The highly sensitive magnetic field sensor based on photonic crystal fiber filled with nano-magnetic fluid

Received: 1 February 2019 DOI: 10.1002/mma.5909

RESEARCH ARTICLE

Low Mach limit to one-dimensional nonisentropic planar compressible magnetohydrodynamic equations Xin Liu

Lijuan Wang

Department of Applied Mathematics, School of Statistics and Information, Shanghai University of International Business and Economics, Songjiang, Shanghai, 201620, China

This paper is concerned with a one-dimensional nonisentropic compressible planar magnetohydrodynamic flow with general initial data, whose behaviors at far fields x → ±∞ are different. The low Mach limit for the system is rigorously justified. The limit relies on the uniform estimates including weighted

Correspondence Xin Liu, Department of Applied Mathematics, School of Statistics and Information, Shanghai University of International Business and Economics, Songjiang, Shanghai 201620, China. Email: [email protected]

time derivatives and an extended convergence lemma. K E Y WO R D S low Mach limit, nonisentropic magnetohydrodynamics (MHD), uniform estimates M S C C L A S S I F I C AT I O N 35Q10; 76N10

Communicated by: M. Groves Funding information National Natural Science Foundation of China, Grant/Award Number: 11671075, 11801357 and 11801358

1

I N T RO DU CT ION

Magnetohydrodynamics (MHD) is concerned with the study of the interaction between magnetic fields and fluid conductors of electricity. The application of magnetohydrodynamics covers a very wide range of physical areas from liquid metals to cosmic plasmas, for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics, and plasma physics. The one-dimensional nonisentropic compressible magnetohydrodynamic flow mentioned above is described by the following equations in the Eulerian coordinate system: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

𝜌t + (𝜌u)x = 0, (𝜌u)t + (𝜌u2 + P + 12 |b|2 )x = (𝜆ux )x , (𝜌w)t + (𝜌uw − b)x = (𝜇wx )x , bt + (ub ) ( (− w)x = (𝜈bx )x), Et + u E + P + 12 |b|2 − w · b + qx = (𝜆uux + 𝜇w · wx + 𝜈b · bx )x ,

(1.1)

x

where 𝜌 denotes the density, u ∈ R the longitudinal velocity, w ∈ R2 the transverse velocity, b ∈ R2 the transverse magnetic field, and P the pressure. 𝜆 and 𝜇 are the bulk and the shear viscosity coefficients, respectively, which satisfy 𝜇 > 0, 2𝜇 + 𝜆 > 0, and 𝜈 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. The total energy E is given by )) ( ( 1 2 1 1 |u| + |w|2 + |b|2 . E =𝜌 e+ 2 2 2

Math Meth Appl Sci. 2019;1–20.

wileyonlinelibrary.com/journal/mma

© 2019 John Wiley & Sons, Ltd.

1

2

LIU AND WANG

We consider the perfect gas, so that the pressure function P and the internal function e are given by P = R𝜌T,

(1.2)

e = Cv T,

the heat flux q be given through Fourier's law by q = −𝜅Tx ,

(1.3)

where the parameters R > 0, Cv > 0, and 𝜅 are the gas constant, heat capacity at the constant volume, and heat conductivity coefficient, respectively. For simplicity, we assume 𝜆, 𝜇, 𝜈, and 𝜅 are constants and normalize R = 1 and Cv = 1. Throughout this paper, let 𝜀 be the compressibility parameter, which represents the maximum Mach number of the fluid. As in Schochet,1 we set u(x, t) = 𝜀u𝜀 (x, 𝜀t),

w(x, t) = 𝜀w𝜀 (x, 𝜀t),

b(x, t) = 𝜀b𝜀 (x, 𝜀t),

𝜇 = 𝜀𝜇𝜀 ,

𝜅 = 𝜀𝜅 𝜀 .

(1.4)

and 𝜆 = 𝜀𝜆𝜀 ,

𝜈 = 𝜀𝜈 𝜀 ,

Under these changes of variables and coefficients, the system (1.1) takes the following equivalent form: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

𝜌𝜀t + (𝜌𝜀 u𝜀 )x = 0, P𝜀 𝜌𝜀 (u𝜀t + u𝜀 u𝜀x ) + b𝜀 · b𝜀x + 𝜀2x = 𝜆𝜀 u𝜀xx , 𝜀 , 𝜌𝜀 (wt𝜀 + u𝜀 wx𝜀 ) − 1𝜀 b𝜀x = 𝜇𝜀 wxx 1 𝜀 𝜀 𝜀 𝜀 𝜀 𝜀 bt + (u b − 𝜀 w )x = 𝜈 bxx , ( ) 𝜀 𝜌𝜀 (Tt𝜀 + u𝜀 Tx𝜀 ) + P𝜀 u𝜀x = 𝜅 𝜀 Txx + 𝜀2 𝜆𝜀 |u𝜀x |2 + 𝜇𝜀 |wx𝜀 |2 + 𝜈 𝜀 |b𝜀x |2 .

(1.5)

As 𝜀 → 0+ , the limit of solutions of (1.5) is called the low Mach limit.2,3 The purpose of the low Mach number approximation is to justify that the compression, due to pressure variations, can be neglected. This is a common assumption when discussing the fluid dynamics of highly subsonic flows. Before our research is started, let us briefly recall the related results. For Euler equations, the first result is due to Klainerman and Majda,2,3 in which they proved the incompressible limit of the isentropic Euler equations to the incompressible Euler equations for local smooth solutions. Ukai4 verified the low Mach limit for the general data by using the fast decay of acoustic waves. Schochet5 obtained the convergence of the nonisentropic compressible Euler equations to the incompressible nonisentropic Euler equations in a bounded domain for local smooth solutions and well-prepared initial data. As mentioned above, Métivier and Schochet6 proved rigorously the incompressible limit of the compressible nonisentropic Euler equations in the whole space with general initial data; see also Alazard and Levermore et al7-9 for further extensions. Métivier and Schochet10 showed the incompressible limit of the one-dimensional nonisentropic Euler equations in a periodic domain with general data. For more related results, we also see Masmoudi and Métivier and Schochet.6,11 For Navier-Stokes equations, Alazard8 justified the low Mach limit in the whole space for the ill-prepared data, by employing a uniform estimate and the convergence lemma of Masmoudi.6 For the bounded domain, the low Mach limit was justified by Jiang and Ou12 and Dou et al.13 Recently, Hang et al14 established the low Mach limit for 1D nonisentropic compressible Navier-Stokes flow with well-prepared and ill-prepared data, whose density and temperature have different asymptotic states at infinity. Liu15 obtained the low Mach number limit for one-dimensional nonisentropic compressible Navier-Stokes system without viscosity. For other interesting results, see Danchin, Feireisl and Novotny, Kim and Lee, Masmoudi, and Schochet.1,11,16-18 For MHD equations, Klainerman and Majda2 studied the low Mach number limit to the compressible isentropic MHD equations in the spatially periodic case with well-prepared initial data. Li19 considered the inviscid, incompressible limit of the viscous isentropic compressible MHD equations, also for well-prepared initial data. Hu and Wang20 obtained the convergence of weak solutions to the compressible viscous MHD equations unbounded domains, periodic domains, and the wholes pace. Fan et al21 studied the low Mach number limit of the nonisentropic MHD equations with zero thermal conductivity under the assumption that the initial data are uniformly bounded with respect to the Mach number in H3 (R3 ) and are well-prepared in H1 (R3 ). Jiang et al22 established the low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data in the whole space R3 . For other interesting results, see Fan et al, Hu and Wang, Jiang et al, and Jiang and Ju.20-25

LIU AND WANG

3

In this paper, we will study the low Mach limit when the background is not constant state, that is, (𝜌𝜀 , T 𝜀 )(x, t) → (𝜌± , T± ), as x → ±∞, with 𝜌− T− = 𝜌+ T+ , where T− may not be equal to T+ and want to know what happens in the limiting process. To this end, we consider system (1.5) in the physical regime: P𝜀 = P + O(𝜀),

(1.6)

where P is a certain given constant, which is normalized to be P = 1, and (1.6) satisfies [ ] 𝜀 + 𝜀2 𝜆|u𝜀x |2 + 𝜇|wx𝜀 |2 + 𝜈|b𝜀x |2 . Pt𝜀 + (P𝜀 u𝜀 )x + P𝜀 u𝜀x = 𝜅Txx

(1.7)

As 𝜀 → 0+ , (1.7) has limiting form (2u − Tx )x = 0, which, with mass equation and state equation, gives ( 𝜌t =

𝜅 𝜌x 2 𝜌

) .

