay rate for the higher order spatial derivatives of magnetic field, and give the lower bound of decay rate for magnetic field. Our main results can be stated as follows: Theorem 1.2. Under the assumptions of Theorem 1.1, then the magnetic field B(x, t) of global solution to the Cauchy problem (1.1) has the time decay rate 5
k∇B(t)kH 1 + k∂t B(t)kL2 6 C3 (1 + t)− 4 ,
t > T1 .
(1.4)
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c0 satisfies |B c0 | ≥ c0 , 0 ≤ |ξ| 1, with c0 > 0 a constant. Furthermore, assume that the Fourier transform F(B0 ) = B Then, the magnetic field B(x, t) obtained in Theorem 1.1 has the decay rates for all t > T2
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C4 (1 + t)−
3+2k 4
≤ k∇k B(t)kL2 6 C3 (1 + t)−
3+2k 4
, k = 0, 1.
(1.5)
Here Ti (i = 1, 2) is a positive large time, and Ci (i = 3, 4) is a constant independent of time. Remark 1.1. Compared with decay rate (1.3), the advantage of decay rate (1.4) not only implies that the second order spatial derivative of magnetic field tends to zero, but also shows that the first and second order spatial derivatives of 5 magnetic tend to zero at the L2 −rate (1 + t)− 4 . Remark 1.2. By the Sobolev interpolation inequality, it is shown that the magnetic field tends to zero at the Lp (2 ≤ p ≤ 3 1 6)−rate (1 + t)− 2 (1− p ) . Remark 1.3. Since the density and velocity satisfy the hyperbolic and parabolic coupled system, we can not obtain the optimal decay rate for the first and second order spatial derivatives of density and velocity just taking the method in this paper. 2
Journal Pre-proof Decay Rate for Compressible MHD Equations
2
Proof of Theorem 1.2
In this section, we will give the proof of Theorem 1.2. To this end, the upper decay rate (1.4) will be established as follows. Lemma 2.1. Under the assumptions in Theorem 1.1, the magnetic field B(x, t) obtained in Theorem 1.1 has the decay rate for all t > T1 5 k∇B(t)kH 1 + k∂t B(t)kL2 6 C(1 + t)− 4 , (2.1) where T1 is defined below. Proof. By virtue of the divergence free condition divB = 0, the third equation of (1.1) can be written as follows ∆
Bt − ν∆B = B · ∇u − u · ∇B − (div u)B = F.
(2.2)
Applying ∇ operator to (2.2), multiplying by ∇B and integrating over R3 , we get Z Z Z 1 d |∇B|2 dx + ν |∇2 B|2 dx = ∇F · ∇Bdx. 2 dt R3 R3 R3
(2.3)
R3
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Integrating by parts and applying H¨older inequality, it is easy to check that Z ∇F · ∇Bdx 6 (kBkL∞ k∇ukL2 + kukL3 k∇BkL6 )k∇2 BkL2 1
(2.4)
3
6 CkBkL4 2 k∇2 BkL4 2 k∇ukL2 k∇2 BkL2 + CkukH 1 k∇2 Bk2L2 .
Here we have used the Sobolev inequalities: kukL3 ≤ CkukH 1 and k∇BkL6 ≤ Ck∇2 BkL2 . Substituting (2.4) into (2.3), and using Young inequality, it follows that Z Z 1 d |∇B|2 dx + ν |∇2 B|2 dx 2 dt R3 3 R 1 2 8 6 CkBkL2 k∇ukL2 + νk∇2 Bk2L2 + CkukH 1 k∇2 Bk2L2 4 3 1 − 15 2 6 C(1 + t) + νk∇2 Bk2L2 + C(1 + t)− 4 k∇2 Bk2L2 , 4 4
where we have used the decay rate (1.3) in the last inequality. Then, for some large time T1 := (4C/ν) 3 , it holds on Z Z 15 d |∇B|2 dx + ν |∇2 B|2 dx 6 C(1 + t)− 2 , t > T1 . (2.5) dt R3 R3 For some constant R defined below, denoting the time sphere(see [19]) ∆
R 1 ) 2 }, 1+t
ur
S0 = {ξ ∈ R3 | |ξ| 6 (