(1.8)

x

Considering the far field condition lim 𝜌 = 𝜌± , one sees that (1.8) admits a unique self-similarity solution (𝜉), 𝜉 = x→±∞

satisfying (±∞, t) = 𝜌± . Let 𝛿 = |𝜌+ − 𝜌− |. Then, (x, t) satisfies O(1)𝛿 − 4d(𝜌 x2)(1+t) ± x (x, t) = √ e , 1+t

x → ±∞,

d(𝜌) =

√

x 1+t

,

𝜅 . 2𝜌

To understand the role of the thermodynamics, as in Alazard,8 we introduce the following transformation to ensure positive of P𝜀 and T𝜀 : 𝜀 𝜀 (1.9) P𝜀 (x, t) = e𝜀p (x,t) , T 𝜀 (x, t) = e𝜃 (x,t) . 𝜀

𝜀

It follows from (1.3) and (1.7) that 𝜌𝜀 = e𝜀p (x,t)−𝜃 (x,t) . Under these changes of variables, the 1D nonisentropic compressible planar MHD system (1.5) takes the following equivalent form: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

) ) 𝜀 𝜀 𝜀 ( − 𝜅 𝜀 e−𝜀p +𝜃 𝜃x𝜀 x = 𝜀e−𝜀p 𝜆𝜀 |u𝜀x |2 + 𝜇𝜀 |wx𝜀 |2 + 𝜈 𝜀 |b𝜀x |2 𝜀 𝜀 +𝜅e−𝜀p +𝜃 px 𝜃x𝜀 , ( 𝜀 𝜀 ) p𝜀x −𝜃 𝜀 𝜀 𝜀 𝜀 −𝜀p𝜀 e (ut + u ux ) + 𝜀 = e 𝜆 uxx − b𝜀 · b𝜀x , ( ) 𝜀 𝜀 𝜀 e−𝜃 (wt𝜀 + u𝜀 wx𝜀 ) = e−𝜀p 𝜇𝜀 wxx + 1𝜀 b𝜀x ,

p𝜀t + u𝜀 px +

1( 2u𝜀 𝜀

(1.10)

b𝜀t + (u𝜀 b𝜀 − 1𝜀 w𝜀 )x = 𝜈 𝜀 b𝜀xx , ) ) 𝜀( 𝜀 𝜀 ( 𝜃t𝜀 + u𝜀 𝜃x𝜀 + u𝜀x = 𝜅 𝜀 e−𝜀p e𝜃 𝜃x𝜀 x + 𝜀2 e−𝜀p 𝜆𝜀 |u𝜀x |2 + 𝜇𝜀 |wx𝜀 |2 + 𝜈 𝜀 |b𝜀x |2 .

We shall study the limit of solutions to system (1.10). Formally, as 𝜀 → 0+ , if the sequence (p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 ) converges strongly to a limit (1, u, w, b, 𝜃) in some sense and (𝜆𝜀 , 𝜇𝜀 , 𝜈 𝜀 , 𝜅 𝜀 ) converges to a constant vector (𝜆, 𝜇, 𝜈, 𝜅), for simplicity, we still use (𝜆, 𝜇, 𝜈, 𝜅) to represent (𝜆, 𝜇, 𝜈, 𝜅). Thus, as 𝜀 → 0+ , (1.7) and 102 − 105 have the following form: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ with some functions 𝜋, 𝜔, and 𝛽.

(2u − 𝜅e𝜃 𝜃 x )x = 0, e−𝜃 (ut + uux ) + b · bx + 𝜋 x = 𝜆uxx , e−𝜃 (wt + uwx ) − 𝛽 x = 𝜇wxx , bt + (ub)x − 𝜔x =(𝜈bxx ,) 𝜃 t + u𝜃 x + ux = 𝜅 e𝜃 𝜃 x , x

(1.11)

4

LIU AND WANG

The purpose of this paper is to establish the above limit process rigorously with general initial data. To this end, we supplement system (1.10) with the following initial conditions: ( ) 𝜀 𝜀 , b𝜀in , 𝜃in , (p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 ) |t=0 = p𝜀in , u𝜀in , win

(1.12)

with 𝜀

0 < a ≤ e−𝜀pin ≤ a,

𝜀

0 < b ≤ e𝜃in ≤ b,

where a, a, b, and b are given constants independent of 𝜀. The layout of this paper is as follows: A few notations and the main results are stated in Section 2. In Section 3, we establish a priori estimates, and with the help of these estimates, we prove Theorem 2.1. In Section 4, we prove Theorem 2.2 by modifying the arguments developed by Métivier and Schochet.6

2

MAIN RESULTS

In this section, we firstly give some notations and then state the main results to the Cauchy problems (1.10) and (1.12). The notation will be chosen as follows. We denote by || · ||B the norm in the space B, especially, by ||f|| the ||𝑓 ||L2 (R) . For a multi-index 𝛼 = (𝛼 0 , 𝛼 1 ), we define ∑ 𝛼 𝛼 ||𝜕 𝛼 𝑓 (t)||. 𝜕 ∶= (𝜀𝜕t )𝛼0 𝜕x 1 and |𝛼| = |𝛼 1 | + |𝛼 2 |. Under this annotation, we set ||𝑓 || s = |𝛼|≤s

Since 𝜃 − may not be equal to 𝜃 + , we need to introduce a background profile 𝜃̃ for 𝜃 𝜀 . Here, we choose 𝜃̃ = − ln Θ satisfying 𝜃̃ → 𝜃± , as x → ±∞. Then, we define the following solution space for s ≥ 4, 2 ̃  (t) ∶ = ||(p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 − 𝜃)(t)|| H s,𝜀

=

s s+1 ∑ ∑ ||𝜕 𝛼 (p𝜀 , u𝜀 , w𝜀 , b𝜀 )(t)||2 + ||𝜕 𝛼 (𝜀p𝜀 , 𝜀u𝜀 , 𝜀w𝜀 , 𝜀b𝜀 )(t)||2 𝛼=0

𝛼=0

2 ̃ + ||(𝜃 𝜀 − 𝜃)(t)|| +

s ∑ 𝛼=0 s+1

+

s+1 ∑ ∑ ||𝜕 𝛼 𝜃 𝜀 (t)||2 +

t

∫0

∫ 𝛼=0 0

𝛼=0

t

||𝜕 𝛼 (p𝜀x , u𝜀x , wx𝜀 , b𝜀x )(𝜏)||2 d𝜏

(2.1)

||𝜕 𝛼 (𝜃x𝜀 , 𝜀u𝜀x , 𝜀wx𝜀 , 𝜀b𝜀x )(𝜏)||2 d𝜏.

Now, we are in a position to state our main results. More detailed proofs will be given in latter sections. Theorem 2.1. (Uniform estimates) Let s ≥ 4 be an integer and assume that the initial data (1.12) satisfy ( ) 2 𝜀 𝜀 || p𝜀in , u𝜀in , win ||H s,𝜀 ≤ C0 < ∞, , b𝜀in , 𝜃in

𝜀 ∈ (0, 1],

(2.2)

where C0 is independent of 𝜀. Then, there exist positive constants T0 and 𝜀0 depending only on C0 and |𝜃 + − 𝜃 − | such that the Cauchy problems (1.10) and (1.12) have a unique smooth solution (p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 ) satisfying ||(p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 )||2H s,𝜀 ≤ C̃0 ,

(2.3)

where C̃0 > 0 depends only on C0 and |𝜃 + − 𝜃 − |. The wave strength |𝜃 + − 𝜃 − | is allowed to be large. Remark 2.1. The wave strength |𝜃 + − 𝜃 − | is allowed to be large. But the constants T0 and 𝜀0 may depend on the wave strength. Then, we have the following low Mach limit.

LIU AND WANG

5

Theorem 2.2. (Low Mach number limit) Under the assumption of Theorem 2.1 and further assume that the initial data (1.12) satisfy ) ( ) ( 𝜀 𝜀 𝜀 𝜀 , b𝜀in , 𝜃in − 𝜃̃ → pin , uin , win , bin , 𝜃in − 𝜃̃ , in H s (R) as 𝜀 → 0+ , (2.4) pin , uin , win |𝜃in − 𝜃+ | ≤ Cx−1−𝜎 ,

for

(2.5)

x ∈ [1, +∞),

s (R)) where 𝜎 and C are positive constants. Then, the solution (p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 ) of (1.10) converges strongly in L2 (0, T0 ; Hloc ′ 𝜀 𝜀 , b𝜀in , 𝜃in ). for all s < s to (0, u, w, b, 𝜃), where (u, w, b, 𝜃) is the unique solution of (1.9) with the initial data (u𝜀in , win ′

Remark 2.2. Condition (2.5) can also be replaced by |𝜃 in − 𝜃 − | ≤ Cx−1−𝜎 , for x ∈ (−∞, −1].

3

UNIFORM EST IMATES

For the sake of convenience, we omit the superscript 𝜀 of the variables and denote 𝜀

e−𝜀p = e−𝜀p = a(𝜀p),

𝜀

e𝜃 = e𝜃 = b(𝜃).

(3.1)

For later use, we define that ⎧ (t) ∶= sup (||(p, u, w, b)|| s + ||(𝜀p, 𝜀u, 𝜀w, 𝜀b)|| s+1 ⎪ 0≤𝜏≤t ̃ L2 + ||((𝜀𝜕t )𝜃, 𝜃x )|| s ) ⎨ +||𝜃 − 𝜃|| ⎪ (t) ∶= ||(px , ux , wx , bx )|| s + ||(𝜃x , 𝜀ux , 𝜀wx , 𝜀bx )|| s+1 . ⎩

(3.2)

Assume the following a priori assumption: 0<

𝜀 1 a ≤ e−𝜀pin ≤ 2a, 2

0<

𝜀 1 b ≤ e𝜃in ≤ 2b, 2

on

t ∈ [0, T].

(3.3)

The proof of Theorem 2.1 depends on the following key proposition. Proposition 3.1. For any given integer s ≥ 4 and 𝜀 ∈ (0, 1], let (px , ux , wx , bx , 𝜃 x ) be the classical solution to the Cauchy problems (1.10) and (1.12). Under the a priori assumption (3.3), it holds that ( 1 )  (t) ≤ C[1 + Λ((0))] + C t 2 + 𝜀 Λ( (t)),

(3.4)

where  (t) is defined in (2.1), the constant C > 0 may depend on a, a, b, and b, and Λ(·) is a finite order polynomial. To prove the proposition, we need to prove some of the following estimates.

3.1

 s estimates of (𝜀p, 𝜀u, 𝜀w, 𝜀b, 𝜃)

Lemma 3.1. For s ≥ 4, it holds that ̃ 2+ ||𝜃 − 𝜃||

s ∑

𝛼

||𝜕 𝜃|| + 2

|𝛼|=1

s ∑ |𝛼|=0

t

∫0

||𝜕 𝛼 𝜃x ||2 d𝜏 ≤ C + (0) +

t

∫0

Λ((𝜏))d𝜏.

(3.5)

Proof. Equation (1.10)5 can be rewritten as ̃ t + u(𝜃 − 𝜃) ̃ x + ux − 𝜅e−𝜀p (e𝜃 (𝜃 − 𝜃) ̃ x ) = 𝜅e−𝜀p (e𝜃 𝜃̃x )x (𝜃 − 𝜃) ( ) +𝜀2 e−𝜀p 𝜆|ux |2 + 𝜇|wx |2 + 𝜈|bx |2 − u𝜃̃x − 𝜃̃t .

(3.6)

6

LIU AND WANG

Multiplying (3.6) by 𝜃 − 𝜃̃ and integrating with respect to x over R, based on the definition of Λ(·), we obtain 1d ̃ x |2 dx + ̃ − 𝜃) ̃ x dx ̃ 2 dx + 𝜅 a(𝜀p)b(𝜃)|(𝜃 − 𝜃) 𝜅a(𝜀p)b(𝜃)𝜀px (𝜃 − 𝜃)(𝜃 |𝜃 − 𝜃| ∫ ∫ 2 dt ∫ ( ) ̃ 2 + C||ux ||2 ||𝜃 − 𝜃|| ̃ + C𝜀2 ||𝜃 − 𝜃|| ̃ L∞ ||ux ||2 + ||wx ||2 + ||bx ||2 ≤ C||ux ||L∞ ||𝜃 − 𝜃|| ( ) ̃ + C||𝜃 − 𝜃|| ̃ 2 + C||𝜃̃t || ̃ L∞ ||ux ||2 + ||px ||2 ||𝜃 − 𝜃|| + ||𝜃||

(3.7)

≤ CΛ((t)). Integrating (3.7) with respect to t, we have ̃ 2+ ||𝜃(t) − 𝜃||

t

∫0

2 ̃ x (𝜏)||2 d𝜏 ≤ C + ||(𝜃 − 𝜃)(0)|| ̃ ||(𝜃 − 𝜃) +

t

∫0

Λ((𝜏))d𝜏.

(3.8)

Let 𝜃 𝛼 = 𝜕 𝛼 𝜃, for 1 ≤ |𝛼| ≤ s. Applying 𝜕 𝛼 to (1.10)5 , we derive ) ( 𝜕t 𝜃𝛼 + u𝜕x 𝜃𝛼 + 𝜕 𝛼 ux − 𝜅a(𝜀p)(b(𝜃)𝜕x 𝜃𝛼 )x = 𝜀2 𝜕 𝛼 a(𝜀p)(𝜆|ux |2 + 𝜇|wx |2 + 𝜈|bx |2 ) − (𝜕 𝛼 , u)𝜃x + 𝜅 {𝜕 𝛼 (b(𝜃)𝜃x )x − a(𝜀p)(b(𝜃)𝜕x 𝜃𝛼 )x } .

(3.9)

Multiplying (3.9) by 𝜃 𝛼 and integrating the resulting equation with respect to x, we arrive at 1d 1 u𝜕x (|𝜃𝛼 |2 )dx + |𝜃𝛼 |2 dx + 𝜅 a(𝜀p)b(𝜃)|𝜕x 𝜃𝛼 |2 dx + 𝜃 𝜕 u dx ∫ ∫ ∫ 𝛼 𝛼 x 2 dt 2∫ | | ) ( ≤ C || a(𝜀p)b(𝜃)𝜀px 𝜃𝛼 𝜕x 𝜃𝛼 dx|| + C||𝜃𝛼 ||||𝜕 𝛼 a(𝜀p)(𝜆|𝜀ux |2 + 𝜇|𝜀wx |2 + 𝜈|𝜀bx |2 ) ∫ | | | [ ]| 𝛼 2 + C||𝜃𝛼 ||||(𝜕 , u)𝜃x || + C || 𝜃𝛼 𝜕 𝛼 (a(𝜀p)(b(𝜃)𝜕x 𝜃)x ) − a(𝜀p)(b(𝜃)𝜕x 𝜃𝛼 )x || dx. |∫ |

(3.10)

Using the Cauchy inequality, we get u𝜕x (|𝜃𝛼 |2 )dx + 𝜃 𝜕 u dx ∫ 𝛼 𝛼 x ( ) ( ) ≤ C ||u||L∞ ||𝜃𝛼 ||2 + ||u𝛼 ||2 + ||𝜕x 𝜃𝛼 ||2 ≤ C ||u||H 2 ||𝜃𝛼 ||2 + ||u𝛼 ||2 + ||𝜕x 𝜃𝛼 ||2 ∫

(3.11)

≤ CΛ((t)), | | | a(𝜀p)b(𝜃)𝜀px 𝜃𝛼 𝜕x 𝜃𝛼 dx| |∫ | | | ( ) ( ) 2 ≤ C ||𝜀px ||L∞ ||𝜃𝛼 || + ||𝜕x 𝜃𝛼 ||2 ≤ C ||𝜀px ||H 2 ||𝜃𝛼 ||2 + ||𝜕x 𝜃𝛼 ||2

(3.12)

≤ CΛ((t)). Notice that (𝜕 𝛼 , u)𝜃x =

∑

C𝛼,𝛽 𝜕 𝛽 u𝜕 𝛼−𝛽 𝜃x .

(3.13)

1≤|𝛽|≤𝛼

Then, we have ||(𝜕 𝛼 , u)𝜃x || ≤ C (||u||W 1,∞ ||𝜃x || s−1 + ||u|| s ||𝜃x ||L∞ ) ≤ Λ((t)).

(3.14)

On the other hand, a direct calculation gives that ) ( ||𝜕 𝛼 a(𝜀p)(𝜆|𝜀ux |2 + 𝜇|𝜀wx |2 + 𝜈|𝜀bx |2 ) || ( ( ) ) ≤ C||𝜕 𝛼 𝜆|𝜀ux |2 + 𝜇|𝜀wx |2 + 𝜈|𝜀bx |2 || + Λ(||𝜀p|| s )|| 𝜆|𝜀ux |2 , 𝜇|𝜀wx |2 , 𝜈|𝜀bx |2 || s−1 ( ( ) ) ≤ C|| 𝜀ux , 𝜀wx , 𝜀b2x || s + Λ(||𝜀p|| s )|| 𝜆|𝜀ux |2 , 𝜇|𝜀wx |2 , 𝜈|𝜀bx |2 || s−1

(3.15)

LIU AND WANG

7

≤ CΛ((t)), ||𝜕 𝛼 (a(𝜀p)(b(𝜃)𝜕x 𝜃)x ) − a(𝜀p)(b(𝜃)𝜕x 𝜃𝛼 )x || ≤ CΛ(||𝜀p|| s ) (||𝜃xx || s−1 + Λ(||𝜃x || s )) + CΛ(||𝜃x || s ) (1 + ||𝜃xx || s−1 )

(3.16)

≤ CΛ((t)). Substituting (3.11), (3.12), (3.14), (3.15), and (3.16) into (3.10), integrating with respect to t, we conclude ||𝜕 𝛼 𝜃||2 +

t

∫0

||𝜕 𝛼 𝜃x ||2 d𝜏 ≤ C +

t

∫0

Λ((𝜏))d𝜏.