(2.6)
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and we have immediately by using the Plancherel theorem that Z Z Z Z Z R R R2 2 2 4 b 2 2 b 2 2 b 2 b 2 dξ. |∇ B| dx > |ξ| |B| dξ > |ξ| |B| dξ > |ξ| |B| dξ − |B| 1 + t R3 /S0 1 + t R3 (1 + t)2 S0 R3 R3 /S0 Combining (2.6) with (2.5) yields directly Z 15 7 d Rν R2 |∇B|2 dx + k∇Bk2L2 6 kBk2L2 + C(1 + t)− 2 6 CR2 (1 + t)− 2 . dt R3 1+t (1 + t)2
Choosing R = ν3 , multiplying the resulting inequality by (1 + t)3 and integrating over [T1 , t], it holds on for t > T1 1
1
5
k∇B(t)k2L2 ≤ C(1 + t)−3 ((1 + T1 )3 k∇B(T1 )k2L2 + (1 + t) 2 − (1 + T1 ) 2 ) 6 C(1 + t)− 2 , where we have used the uniform bound (1.2). Using equality (2.2), straightforward computation shows that Z Z Z 1 d |∇2 B|2 dx + ν |∇3 B|2 dx = ∇2 F · ∇2 Bdx. 2 dt R3 R3 R3 3
(2.7)
Journal Pre-proof J.C.Gao, Z.Z.Wei, Z.A.Yao
Integrating by part, applying H¨older inequality and decay rate (1.3), it is easy to obtain Z ∇2 F · ∇2 Bdx 6 C(k∇BkL2 k∇ukL∞ + kBkL6 k∇2 ukL3 + kukL3 k∇2 BkL6 )k∇3 BkL2 R3
1 6 Ck∇Bk2L2 k∇2 uk2H 1 + νk∇3 Bk2L2 + CkukH 1 k∇3 Bk2L2 4 3 1 6 Ck∇Bk2L2 k∇2 uk2H 1 + νk∇3 Bk2L2 + C(1 + t)− 4 k∇3 Bk2L2 . 4
Using the decay rate (2.7), then it holds on for all t ≥ T1 Z Z 5 d |∇2 B|2 dx + ν |∇3 B|2 dx 6 C(1 + t)− 2 k∇2 uk2H 1 . dt R3 R3 Similar to the inequality (2.6), it follows that
and hence, we have d dt
Z
R3
9 R R R2 k∇Bk2L2 ≥ k∇2 Bk2L2 − k∇2 Bk2L2 − CR2 (1 + t)− 2 , 1+t (1 + t)2 1+t
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k∇3 Bk2L2 >
|∇2 B|2 dx +
Rν 1+t
Z
9
R3
5
|∇2 B|2 dx 6 CR2 (1 + t)− 2 + C(1 + t)− 2 k∇2 uk2H 1 .
Choosing R = ν3 , multiplying the resulting inequality by (1 + t)3 , and integrating over [T1 , t], we obtain for t ≥ T1 5
k∇2 B(t)k2L2 6 C(1 + t)− 2 ,
(2.8)
where we have used uniform estimate (1.2). Using the equation (2.2), we get 5
k∂t BkL2 6 νk4BkL2 + kF kL2 6 νk4BkL2 + k∇BkH 1 kukH 1 6 C(1 + t)− 4 . This and the decay rates (2.7) and (2.8) imply (2.1). Therefore, we complete the proof of this lemma. 5
Next, we will establish the lower bound of decay rate (1.5), which shows that the L2 −rate (1 + t)− 4 for the first order spatial derivative of magnetic field will be optimal. c0 satisfies |B c0 | ≥ c0 , 0 ≤ |ξ| 1, with c0 > 0 a constant. Lemma 2.2. Assume that the Fourier transform F(B0 ) = B Then, the magnetic field B(x, t) obtained in Theorem 1.1 has the decay rate for all t > T2 k∇k B(t)kL2 ≥ C(1 + t)−
,
k = 0, 1,
ur
where T2 is defined below.
3+2k 4
(2.9)
Proof. Step 1: Consider the linear equation corresponding to the third equation of (1.1)
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∂t Bl (x, t) − ν∆Bl (x, t) = 0, divBl (x, t) = 0, (x, t) ∈ R3 × R+ ;
Bl (x, t) |t=0 = B0 (x), x ∈ R3 .
Using the semigroup method, we can obtain the solution: Bl (x, t) = e−ν∆t B0 (x). It is easy to check that Z Z Z 2 3 b0 |2 e−2ν|ξ|2 t dξ > c20 |Bl |2 dx = |B e−2ν|ξ| t dξ > C(1 + t)− 2 . R3
Similarly, we also have
R3
Z
R3
∆
|∇Bl |2 dx =
(2.10)
(2.11)
|ξ| 1
Z
R3
2
5
b0 |2 |ξ|2 e−2ν|ξ| t dξ > C(1 + t)− 2 . |B
(2.12)
Step 2: Define Bδ (x, t) = B(x, t) − Bl (x, t), and hence, Bδ (x, t) will satisfy the following equation ∆
∂t Bδ − ν∆Bδ = ∇ × (u × B) = F, 4
divBδ = 0,
Bδ |t=0 = 0.