(3.17)

Combining (3.8) and (3.17), we get (3.5). The proof is complete. Lemma 3.2. For s ≥ 4, it holds that t

||(𝜀p, 𝜀u, 𝜀w, 𝜀b)||2 s +

∫0

t

||(𝜀ux , 𝜀wx , 𝜀bx )||2 s d𝜏 ≤ C2 (0) + C

∫0

Λ((𝜏)) [1 + (𝜏)] d𝜏.

(3.18)

Proof. For any multi-index 𝛼 satisfying 0 ≤ 𝛼 ≤ s, let ̌ = 𝜀w, b̌ = 𝜀b, p̌ = 𝜀p, ǔ = 𝜀u, w

(

) ( ) ̌ b̌ . ̌ 𝛼 , b̌ 𝛼 = 𝜕 𝛼 p, ̌ u, ̌ w, p̌ 𝛼 , ǔ 𝛼 , w

Then, we easily obtain ̌ ̌ ̌ 𝜕t p +̌ u𝜕x p+(2u 𝜕x p̌ −(𝜅a(p)b(𝜃)𝜃 x )x = 𝜅a(p)b(𝜃)𝜃 x) 2 2 2 ̌ ̌ + 𝜈|𝜕x b| , ̌ 𝜆|𝜕x u| ̌ + 𝜇|𝜕x w| +a(p) ̌ · 𝜕x b̌ = 𝜆a(p)𝜕 ̌ xx u, ̌ + px + a(p)b ̌ b(−𝜃)(𝜕t u +̌ u𝜕x u) ̌ + u𝜕x w) ̌ − a(p)𝜕 ̌ ̌ x b = 𝜇a(p)𝜕 ̌ xx w, b(−𝜃)(𝜕t w ̌ 𝜕t b̌ + u𝜕x b̌ + b𝜕x ǔ − 𝜕x w = 𝜈𝜕xx b.

(3.19)

𝜕t p̌ 𝛼 + u𝜕x p̌ 𝛼 = h1 + h2 +h3 +h4 , ̌ xx ǔ 𝛼 = h5 +h6 +h7 +h8 , b(−𝜃)(𝜕t ǔ 𝛼 + u𝜕x ǔ 𝛼 ) + 𝜕 𝛼 px − 𝜆a(p)𝜕 ̌ 𝛼 + u𝜕x w ̌ 𝛼 ) − 𝜇a(p)𝜕 ̌ 𝛼 = h9 +h10 +h11 +h12 , ̌ xx w b(−𝜃)(𝜕t w 𝜕t b̌ 𝛼 + u𝜕x b̌ 𝛼 − 𝜈𝜕xx b̌ 𝛼 = h13 +h14 +h15 ,

(3.20)

𝛼 ̌ h2 = −𝜕 𝛼 (2u − 𝜅a(p)b(𝜃)𝜃 ̌ ̌ ̌ h1 = −[𝜕[𝛼 , u]𝜕(x p, x )x h3 = 𝜕 (𝜅a(p)b(𝜃)𝜃 x 𝜕x p), )] 𝛼 2 2 2 ̌ ̌ + 𝜈|𝜕x b| ̌ 𝜆|𝜕x u| ̌ + 𝜇|𝜕x w| , h4 = 𝜕 a(p) 𝛼 ̌ h6 = −[𝜕( ̌ u, h5 = −[𝜕 𝛼 , b(−𝜃)]𝜕t u, , b(−𝜃)u]𝜕 ) x ̌ xx u, ̌ h8 = −𝜕 𝛼 b · 𝜕x b̌ , h7 = 𝜆[𝜕 𝛼 , a(p)]𝜕 ̌ h10 = −[𝜕 𝛼 , b(−𝜃)u]𝜕x w, ̌ h9 = −[𝜕 𝛼 , b(−𝜃)]𝜕t w, ̌ h12 = 𝜕 𝛼 (a(p)𝜕 ̌ xx w, ̌ x b) , h11 = 𝜆[𝜕 𝛼 , a(p)]𝜕 ̌ h14 = −𝜕 𝛼 (b𝜕x u) ̌ , h15 = 𝜕 𝛼 𝜕x w. h13 = −[𝜕 𝛼 , u]𝜕x b,

(3.21)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Applying 𝜕 𝛼 to (3.19), we have ⎧ ⎪ ⎨ ⎪ ⎩ where ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

Multiplying (3.20)1 by p̌ 𝛼 and integrating the resulting equation with respect to x, we obtain 1d 1 ux |p̌ 𝛼 |2dx + |p̌ 𝛼 |2 dx = (h1 + h2 + h3 + h4 ) p̌ 𝛼 dx. ∫ 2 dt ∫ 2∫

(3.22)

8

LIU AND WANG

With the help of (3.2), we easily derive | | | ux |p̌ 𝛼 |2 dx| ≤ C||u||L∞ ||p̌ 𝛼 ||2 ≤ CΛ((t)). |∫ | | |

(3.23)

̌ ≤ C (||u||W 1,∞ ||𝜕x p|| ̌  s−1 + ||u|| s ||𝜕x p|| ̌ L∞ ) ≤ CΛ((t)). ||h1 || = ||[𝜕 𝛼 , u]𝜕x p||

(3.24)

Similarly, we have ||(h3 , h4 )|| ≤ CΛ((t)), which, together with (3.24), yields that | | | (h1 + h3 + h4 ) p̌ 𝛼 dx| ≤ CΛ((t)). |∫ | | |

(3.25)

According to Sobolev's inequality, we arrive at ||h2 || ≤ C (||u|| s + Λ((t))) (||𝜀p|| s+1 + ||𝜃x || s+1 ) ≤ C ((t) + Λ((t))) , which implies that | | | h2 p̌ 𝛼 dx| ≤ C ((t) + Λ((t))) . | |∫ | |

(3.26)

From (3.22), (3.23), (3.25), and (3.26), we conclude ||p̌ 𝛼 (t)||2 ≤ ||p̌ 𝛼 (0)||2 + C

t

∫0

Λ((𝜏)) (1 + (𝜏)) d𝜏.

(3.27)

Multiplying (3.20)2 by ǔ 𝛼 and integrating the resulting equation, we have 1d ̌ ǔ 𝛼 𝜕xx ǔ 𝛼 dx ≤ b(−𝜃)|ǔ 𝛼 |2 dx − 𝜆 a(p) (h5 + h6 + h7 + h8 ) ǔ 𝛼 dx, ∫ ∫ 2 dt ∫

(3.28)

where we have used | | | | | 𝜕t b(−𝜃)|ǔ 𝛼 |2 dx| + | b(−𝜃)u𝜕x (||ǔ 𝛼 |2 )dx| ≤ CΛ((t)), | |∫ | |∫ | | | | | | | | 𝛼p |− 𝜕x ǔ 𝛼 dx| = | p𝛼 𝜕x ǔ 𝛼 dx| ≤ ||p𝛼 ||||𝜕x ǔ 𝛼 || ≤ CΛ((t)), | ∫ | |∫ | | | | | By (3.2), we directly obtain ̌ = ||[𝜕 𝛼 , b(−𝜃)](𝜀𝜕)t u|| ||h5 || = ||[𝜕 𝛼 , b(−𝜃)]𝜕t u|| ≤ C (||𝜃||H 2 ||(𝜀𝜕)t u|| s−1 + Λ((t))||(𝜀𝜕)t u||L∞ ) ≤ CΛ((t)), ||h6 || ≤ C (||b(−𝜃)u||W 1,∞ ||𝜕x ǔ 𝛼 || s−1 + ||b(−𝜃)u|| s ||𝜕x ǔ 𝛼 ||L∞ ) ≤ CΛ((t)), ̌ W 1,∞ ||𝜕xx ǔ 𝛼 || s−1 + ||a(p)|| ̌  s ||𝜕xx ǔ 𝛼 ||L∞ ) ≤ CΛ((t)), ||h7 || ≤ C (||a(p)|| ) ( ||h8 || ≤ C ||b||2L∞ ||𝜕 𝛼 b̌ x || + ||b||2W 1,∞ ||b̌ x || s−1 + ||b|| s ||b̌ x ||L∞ ≤ CΛ((t)), which implies that | | | (h5 + h6 + h7 + h8 ) ǔ 𝛼 dx| ≤ CΛ((t)). |∫ | | |

(3.29)

LIU AND WANG

9

On the other hand, we have −𝜆

̌ ǔ 𝛼 𝜕xx ǔ 𝛼 dx = 𝜆 a(p)

∫

≥

∫

̌ x ǔ 𝛼 |2 dx + 𝜆 a(p)|𝜕

∫

̌ ǔ 𝛼 𝜕x ǔ 𝛼 dx 𝜕x a(p) (3.30)

3 ̌ x ǔ 𝛼 |2 dx − CΛ((t)), 𝜆 a(p)|𝜕 4 ∫

which, together with (3.28) and (3.29), yields t

||𝜀u(t)||2 s +

t

||𝜀ux (t)||2 s (𝜏))d𝜏 ≤ C||𝜀u(0)||2 s + C

∫0

∫0

Λ((𝜏))d𝜏.

(3.31)

Λ((𝜏)))d𝜏,

(3.32)

Analogously, we have t

||𝜀w(t)||2 s +

t

||𝜀wx (t)||2 s (𝜏))d𝜏 ≤ C||𝜀w(0)||2 s + C

∫0

∫0

t

||𝜀b(t)||2 s +

∫0

t

||𝜀bx (t)||2 s (𝜏))d𝜏 ≤ C||𝜀b(0)||2 s + C

∫0

Λ((𝜏)))d𝜏.

(3.33)

Combining (3.27), (3.31), (3.32), and (3.33), we conclude (3.18). Next, we shall establish the estimates for ||(p, u, w, b)|| s . To achieve this goal, the key is to control 𝜕 s+1 (𝜀p, 𝜀u, 𝜀w, 𝜀b, 𝜃) and (𝜕 t )s (p, u, w, b). We first need to estimate ||(𝜀p, 𝜀u, 𝜀w, 𝜀b, 𝜃)|| s+1 .

3.2

 s+1 estimates of (𝜀p, 𝜀u, 𝜀w, 𝜀b, 𝜃)

Lemma 3.3. For s ≥ 4, it holds that ∑

||𝜕 𝛼 (𝜀p, 𝜀u, 𝜀w, 𝜀b, 𝜃)(t)||2 +

|𝛼|=s+1

∑ ∫ |𝛼|=s+1 0

t

||𝜕 𝛼 (𝜀px , 𝜀ux , 𝜀wx , 𝜀bx , 𝜃x )(t)||2 d𝜏

t

≤ C2 (0) + C

∫0

(3.34)

[1 + (𝜏)] Λ((𝜏))d𝜏.

̌ 𝜃) satisfies the following system: ̌ b, ̂ u, ̌ w, Proof. Let p̂ = 𝜀p − 𝜃. Then, (p, ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜕t p̂ + u𝜕x p̂ + 1𝜀 𝜕x ǔ = 0, ̌ + 1𝜀 (𝜕x p̂ + 𝜃x ) + a(𝜀p)b · 𝜕x b̌ = 𝜆a(𝜀p)𝜕xx u, ̌ b(−𝜃)(𝜕t ǔ + u𝜕x u) ̌ + u𝜕x w) ̌ − a(𝜀p)𝜕x b = 𝜇a(𝜀p)𝜕xx w, ̌ b(−𝜃)(𝜕t w ̌ 𝜕t b̌ + u𝜕x b̌ + b𝜕x ǔ − 𝜕x w = 𝜈𝜕xx b, ) ( ̌ 2 . ̌ 2 + 𝜈|𝜕x b| ̌ 2 + 𝜇|𝜕x w| 𝜃t + u𝜃x + 1 𝜕x ǔ = a(𝜀p)(b(𝜃)𝜃x )x + 𝜀2 a(𝜀p) 𝜆|𝜕x u|

(3.35)

𝜀

Let 𝛼 be a multi-index with |𝛼| = s + 1 and denote (

) ( ) ̌ 𝜃 . ̌ b, ̌ 𝛼 , b̌ 𝛼 , 𝜃𝛼 = 𝜕 𝛼 p, ̂ u, ̌ w, p̂ 𝛼 , ǔ 𝛼 , w

Applying 𝜕 𝛼 to (3.35), we have ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

𝜕t p̂ 𝛼 + u𝜕x p̂ 𝛼 + 1𝜀 𝜕x ǔ 𝛼 = h16 , b(−𝜃)(𝜕t ǔ 𝛼 + u𝜕x ǔ 𝛼 ) + 1𝜀 (𝜕x p̂ 𝛼 + 𝜕x 𝜃𝛼 ) = 𝜆a(𝜀p)𝜕xx ǔ 𝛼 + h17 , ̌ 𝛼 + u𝜕x w ̌ 𝛼 ) = 𝜇a(𝜀p)𝜕xx w ̌ + h18 , b(−𝜃)(𝜕t w 𝜕t b̌ 𝛼 + u𝜕x b̌ 𝛼 + b𝜕x ǔ 𝛼 = 𝜈𝜕xx b̌ 𝛼 + h19 , 𝜕t 𝜃𝛼 + u𝜕x 𝜃𝛼 + 1𝜀 𝜕x ǔ 𝛼 = a(𝜀p)(b(𝜃)𝜕x 𝜃𝛼 )x + h20 ,

(3.36)

10

LIU AND WANG

where ̂ h16 = −[𝜕 𝛼 , u]𝜕x p, ̌ h17 = −[𝜕 𝛼 , b(−𝜃)]𝜕t ǔ − [𝜕 𝛼 , b(−𝜃)u]𝜕x ǔ + [𝜕 𝛼 , 𝜆a(𝜀p)]𝜕xx ǔ − 𝜕 𝛼 (b · 𝜕x b), ̌ − [𝜕 𝛼 , b(−𝜃)u]𝜕x w ̌ + [𝜕 𝛼 , 𝜆a(𝜀p)]𝜕xx w ̌ + 𝜕 𝛼 (a(𝜀p)𝜕x b) , h18 = −[𝜕 𝛼 , b(−𝜃)]𝜕t w h19 = −[𝜕 𝛼 , u]𝜕x b̌ − [𝜕 𝛼 , b]𝜕x ǔ − b𝜕x ǔ 𝛼 + 𝜕x w𝛼 , ( ( )) ̌ 2 ̌ 2 + 𝜈|𝜕x b| ̌ 2 + 𝜇|𝜕x w| h20 = −[𝜕 𝛼 , u]𝜕x 𝜃 + 𝜕 𝛼 a(𝜀p) 𝜆|𝜕x u| [ ] + 𝜕 𝛼 (a(𝜀p)(b(𝜃)𝜃x )x ) − a(𝜀p)(b(𝜃)𝜕x 𝜃𝛼 )x . ̌ 𝛼 , b̌ 𝛼 , and 𝜃 𝛼 , respectively, adding the resulting equations, and integrating Multiplying (3.36)1 to (3.36)5 by p̂ 𝛼 , ǔ 𝛼 , w with respect to x, we obtain ( ) 1d ̌ 𝛼 |2 + |b̌ 𝛼 |2 + |𝜃𝛼 |2 dx |p̂ 𝛼 |2 + b(−𝜃)|ǔ 𝛼 |2 + b(−𝜃)|w 2 dt ∫ [ ] ( ) 3 ̌ 𝛼 |2 + 𝜈|𝜕x b̌ 𝛼 |2 + a(𝜀p)b(𝜃)|𝜕x 𝜃𝛼 |2 dx a(𝜀p) 𝜆|𝜕x ǔ 𝛼 |2 + 𝜇|𝜕x w + 4∫ ( ) ̌ 𝛼 + +h19 b̌ 𝛼 + +h20 𝜃𝛼 dx, ≤ CΛ((t)) + h16 p̂ 𝛼 + h17 ǔ 𝛼 + h18 w ∫ where we have used the following estimates: ( ) | | | ( ) | | u p̂ 𝛼 𝜕x p̂ 𝛼 + b̌ 𝛼 𝜕x b̌ 𝛼 dx| + | 𝜕tb (−𝜃) |ǔ 𝛼 |2 + |w ̌ 𝛼 |2 dx|| | |∫ |∫ | | | | | | ̌ 𝛼 ) dx|| ≤ CΛ((t)), ̌ 𝛼 𝜕x w + || b(−𝜃)u (ǔ 𝛼 𝜕x ǔ 𝛼 + w |∫ | ∫