(2.13)
Journal Pre-proof Decay Rate for Compressible MHD Equations
Using the semigroup method and Duhamel principle formula, it holds on 12 Z t Z −2ν|ξ|2 (t−τ ) b 2 e |F | dξ dτ kBδ (t)kL2 ≤ R3
0
≤
Z t (Z
|ξ|61
0
≤C ≤C
Z
t
0
Z
t
0
2 −2ν|ξ|2 (t−τ )
|ξ| e
−2
|ξ|
5
Z
|Fb| dξ + 2
e
−2ν|ξ|2 (t−τ )
|ξ|>1
(1 + t − τ )− 4 (k|ξ|−1 FbkL∞ + kF kL2 )dτ
|Fb| dξ 2
) 12
dτ
5
(1 + t − τ )− 4 (k(u, B)k2L2 + k∇ × (u × B)kL2 )dτ.
Using the decay rate (1.3), we get Z t Z t 5 3 5 5 (1 + t − τ )− 4 (1 + τ )− 2 dτ ≤ C(1 + t)− 4 . (1 + t − τ )− 4 k(u, B)k2L2 dτ ≤ C 0
0
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Using the Sobolev inequality, H¨older inequality, uniform bound (1.2) and decay rate (1.3), it follows that Z t 5 (1 + t − τ )− 4 k∇ × (u × B)kL2 dτ 0 Z t 1 3 5 ≤C (1 + t − τ )− 4 k∇(u, B)kL2 2 k∇2 (u, B)kL2 2 dτ 0
≤C
≤C
Z
t
(1 + t − τ )
0
Z
0
− 53
t
k∇(u, B)k2L2 dτ 3
5
(1 + t − τ )− 3 (1 + τ )− 2 dτ
43 Z
34
t
0
k∇
2
(u, B)k2L2 dτ
14
9
≤ C(1 + t)− 8 .
Thus, the decay rate holds on for all t > 0
9
kBδ (t)kL2 ≤ C(1 + t)− 8 .
Using the equation (2.13), Sobolev inequality and Cauchy inequality, we get Z Z 1 d 2 |∇Bδ | dx + ν |∇2 Bδ |2 dx 2 dt R3 R3 1
3
≤ CkBkL2 2 k∇2 BkL2 2 k∇uk2L2 + Ckuk2H 1 k∇2 Bk2L2 + 9
≤ C(1 + t)− 2 +
ν k∆Bδ k2L2 , 2
(2.14)
ν k∆Bδ k2L2 2
and hence, similar to the decay rate (2.7), it holds on for all t ≥ T1
13
ur
k∇Bδ (t)kL2 ≤ C(1 + t)− 8 .
(2.15)
Step 3: The combination of (2.11) and (2.14) yields 3
9
3
Jo
kBkL2 > kBl kL2 − kBδ kL2 > C(1 + t)− 4 − C(1 + t)− 8 = C(1 + t)− 4 −
C
3
3
(1 + t) 8
(1 + t)− 4 .
Similarly, we combine the decay rates (2.12) and (2.15) to get 5
k∇BkL2 > C(1 + t)− 4 −
C
5
3
(1 + t) 8
(1 + t)− 4 .
If we show that decay rate (2.9) holds on for t ≥ T2 (T2 being large enough). Thus, we complete the proof of this lemma.
Acknowledgements Jincheng Gao’s research was partially supported by Fundamental Research Funds for the Central Universities(Grant No.18lgpy66) and NNSF of China(Grant Nos. 11571380 and 11801586). Zheng-an Yao’s research was partially supported by NNSF of China(Grant Nos.11431015 and 11971496). 5
Journal Pre-proof J.C.Gao, Z.Z.Wei, Z.A.Yao
References [1] G.Q.Chen, D.H.Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations 182(2) (2002) 344-376. [2] G.Q.Chen, D.H.Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys. 54(4) (2003) 608-632. [3] Q.Chen, Z.Tan, Global existence and asymptotic analysis of weak solutions to the equations of magnetohydrodynamics, Nonlinear Anal. 72(12) (2010) 4438-4451. [4] Y.H.Chen, J.C.Huang, H.Y.Xu, Global stability of large solutions of the 3-D compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl. 47 (2019) 272-290. [5] B.Ducomet, E.Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. 266(3) (2006) 595-629.
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[6] J.S.Fan, S.Jiang, G.Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm.Math.Phys. 270(3) (2007) 691-708. [7] J.S.Fan, W.H.Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal. 69(10) (2008) 3637-3660. [8] J.C.Gao, Y.H.Chen, Z.A.Yao, Long-time behavior of solution to the compressible magnetohydrodynamic equations, Nonlinear Anal. 128 (2015) 122-135. [9] J.C.Gao, Q.Tao, Z.A.Yao, Optimal decay rates of classical solutions for the full compressible MHD equations, Z. Angew. Math. Phys. 67(2) (2016), Art. 23, 22 pp. [10] D.Hoff, E.Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z.Angew.Math.Phys. 56(5) (2005) 791-804. [11] X.P.Hu, D.H.Wang, Global solutions to the three-dimensional full compressible magnetohydroynamics flows, Comm. Math. Phys. 283(1) (2008) 253-284. [12] X.P.Hu, D.H.Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal. 197(1) (2010) 203-238. [13] S.Kawashima, Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl.Math. 1(1) (1984) 207-222. [14] S.Kawashima, M.Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc.Japan Acad.Ser.A Math. Sci. 58(9) (1982) 384-387.
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[15] F.C.Li, H.J.Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A 141(1) (2011) 109-126.
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[16] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heatconductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55(9) (1979) 337-342. [17] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20(1) (1980) 67-104. [18] X.K.Pu, B.L.Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys. 64(3) (2013) 519-538. [19] M.E.Schonbek, L2 decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88(3) (1985) 209-222. [20] Z.Tan, H.Q.Wang, Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl. 14(1) (2013) 188-201.
6
Jincheng Gao, Zhengzhen Wei, Zheng-an Yao
PII: DOI: Reference:
S0893-9659(19)30424-0 https://doi.org/10.1016/j.aml.2019.106100 AML 106100
To appear in:
Applied Mathematics Letters
Received date : 3 September 2019 Revised date : 14 October 2019 Accepted date : 14 October 2019 Please cite this article as: J.C. Gao, Z.Z. Wei and Z.A. Yao, Decay rate of strong solution for the compressible magnetohydrodynamic equations with large initial data, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106100. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
Journal Pre-proof *Manuscript Click here to view linked References
Decay Rate of Strong Solution for the Compressible Magnetohydrodynamic Equations with Large Initial Data Jincheng Gao†
Zhengzhen Wei‡
Zheng-an Yao]
School of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, P. R. China
Abstract
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In this paper, we study the decay rate of strong solution for the compressible magnetohydrodynamics(MHD) equations with large initial data in three-dimensional whole space. First of all, it is shown that both the first and the second 5 order spatial derivatives of magnetic field converge to zero at the same L2 −rate (1 + t)− 4 . Furthermore, we also prove that the lower bounds of decay rate for the magnetic field itself and first order spatial derivative at the L2 −rate are 5 3 (1 + t)− 4 and (1 + t)− 4 respectively. Keywords: Compressible MHD equations; Decay rate; Large initial data.
1
Introduction
In this paper, we investigate the lower and upper bounds of decay rate for the global solution of compressible magnetohydrodynamic(MHD) equations, which describe the motion of electrically conducting media in the presence of a magnetic field. The compressible MHD equations, which describe the coupling between the compressible Navier-Stokes equations and the magnetic equations, can be written as ∂t ρ + div(ρu) = 0, (1.1) ∂t (ρu) + div(ρu ⊗ u) − µ∆u − (µ + λ)∇ div u + ∇P = (∇ × B) × B, ∂t B − ∇ × (u × B) = −∇ × (ν∇ × B), div B = 0,
ur
where (x, t) ∈ R3 × R+ . The unknown functions ρ, u = (u1 , u2 , u3 ), B = (B1 , B2 , B3 ) and P represent the density, velocity, magnetic field and pressure respectively. The pressure P is given by smooth function P = P (ρ) = ργ with the adiabatic exponent γ ≥ 1. The constants µ and λ are the viscosity coefficients, ν is the magnetic diffusion coefficient, which satisfy the following conditions: µ > 0, 2µ + 3λ > 0, ν > 0. Although the electric field E does not appear in (1.1), it can be written in terms of the magnetic field and velocity as follows: E = ν∇ × B − u × B. To complete the system (1.1), initial data is given by (ρ, u, B)(x, t)|t=0 = (ρ0 (x), u0 (x), B0 (x)).