𝜆a(𝜀p)ǔ 𝛼 𝜕xx ǔ 𝛼 dx = − ≤−

∫

∫

∫

̌ 𝛼 |2 dx − 𝜇a(𝜀p)|𝜕x w

𝜆𝜕x a(𝜀p)ǔ 𝛼 𝜕x ǔ 𝛼 dx

∫

̌ 𝛼 dx ̌ 𝛼 𝜕x w 𝜆𝜕x a(𝜀p)w

3 ̌ 𝛼 |2 dx + CΛ((t)), 𝜇a(𝜀p)|𝜕x w 4∫

a(𝜀p)𝜃𝛼 𝜕x (b(𝜃)𝜕x 𝜃𝛼 )dx = − ≤−

∫

3 𝜆a(𝜀p)|𝜕x ǔ 𝛼 |2 dx + CΛ((t)), 4∫

̌ 𝛼 dx = − ̌ 𝛼 𝜕xx w 𝜇a(𝜀p)w ≤−

𝜆a(𝜀p)|𝜕x ǔ 𝛼 |2 dx −

∫

∫

a(𝜀p)b(𝜃)|𝜕x 𝜃𝛼 |2 dx −

∫

𝜕x a(𝜀p)b(𝜃)𝜃𝛼 𝜕x 𝜃𝛼 dx

3 a(𝜀p)b(𝜃)|𝜕x 𝜃𝛼 |2 dx + CΛ((t)), 4∫

and ∫

( ) ̌ 𝛼 dx = 0. ̌ 𝛼 · 𝜕x b̌ 𝛼 + b̌ 𝛼 · 𝜕x w ǔ 𝛼 𝜕x p̂ 𝛼 + p̂ 𝛼 𝜕x ǔ 𝛼 + ǔ 𝛼 𝜕x 𝜃𝛼 + 𝜃𝛼 𝜕x ǔ 𝛼 + w

It infer from (1.10) that 𝜀𝜕t u = −u𝜕x (𝜀u) − b(𝜃)𝜕x p − a(𝜀p)b(𝜃)b𝜕x (𝜀b) + 𝜆a(𝜀p)b(𝜃)𝜕xx (𝜀u), 𝜀𝜕t w = −u𝜕x (𝜀w) + a(𝜀p)b(𝜃)𝜕x (b) + 𝜇a(𝜀p)b(𝜃)𝜕xx (𝜀w), 𝜀𝜕t b = −𝜕x (𝜀ub) − 𝜕x (w) + 𝜈𝜕xx (𝜀b),

(3.37)

LIU AND WANG

11

which gives [ ] ||𝜀𝜕t u(t)||2 s ≤ CΛ((t)) 1 + ||𝜕x (𝜀u)(t)||2 s + ||𝜕x p(t)||2 s + ||𝜕xx (𝜀u)(t)||2 s ≤ CΛ((t)) [1 + (t)] , ||(𝜀𝜕t w, 𝜀𝜕t b)(t)||2 s ≤ CΛ((t)) [1 + (t)] .

(3.38)

(3.39)

On the basis of (3.2) and (3.38), we obtain ||h16 || ≤ C (||u||W 1,∞ ||𝜀𝜕x p − 𝜕x 𝜃|| s + ||u|| s+1 ||𝜀𝜕x p − 𝜕x 𝜃||L∞ ) ≤ CΛ((t)) (||𝜀𝜕t u|| s + ||𝜕x u|| s + ||𝜀𝜕x p|| s ) ≤ CΛ((t)) [1 + (t)] , which, together with Hölder's inequality, implies | | | h16 p̂ 𝛼 dx| ≤ CΛ((t)) [1 + (t)] . |∫ | | |

(3.40)

On the other hand, we have ||[𝜕 𝛼 , b(−𝜃)u]𝜕x (𝜀u)|| + ||[𝜕 𝛼 , 𝜆a(𝜀p)]𝜕xx (𝜀u)|| ≤ CΛ((t)) [1 + ||𝜀u|| s+1 + ||𝜀𝜕xx u|| s ] ≤ CΛ((t)) [1 + (t)] , ||[𝜕 𝛼 , b(−𝜃)u]𝜕x (𝜀u)|| ≤ CΛ((t)) [1 + ||𝜀𝜕t u|| s ] ≤ CΛ((t)) [1 + (t)] , ||[𝜕 𝛼 (b𝜕x (𝜀b))]𝜕x (𝜀u)|| ≤ CΛ((t)) [1 + ||𝜀𝜕x b|| s+1 ] ≤ CΛ((t)) [1 + (t)] , which, together with (3.39), yields | | | h17 ǔ 𝛼 dx| ≤ CΛ((t)) [1 + (t)] . | |∫ | |

(3.41)

) | | ( | ̌ 𝛼 + h19 b̌ 𝛼 + h20 𝜃𝛼 dx|| ≤ CΛ((t)) [1 + (t)] . |∫ h18 w | |

(3.42)

Similarly, we have

Combining (3.37), (3.40), (3.41), and (3.42), we conclude (3.34).

3.3

 s estimates of (𝜀𝜕 t p, 𝜀𝜕 t u, 𝜀𝜕 t w, 𝜀𝜕 t b)

In this subsection, we shall control the term (𝜕 t )s (p, u, w, b). To this end, for a given state (p, u, w, b, 𝜃), we consider an L2 estimates for the following linearized system of (1.10): ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

) 2u − 𝜅a(𝜀p)b(𝜃)𝜃x = 𝜅a(𝜀p)b(𝜃)p 𝜃x x ( ) x +𝜀a(𝜀p) 𝜆ux ux + 𝜇wx · wx + 𝜈bx · bx + 𝑓1 , b(−𝜃)(𝜕t u + u𝜕x u) + 1𝜀 px + a(𝜀p)b · 𝜕x b = 𝜆a(𝜀p)𝜕xx u + 𝑓2 , b(−𝜃)(𝜕t w + u𝜕x w) − 1𝜀 a(𝜀p)𝜕x b = 𝜇a(𝜀p)𝜕xx w + 𝑓3 , 𝜕t b + u𝜕x b + b𝜕x u − 1𝜀 𝜕x w = 𝜈𝜕xx b + 𝑓4 , ( ) 𝜕t 𝜃 + u𝜕x 𝜃 + 𝜕x u = 𝜅a(𝜀p)(b(−𝜃)𝜃x )x + 𝜀2 a(𝜀p) 𝜆ux ux + 𝜇wx · wx + 𝜈bx · bx + 𝑓5 ,

𝜕t p + u𝜕x p +

1 𝜀

(

where fi , i = 1, 2, … , 5, are source terms. We start with the following lemma.

(3.43)

12

LIU AND WANG

Lemma 3.4. Let (p, u, w, b, 𝜃) be the solution to system (3.44) and assume that

0<

1 a ≤ a(𝜀p) ≤ 2a, 2

0<

1 b ≤ b(𝜃) ≤ 2b, 2

on t ∈ [0, T].

(3.44)

Then, it holds that, for 0 < t ≤ T,

t

||(p, u, w, b)(t)||2 + {

∫0

||(ux , wx , bx )(𝜏)||2 d𝜏 ≤ ||(p, u, w, b)(0)||2 + C sup ||𝜃x (𝜏)||2 0≤𝜏≤t

t

t

||(p, u, w, b, 𝜃x )(𝜏)||2 d𝜏 ∫0 ( t ) 12 ( t ) 12 } t 2 2 2 ||𝑓5 (𝜏)|| d𝜏 + ||(p, u, w, b, 𝜃x )(𝜏)|| d𝜏 ||(𝑓1 , 𝑓2 , 𝑓3 , 𝑓4 )(𝜏)|| d𝜏 , + ∫0 ∫0 ∫0

+ CΛ(0 )

where 0 = sup

0≤𝜏≤t

∫0

||(𝜀ux , 𝜀wx , 𝜀bx , 𝜃xx )(𝜏)||2 d𝜏 +

(3.45)

{ } ||(𝜕t 𝜃, 𝜕t p)(𝜏)||L∞ + {||(p, u, w, b, 𝜃)(𝜏)||W 1,∞ .

Proof. Set u ∶= 2u − 𝜅a(𝜀p)b(𝜃)𝜃x . Then, (3.43) becomes

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

( ) pt + upx + 1𝜀 ux = 𝜅a(𝜀p)b(𝜃)p 𝜃x + 𝜆2 𝜀a(𝜀p)uux + 𝜆2 𝜀a(𝜀p)ux 𝜅a(𝜀p)b(𝜃)𝜃x x x ( ) +𝜀a(𝜀p) 𝜇wx · wx + 𝜈bx · bx + 𝑓1 , ) ( b(−𝜃)(ut + uux ) + 1𝜀 px + a(𝜀p)b · bx − 𝜆2 a(𝜀p)uxx = 12 b(𝜃) 𝜅a(𝜀p)b(𝜃)𝜃x xx ] [ − 12 b(−𝜃) (𝜅a(𝜀p)b(𝜃)𝜃x )t + u(𝜅a(𝜀p)b(𝜃)𝜃x )x + 𝑓2 ,

b(−𝜃)(wt + uwx ) − 1𝜀 a(𝜀p)bx = 𝜇a(𝜀p)wxx + 𝑓3 , a(𝜀p)bt + a(𝜀p)ubx + 12 a(𝜀p)bux − 1𝜀 a(𝜀p)wx − 𝜈a(𝜀p)bxx [ ] = − 12 a(𝜀p)b a(𝜀p)b(𝜃)𝜃x + a(𝜀p)𝑓4 , x] [ ( ) 𝜃t + u𝜃x + 12 ux + 𝜅2 a(𝜀p)b(𝜃)𝜃x − 𝜅a(𝜀p) b(𝜃)𝜃x x = 𝜆2 𝜀2 a(𝜀p)uux ( )x ( ) + 𝜆2 𝜀2 a(𝜀p)ux 𝜅a(𝜀p)b(𝜃)𝜃x + 𝜀2 a(𝜀p) 𝜇wx · wx + 𝜈bx · bx + 𝑓5 .