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Furthermore, we also assume that the initial data satisfies the conditions divB0 (x) = 0,
lim (ρ0 , u0 , B0 )(x) = (1, 0, 0).
|x|→∞
Due to their physical importance and mathematical challenges, there are large literature on compressible MHD system (1.1). For the case of dimension one, there are many mathematical results about the non-isentropic compressible MHD equations, for uniqueness and continuous dependence on initial data of weak solution [10], vanishing shear viscosity limit [6], global well-posedness with large initial data on whole space [14], free boundary [1] and bounded domain [2]. For the two-dimensional case, Kawashima [13] obtained the global existence of smooth solution to the general electromagnetic fluid equations when the initial data is small perturbation of some given constant state. For the dimension three, Ducomet and Feireisl [5] obtained the existence of global weak solution on a bounded spatial domain supplemented with conservative boundary conditions. Hu and Wang [11, 12] and Fan and Yu [7] obtained the global weak solution to the nonlinear Email: † [email protected], ‡ [email protected], ] [email protected]
Journal Pre-proof J.C.Gao, Z.Z.Wei, Z.A.Yao
compressible MHD equations with general initial data respectively. Motivated by the work for the compressible NavierStokes equations [16, 17], the recent work [3, 15, 18] not only established the global existence of small solution under the H 3 −framework, but also obtained the decay rate if the initial data belongs to Lq (q ∈ [1, 6/5)) additionally. Later, these work were improved by [8, 9] in the sense that the higher order spatial derivatives of solution obey optimal decay rate. The decay rate of small solution for the compressible MHD equations was also studied [20] if the initial data belongs to some negative Sobolev space rather than Lq (q ∈ [1, 6/5)). Very recently, Chen, Huang and Xu [4] investigated the global-in-time stability of large solution to the compressible MHD equations (1.1), and established the upper decay rate: 3
k(ρ − 1)(t)kH 1 + ku(t)kH 1 + kB(t)kH 1 6 C(1 + t)− 4 .
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Here they required the condition supt∈R+ kB(t, ·)kL∞ + supt∈R+ kρ(t, ·)kC α 6 M for small 0 < α < 1, and the initial data (ρ0 − 1, u0 , B0 ) belongs to L1 (R3 ) ∩ H 2 (R3 ). Based on the recent work [4], we are interested in the decay rate of magnetic field for the compressible MHD equations (1.1) with large initial data. And hence, we hope to address the following three problems: 3 (i)The first order spatial derivative of magnetic field converging to zero has faster decay rate than (1 + t)− 4 or not? (ii)The second order spatial derivative of magnetic field will converge to zero or not? (iii)Can we provide some lower bounds of decay rate for the magnetic field itself and higher order spatial derivatives? The main purpose of this paper is to provide an affirmative answer along this direction. Notation: In this paper, we use H s (s ∈ R3 ) to denote the usual Sobolev space with norm k · kH s and Lp (R3 ) to denote the usual Lp space with norm k · kLp . F(f ) := fb represents the usual Fourier transform of the function f . The constant C denotes the generic positive constant independent of time, and may change from line to line. First of all, we recall the following results obtained in [4], which will be used in this paper frequently. Theorem 1.1 (see [4]). Let (ρ, u, B) be a global and smooth solution of (1.1), if (ρ0 − 1, u0 , B0 ) ∈ L1 (R3 ) ∩ H 2 (R3 ), 0 6 ρ 6 ρ¯, and supt∈R+ kB(t, ·)kL∞ + supt∈R+ kρ(t, ·)kC α 6 M , then the global solution (ρ, u, B) has the uniform-in-time bounds and time decay as follows Z t k(ρ − 1)(t)k2H 2 + ku(t)k2H 2 + kB(t)k2H 2 + (k∇ρ(τ )k2H 1 + k∇u(τ )k2H 2 + k∇B(τ )k2H 2 )dτ 6 C1 , (1.2) 0
3
k(ρ − 1)(t)kH 1 + ku(t)kH 1 + kB(t)kH 1 6 C2 (1 + t)− 4 ,
(1.3)
where the constants C1 and C2 are independent of time.
In this paper, we hope to establish optimal decay rate for the higher order spatial derivatives of magnetic field, and give the lower bound of decay rate for magnetic field. Our main results can be stated as follows: Theorem 1.2. Under the assumptions of Theorem 1.1, then the magnetic field B(x, t) of global solution to the Cauchy problem (1.1) has the time decay rate 5
k∇B(t)kH 1 + k∂t B(t)kL2 6 C3 (1 + t)− 4 ,
t > T1 .
(1.4)
ur
c0 satisfies |B c0 | ≥ c0 , 0 ≤ |ξ| 1, with c0 > 0 a constant. Furthermore, assume that the Fourier transform F(B0 ) = B Then, the magnetic field B(x, t) obtained in Theorem 1.1 has the decay rates for all t > T2
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C4 (1 + t)−
3+2k 4
≤ k∇k B(t)kL2 6 C3 (1 + t)−
3+2k 4
, k = 0, 1.
(1.5)
Here Ti (i = 1, 2) is a positive large time, and Ci (i = 3, 4) is a constant independent of time. Remark 1.1. Compared with decay rate (1.3), the advantage of decay rate (1.4) not only implies that the second order spatial derivative of magnetic field tends to zero, but also shows that the first and second order spatial derivatives of 5 magnetic tend to zero at the L2 −rate (1 + t)− 4 . Remark 1.2. By the Sobolev interpolation inequality, it is shown that the magnetic field tends to zero at the Lp (2 ≤ p ≤ 3 1 6)−rate (1 + t)− 2 (1− p ) . Remark 1.3. Since the density and velocity satisfy the hyperbolic and parabolic coupled system, we can not obtain the optimal decay rate for the first and second order spatial derivatives of density and velocity just taking the method in this paper. 2
Journal Pre-proof Decay Rate for Compressible MHD Equations
2
Proof of Theorem 1.2
In this section, we will give the proof of Theorem 1.2. To this end, the upper decay rate (1.4) will be established as follows. Lemma 2.1. Under the assumptions in Theorem 1.1, the magnetic field B(x, t) obtained in Theorem 1.1 has the decay rate for all t > T1 5 k∇B(t)kH 1 + k∂t B(t)kL2 6 C(1 + t)− 4 , (2.1) where T1 is defined below. Proof. By virtue of the divergence free condition divB = 0, the third equation of (1.1) can be written as follows ∆
Bt − ν∆B = B · ∇u − u · ∇B − (div u)B = F.
(2.2)
Applying ∇ operator to (2.2), multiplying by ∇B and integrating over R3 , we get Z Z Z 1 d |∇B|2 dx + ν |∇2 B|2 dx = ∇F · ∇Bdx. 2 dt R3 R3 R3
(2.3)
R3
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Integrating by parts and applying H¨older inequality, it is easy to check that Z ∇F · ∇Bdx 6 (kBkL∞ k∇ukL2 + kukL3 k∇BkL6 )k∇2 BkL2 1
(2.4)
3
6 CkBkL4 2 k∇2 BkL4 2 k∇ukL2 k∇2 BkL2 + CkukH 1 k∇2 Bk2L2 .
Here we have used the Sobolev inequalities: kukL3 ≤ CkukH 1 and k∇BkL6 ≤ Ck∇2 BkL2 . Substituting (2.4) into (2.3), and using Young inequality, it follows that Z Z 1 d |∇B|2 dx + ν |∇2 B|2 dx 2 dt R3 3 R 1 2 8 6 CkBkL2 k∇ukL2 + νk∇2 Bk2L2 + CkukH 1 k∇2 Bk2L2 4 3 1 − 15 2 6 C(1 + t) + νk∇2 Bk2L2 + C(1 + t)− 4 k∇2 Bk2L2 , 4 4
where we have used the decay rate (1.3) in the last inequality. Then, for some large time T1 := (4C/ν) 3 , it holds on Z Z 15 d |∇B|2 dx + ν |∇2 B|2 dx 6 C(1 + t)− 2 , t > T1 . (2.5) dt R3 R3 For some constant R defined below, denoting the time sphere(see [19]) ∆
R 1 ) 2 }, 1+t
ur
S0 = {ξ ∈ R3 | |ξ| 6 (
(2.6)
Jo
and we have immediately by using the Plancherel theorem that Z Z Z Z Z R R R2 2 2 4 b 2 2 b 2 2 b 2 b 2 dξ. |∇ B| dx > |ξ| |B| dξ > |ξ| |B| dξ > |ξ| |B| dξ − |B| 1 + t R3 /S0 1 + t R3 (1 + t)2 S0 R3 R3 /S0 Combining (2.6) with (2.5) yields directly Z 15 7 d Rν R2 |∇B|2 dx + k∇Bk2L2 6 kBk2L2 + C(1 + t)− 2 6 CR2 (1 + t)− 2 . dt R3 1+t (1 + t)2
Choosing R = ν3 , multiplying the resulting inequality by (1 + t)3 and integrating over [T1 , t], it holds on for t > T1 1
1
5
k∇B(t)k2L2 ≤ C(1 + t)−3 ((1 + T1 )3 k∇B(T1 )k2L2 + (1 + t) 2 − (1 + T1 ) 2 ) 6 C(1 + t)− 2 , where we have used the uniform bound (1.2). Using equality (2.2), straightforward computation shows that Z Z Z 1 d |∇2 B|2 dx + ν |∇3 B|2 dx = ∇2 F · ∇2 Bdx. 2 dt R3 R3 R3 3
(2.7)
Journal Pre-proof J.C.Gao, Z.Z.Wei, Z.A.Yao
Integrating by part, applying H¨older inequality and decay rate (1.3), it is easy to obtain Z ∇2 F · ∇2 Bdx 6 C(k∇BkL2 k∇ukL∞ + kBkL6 k∇2 ukL3 + kukL3 k∇2 BkL6 )k∇3 BkL2 R3
1 6 Ck∇Bk2L2 k∇2 uk2H 1 + νk∇3 Bk2L2 + CkukH 1 k∇3 Bk2L2 4 3 1 6 Ck∇Bk2L2 k∇2 uk2H 1 + νk∇3 Bk2L2 + C(1 + t)− 4 k∇3 Bk2L2 . 4
Using the decay rate (2.7), then it holds on for all t ≥ T1 Z Z 5 d |∇2 B|2 dx + ν |∇3 B|2 dx 6 C(1 + t)− 2 k∇2 uk2H 1 . dt R3 R3 Similar to the inequality (2.6), it follows that
and hence, we have d dt
Z
R3
9 R R R2 k∇Bk2L2 ≥ k∇2 Bk2L2 − k∇2 Bk2L2 − CR2 (1 + t)− 2 , 1+t (1 + t)2 1+t
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k∇3 Bk2L2 >
|∇2 B|2 dx +
Rν 1+t
Z
9
R3
5
|∇2 B|2 dx 6 CR2 (1 + t)− 2 + C(1 + t)− 2 k∇2 uk2H 1 .