(3.46)

x

Multiplying 𝜕 x (3.46)5 by 𝜅2 a(𝜀p) and then adding the result to (3.46)2 , we have 1 1 𝜅 𝜆 b(−𝜃)(ut + uux ) + px − a(𝜀p)uxx − a(𝜀p)uxx 2 𝜀 4 2 ) ] ( [ 𝜆 1 = a(𝜀p) 𝜅a(𝜀p)b(𝜃)𝜃x + 𝑓2 − b(−𝜃) (𝜅a(𝜀p)b(𝜃)𝜃x )t + u(𝜅a(𝜀p)b(𝜃)𝜃x )x 2 2 ] ] [ xx [ 𝜅 𝜅 𝜅 − a(𝜀p)b · bx − a(𝜀p) 𝜅a(𝜀p)(b(𝜃)𝜃x )x − a(𝜀p) 𝜆𝜀2 a(𝜀p)ux ux )x − a(𝜀p)𝜕x 𝑓5 2 4 2 x ] ] [ [ 𝜆 2 𝜅 1 − 𝜀 𝜅a(𝜀p) ux (𝜅a(𝜀p)b(𝜃)𝜃x )x + a(𝜀p) 𝜅a(𝜀p)b(𝜃)𝜃x + a(𝜀p)ux 𝜃x 4 4 2 x xx 𝜅 = g + 𝑓2 − a(𝜀p)𝜕x 𝑓5 . 2

(3.47)

LIU AND WANG

13

Multiplying (3.46)1 to (3.46)4 by p, u, w, and b, respectively, adding the resulting equations, and integrating with respect to x, we obtain ) ) 1 2 1 1 1 |p| + b(−𝜃)|u|2 + b(−𝜃)|w|2 + a(𝜀p)|b|2 dx + ∫ 2 4 2 2 [ ( )] 𝜅 𝜆 ( + )a(𝜀p)|ux |2 + a(𝜀p) |wx |2 + |bx |2 dx ∫ 4 2 ( ) )] ( 𝜅 𝜆 1 1 ( + a(𝜀p)|ux |2 dx + a(𝜀p) |wx |2 + |bx |2 dx + ||𝑓5 ||2 + ≤ gudx ∫ 8∫ 4 2 2∫ d dt

+

∫

(

(

(3.48)

(𝑓1 p + 𝑓2 u + 𝑓3 w + 𝑓4 b) dx + CΛ(0 )||(p, u, w, b, 𝜃x , 𝜃xx )||2 .

A straightforward calculation gives that gudx ≤

∫

1 8∫

( (

) 𝜅 𝜆 a(𝜀p)|ux |2 dx + CΛ(0 )||(p, u, w, b, 𝜃x , 𝜃xx )||2 , + 4 2

which, together with (3.48), yields (3.45). Next, we use Lemma 3.4 to estimate ||(𝜀𝜕 t )k (p, u, w, b)||, 1 ≤ k ≤ s. Lemma 3.5. Let s ≥ 4 and 0 ≤ k ≤ s. It holds that t

||(𝜀𝜕t )k (p, u, w, b)(t)||2 + + C sup 0≤𝜏≤t

||𝜃x (𝜏)||2 k

∫0

||(𝜀𝜕t )k (ux , wx , bx )(𝜏)||2 d𝜏 ≤ ||(𝜀𝜕t )k (p, u, w, b)(0)||2

1 2

+ Ct Λ(0 ) + CΛ()

(3.49)

t

∫0

||𝜃xx (𝜏)||2 k d𝜏,

where  = sup {||(𝜕t 𝜃, 𝜕t p)(𝜏)||L∞ + {||(p, u, w, b, 𝜃)(𝜏)||W 1,∞ }. 0≤𝜏≤t

Proof. Set (pk , uk , wk , bk , 𝜃 k ) = (𝜀𝜕 t )k (p, u, w, b, 𝜃). Applying (𝜀𝜕 t )k to (1.10), we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

𝜕t pk + u𝜕x pk + 1𝜀 (2uk − 𝜅a(𝜀p)b(𝜃)𝜕x 𝜃k )x = 𝜅a(𝜀p)b(𝜃)px 𝜃x +𝜀a(𝜀p) (𝜆𝜕x u𝜕x uk + 𝜇𝜕x w · 𝜕x wk + 𝜈𝜕x b · 𝜕x bk ) + 𝑓k1 , b(−𝜃)(𝜕t uk + u𝜕x uk ) + 1𝜀 𝜕x pk + a(𝜀p)b · 𝜕x bk = 𝜆a(𝜀p)𝜕xx uk + 𝑓k2 , b(−𝜃)(𝜕t wk + u𝜕x wk ) − 1𝜀 a(𝜀p)𝜕x bk = 𝜇a(𝜀p)𝜕xx wk + 𝑓k3 , 𝜕t bk + u𝜕x bk + b𝜕x uk − 1𝜀 𝜕x wk = 𝜈𝜕xx bk + 𝑓k4 , 𝜕t 𝜃k + u𝜕x 𝜃k + 𝜕x uk = 𝜅a(𝜀p)(b(−𝜃)𝜕x 𝜃k )x +𝜀2 a(𝜀p) (𝜆𝜕x u𝜕x uk + 𝜇𝜕x w · 𝜕x wk + 𝜈𝜕x b · 𝜕x bk ) + 𝑓k5 ,

(3.50)

which, together with (3.2) and Lemma 3.4, yields t

||(𝜀𝜕t )k (p, u, w, b)(t)||2 +

||(𝜀𝜕t )k (ux , wx , bx )(𝜏)||2 d𝜏 ≤ ||(𝜀𝜕t )k (p, u, w, b)(0)||2 [ t t 2 + C sup ||𝜃x (𝜏)|| k + CΛ() ||(𝜀𝜕t )k 𝜃xx (𝜏)||2 d𝜏 + ||(pk , uk , wk , bk , 𝜕x 𝜃k )||2 d𝜏 ∫0 ∫0 0≤𝜏≤t ( t ) 12 ] t 1 2 2 ||𝑓k5 (𝜏)|| d𝜏 + t 2 Λ((𝜏)) ||(𝑓k1 , 𝑓k2 , 𝑓k3 , 𝑓k4 )(𝜏)|| d𝜏 + , ∫0 ∫0 ∫0

(3.51)

14

LIU AND WANG

where

] ) [ ] 1 ([ 𝑓k1 = −[(𝜀𝜕t )k , u]px + 𝜅 (𝜀𝜕t )k , a(𝜀p)b(𝜃) 𝜃x x + (𝜀𝜕t )k , 𝜅a(𝜀p)b(𝜃)px 𝜃x 𝜀 ] ] ] [ [ [ + 𝜀 (𝜀𝜕t )k , 𝜆𝜀a(𝜀p)ux ux + 𝜀 (𝜀𝜕t )k , 𝜇𝜀a(𝜀p)wx wx + 𝜀 (𝜀𝜕t )k , 𝜈𝜀a(𝜀p)bx bx , [ ] [ ] 𝑓k2 = −[(𝜀𝜕t )k , b(−𝜃)]ut − [(𝜀𝜕t )k , b(−𝜃)u]ux + 𝜆 (𝜀𝜕t )k , a(𝜀p) uxx − (𝜀𝜕t )k , a(𝜀p)b bx , [ ] [ ] 𝑓k3 = −[(𝜀𝜕t )k , b(−𝜃)]wt − [(𝜀𝜕t )k , b(−𝜃)u]wx + 𝜇 (𝜀𝜕t )k , a(𝜀p) wxx − (𝜀𝜕t )k , a(𝜀p) bx , 𝑓k4 = −[(𝜀𝜕t )k , u]bx − [(𝜀𝜕t )k , b]ux ,

( ) 𝑓k5 = −[(𝜀𝜕t )k , u]𝜃x + 𝜅[(𝜀𝜕t )k , a𝜀p](b(𝜃)𝜃x )x + 𝜅a(𝜀p) [(𝜀𝜕t )k , b𝜃]𝜃x x [ [ [ ] ] ] + 𝜀 (𝜀𝜕t )k , 𝜆𝜀a(𝜀p)ux ux + 𝜀 (𝜀𝜕t )k , 𝜇𝜀a(𝜀p)wx wx + 𝜀 (𝜀𝜕t )k , 𝜈𝜀a(𝜀p)bx bx . It remains to estimate the terms involving (fk1 , fk2 , fk3 , fk4 , fk5 ). Firstly, for fk1 , we notice that ] ) [ ] [ ] ([ (𝜀𝜕t )k , a(𝜀p)b(𝜃) 𝜃x x = (𝜀𝜕t )k , a(𝜀p)b(𝜃) 𝜃xx + (𝜀𝜕t )k , (a(𝜀p)b(𝜃))x 𝜃x , where