Choosing R = ν3 , multiplying the resulting inequality by (1 + t)3 , and integrating over [T1 , t], we obtain for t ≥ T1 5
k∇2 B(t)k2L2 6 C(1 + t)− 2 ,
(2.8)
where we have used uniform estimate (1.2). Using the equation (2.2), we get 5
k∂t BkL2 6 νk4BkL2 + kF kL2 6 νk4BkL2 + k∇BkH 1 kukH 1 6 C(1 + t)− 4 . This and the decay rates (2.7) and (2.8) imply (2.1). Therefore, we complete the proof of this lemma. 5
Next, we will establish the lower bound of decay rate (1.5), which shows that the L2 −rate (1 + t)− 4 for the first order spatial derivative of magnetic field will be optimal. c0 satisfies |B c0 | ≥ c0 , 0 ≤ |ξ| 1, with c0 > 0 a constant. Lemma 2.2. Assume that the Fourier transform F(B0 ) = B Then, the magnetic field B(x, t) obtained in Theorem 1.1 has the decay rate for all t > T2 k∇k B(t)kL2 ≥ C(1 + t)−
,
k = 0, 1,
ur
where T2 is defined below.
3+2k 4
(2.9)
Proof. Step 1: Consider the linear equation corresponding to the third equation of (1.1)
Jo
∂t Bl (x, t) − ν∆Bl (x, t) = 0, divBl (x, t) = 0, (x, t) ∈ R3 × R+ ;
Bl (x, t) |t=0 = B0 (x), x ∈ R3 .
Using the semigroup method, we can obtain the solution: Bl (x, t) = e−ν∆t B0 (x). It is easy to check that Z Z Z 2 3 b0 |2 e−2ν|ξ|2 t dξ > c20 |Bl |2 dx = |B e−2ν|ξ| t dξ > C(1 + t)− 2 . R3
Similarly, we also have
R3
Z
R3
∆
|∇Bl |2 dx =
(2.10)
(2.11)
|ξ| 1
Z
R3
2
5
b0 |2 |ξ|2 e−2ν|ξ| t dξ > C(1 + t)− 2 . |B
(2.12)
Step 2: Define Bδ (x, t) = B(x, t) − Bl (x, t), and hence, Bδ (x, t) will satisfy the following equation ∆
∂t Bδ − ν∆Bδ = ∇ × (u × B) = F, 4
divBδ = 0,
Bδ |t=0 = 0.
(2.13)
Journal Pre-proof Decay Rate for Compressible MHD Equations
Using the semigroup method and Duhamel principle formula, it holds on 12 Z t Z −2ν|ξ|2 (t−τ ) b 2 e |F | dξ dτ kBδ (t)kL2 ≤ R3
0
≤
Z t (Z
|ξ|61
0
≤C ≤C
Z
t
0
Z
t
0
2 −2ν|ξ|2 (t−τ )
|ξ| e
−2
|ξ|
5
Z
|Fb| dξ + 2
e
−2ν|ξ|2 (t−τ )
|ξ|>1
(1 + t − τ )− 4 (k|ξ|−1 FbkL∞ + kF kL2 )dτ
|Fb| dξ 2
) 12
dτ
5
(1 + t − τ )− 4 (k(u, B)k2L2 + k∇ × (u × B)kL2 )dτ.
Using the decay rate (1.3), we get Z t Z t 5 3 5 5 (1 + t − τ )− 4 (1 + τ )− 2 dτ ≤ C(1 + t)− 4 . (1 + t − τ )− 4 k(u, B)k2L2 dτ ≤ C 0
0
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Using the Sobolev inequality, H¨older inequality, uniform bound (1.2) and decay rate (1.3), it follows that Z t 5 (1 + t − τ )− 4 k∇ × (u × B)kL2 dτ 0 Z t 1 3 5 ≤C (1 + t − τ )− 4 k∇(u, B)kL2 2 k∇2 (u, B)kL2 2 dτ 0
≤C
≤C
Z
t
(1 + t − τ )
0
Z
0
− 53
t
k∇(u, B)k2L2 dτ 3
5
(1 + t − τ )− 3 (1 + τ )− 2 dτ
43 Z
34
t
0
k∇
2
(u, B)k2L2 dτ
14
9
≤ C(1 + t)− 8 .
Thus, the decay rate holds on for all t > 0
9
kBδ (t)kL2 ≤ C(1 + t)− 8 .
Using the equation (2.13), Sobolev inequality and Cauchy inequality, we get Z Z 1 d 2 |∇Bδ | dx + ν |∇2 Bδ |2 dx 2 dt R3 R3 1
3
≤ CkBkL2 2 k∇2 BkL2 2 k∇uk2L2 + Ckuk2H 1 k∇2 Bk2L2 + 9
≤ C(1 + t)− 2 +
ν k∆Bδ k2L2 , 2
(2.14)
ν k∆Bδ k2L2 2
and hence, similar to the decay rate (2.7), it holds on for all t ≥ T1
13
ur
k∇Bδ (t)kL2 ≤ C(1 + t)− 8 .
(2.15)
Step 3: The combination of (2.11) and (2.14) yields 3
9
3
Jo
kBkL2 > kBl kL2 − kBδ kL2 > C(1 + t)− 4 − C(1 + t)− 8 = C(1 + t)− 4 −
C
3
3
(1 + t) 8
(1 + t)− 4 .
Similarly, we combine the decay rates (2.12) and (2.15) to get 5
k∇BkL2 > C(1 + t)− 4 −
C
5
3
(1 + t) 8
(1 + t)− 4 .
If we show that decay rate (2.9) holds on for t ≥ T2 (T2 being large enough). Thus, we complete the proof of this lemma.
Acknowledgements Jincheng Gao’s research was partially supported by Fundamental Research Funds for the Central Universities(Grant No.18lgpy66) and NNSF of China(Grant Nos. 11571380 and 11801586). Zheng-an Yao’s research was partially supported by NNSF of China(Grant Nos.11431015 and 11971496). 5
Journal Pre-proof J.C.Gao, Z.Z.Wei, Z.A.Yao
References [1] G.Q.Chen, D.H.Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations 182(2) (2002) 344-376. [2] G.Q.Chen, D.H.Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys. 54(4) (2003) 608-632. [3] Q.Chen, Z.Tan, Global existence and asymptotic analysis of weak solutions to the equations of magnetohydrodynamics, Nonlinear Anal. 72(12) (2010) 4438-4451. [4] Y.H.Chen, J.C.Huang, H.Y.Xu, Global stability of large solutions of the 3-D compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl. 47 (2019) 272-290. [5] B.Ducomet, E.Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. 266(3) (2006) 595-629.
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[6] J.S.Fan, S.Jiang, G.Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm.Math.Phys. 270(3) (2007) 691-708. [7] J.S.Fan, W.H.Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal. 69(10) (2008) 3637-3660. [8] J.C.Gao, Y.H.Chen, Z.A.Yao, Long-time behavior of solution to the compressible magnetohydrodynamic equations, Nonlinear Anal. 128 (2015) 122-135. [9] J.C.Gao, Q.Tao, Z.A.Yao, Optimal decay rates of classical solutions for the full compressible MHD equations, Z. Angew. Math. Phys. 67(2) (2016), Art. 23, 22 pp. [10] D.Hoff, E.Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z.Angew.Math.Phys. 56(5) (2005) 791-804. [11] X.P.Hu, D.H.Wang, Global solutions to the three-dimensional full compressible magnetohydroynamics flows, Comm. Math. Phys. 283(1) (2008) 253-284. [12] X.P.Hu, D.H.Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal. 197(1) (2010) 203-238. [13] S.Kawashima, Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl.Math. 1(1) (1984) 207-222. [14] S.Kawashima, M.Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc.Japan Acad.Ser.A Math. Sci. 58(9) (1982) 384-387.
ur
[15] F.C.Li, H.J.Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A 141(1) (2011) 109-126.
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[16] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heatconductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55(9) (1979) 337-342. [17] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20(1) (1980) 67-104. [18] X.K.Pu, B.L.Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys. 64(3) (2013) 519-538. [19] M.E.Schonbek, L2 decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88(3) (1985) 209-222. [20] Z.Tan, H.Q.Wang, Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl. 14(1) (2013) 188-201.
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