] [ ] 1 [ || (𝜀𝜕t )k , a(𝜀p)b(𝜃) 𝜃xx || = || (𝜀𝜕t )k−1 𝜕t , a(𝜀p)b(𝜃) 𝜃xx || 𝜀 [ ] [ ] ≤ || (𝜀𝜕t )k−1 , a(𝜀p)b(𝜃) 𝜃txx || + || (𝜀𝜕t )k−1 , (a(𝜀p)b(𝜃))t 𝜃xx || ≤ CΛ((t)) (1 + ||𝜃tx || s−1 + ||𝜃x || s ) , [ ] ] 1 [ || (𝜀𝜕t )k , (a(𝜀p)b(𝜃))x 𝜃x || = || (𝜀𝜕t )k−1 𝜕t , (a(𝜀p)b(𝜃))x 𝜃x || 𝜀 [ [ ] ] ≤ || (𝜀𝜕t )k−1 , (a(𝜀p)b(𝜃))tx 𝜃x || + || (𝜀𝜕t )k−1 , (a(𝜀p)b(𝜃))x 𝜃tx ||

(3.52)

(3.53)

(3.54)

≤ CΛ((t)) (1 + ||𝜃tx || s−1 + ||px || s ) . On the other hand, we easily obtain [ ] [ [ ] ] || (𝜀𝜕t )k , u px || + || (𝜀𝜕t )k , 𝜅a(𝜀p)b(𝜃)px 𝜃x || + ||𝜀 (𝜀𝜕t )k , 𝜆𝜀a(𝜀p)ux ux || [ ] ] [ + ||𝜀 (𝜀𝜕t )k , 𝜇𝜀a(𝜀p)wx wx || + ||𝜀 (𝜀𝜕t )k , 𝜈𝜀a(𝜀p)bx bx ||

(3.55)

≤ CΛ((t)) (1 + ||px || s ) , which, together with (3.52) to (3.54), gives ||𝑓k1 || ≤ CΛ((t)) (1 + ||𝜃tx || s−1 + ||px || s ) .

(3.56)

For ||fk2 ||, we notice that k k ∑ ∑ ] [ Ck,i (𝜀𝜕t )i b(−𝜃) · (𝜀𝜕t )k−i ut = Ck,i (𝜀𝜕t )i−1 𝜕t b(−𝜃) · (𝜀𝜕t )k−i+1 u, (𝜀𝜕t )k , b(−𝜃) ut = i=1

which is easy to obtain

i=1

[ ] || (𝜀𝜕t )k , b(−𝜃) ut || ≤ CΛ((t)) (1 + ||𝜃t || k−1 + Λ (||𝜃t || k−2 )) .

(3.57)

Next, a straightforward calculation gets that [ ] [ ] || (𝜀𝜕t )k , b(−𝜃)u ux || + ||𝜆 (𝜀𝜕t )k , a(𝜀p) uxx || ≤ CΛ((t)), which, together with (3.57), yields ||𝑓k2 || ≤ CΛ((t)) (1 + ||𝜃t || k−1 + Λ (||𝜃t || k−2 )) .

(3.58)

LIU AND WANG

15

Similarly, we have ||𝑓k3 || + ||𝑓k4 || ≤ CΛ((t)) (1 + ||𝜃t || k−1 + Λ (||𝜃t || k−2 )) ,

(3.59)

||𝑓k5 || ≤ CΛ((t)).

(3.60)

It follows from (1.10)5 and (3.2) that ||𝜃tx || s−1 ≤ CΛ((t)) (1 + ||(ux , wx , bx , 𝜃xx )|| s ) .

||𝜃t || s−1 ≤ CΛ((t)),

(3.61)

Then, combining (3.56) with (3.58) to (3.61), we arrive at ||(𝑓k1 , 𝑓k2 , 𝑓k3 , 𝑓k4 )|| ≤ CΛ((t)) (1 + ||(px , ux , wx , bx , 𝜃xx )|| s ) .

(3.62)

Substituting (3.62) into (3.51), we conclude (3.49). Now, the proof of Lemma 3.5 is complete. We infer from Lemmas 3.1, 3.3, and 3.5 that Corollary 3.1. For s ≥ 4, 0 ≤ k ≤ s − 1, it holds that t

||(𝜀𝜕t )k (p, u, w, b)(t)||2 + (

)

||(𝜀𝜕t )k (ux , wx , bx )(𝜏)||2 d𝜏

∫0

(3.63)

1 2

≤ C 1 +  (0) + Ct Λ( (t)), 2

and

t

||(𝜀𝜕t )s (p, u, w, b)(t)||2 +

∫0

||(𝜀𝜕t )s (ux , wx , bx )(𝜏)||2 d𝜏

(3.64)

1 2

≤ C (1 + Λ((0))) + Ct Λ( (t)) + CΛ((t)).

3.4

 s estimates of (p, u, w, b)

In this subsection, we shall use Corollary 3.1 to estimate ||(p, u, w, b)|| s . It follows from (1.10) that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2ux = 𝜀pt + 𝜀upx (+ (𝜅a(𝜀p)b(𝜃)𝜃x )x + 𝜅a(𝜀p)b(𝜃)𝜀p x 𝜃x ) +a(𝜀p) 𝜆|ux |2 + 𝜇|wx |2 + 𝜈|bx |2 , px = −b(−𝜃)𝜀(u t + uux ) − 𝜀a(𝜀p)b · bx + 𝜆𝜀a(𝜀p)u xx , [ ] bx = 𝜀 b(−𝜃)a(𝜀p)(wt + uwx ) − 𝜇a(𝜀p)wxx , wx = 𝜀 [bt + ux b + ubx − 𝜈bxx ] .

(3.65)

Lemma 3.6. For s ≥ 4, it holds that ( 1 ) ||(p, u, w, b)|| s ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)).

(3.66)

Proof. It follows 821 , Lemmas 3.1 to 3.3, and Corollary 3.1 that { ||ux || ≤ C ||𝜀pt || + 𝜀||u||L∞ ||px || + ||a(𝜀p)b(𝜃)||L∞ ||𝜃xx || + ||a(𝜀p)b(𝜃)||L∞ ||𝜃x ||2 } +||a(𝜀p)b(𝜃)||L∞ ||px ||||𝜃x || + ||(𝜆ux , 𝜇wx , 𝜈bx )||2 ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)). For 0 ≤ k ≤ s − 2, we have } { ||(𝜀𝜕t )k ux || ≤ C ||(𝜀𝜕t )k+1 p|| + Λ (||(𝜀px , 𝜀ux , 𝜀wx , 𝜀bx )|| k + ||𝜃xx || k ) ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)),

(3.67)

16

LIU AND WANG

and

} { ||(𝜀𝜕t )s−1 ux || ≤ C ||(𝜀𝜕t )s p|| + Λ (||(𝜀p, 𝜀u, 𝜀w, 𝜀b)|| s+1 + ||(𝜀𝜃t , 𝜃xx )|| s ) ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)) + CΛ((t)).

(3.68)

Similarly, for 0 ≤ k ≤ s − 2, we have { } ||(𝜀𝜕t )k px || ≤ C ||(𝜀𝜕t )k+1 u|| + Λ (||(𝜀p, 𝜀u, , 𝜀b)|| k+1 + ||𝜀𝜃t , uxx || k ) ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)), { } ||(𝜀𝜕t )k wx || ≤ C ||(𝜀𝜕t )k+1 b|| + Λ (||(𝜀u, 𝜀b)|| k+1 + ||𝜀bxx || k ) ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)), { } ||(𝜀𝜕t )k bx || ≤ C ||(𝜀𝜕t )k+1 w|| + Λ (||(𝜀p, 𝜀u, 𝜀w)|| k+1 + ||𝜀wxx || k ) ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)), and

{ } ||(𝜀𝜕t )s−1 px || ≤ C ||(𝜀𝜕t )s u|| + Λ (||(𝜀p, 𝜀u, , 𝜀b)|| s + ||𝜀𝜃t , uxx || s−1 ) ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)), { } ||(𝜀𝜕t )s−1 wx || ≤ C ||(𝜀𝜕t )s b|| + Λ (||(𝜀u, 𝜀b)|| s + ||𝜀bxx || s−1 ) ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)), } { ||(𝜀𝜕t )s−1 bx || ≤ C ||(𝜀𝜕t )s w|| + Λ (||(𝜀p, 𝜀u, 𝜀w)|| s + ||𝜀wxx || s−1 ) ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)).

(3.69)

(3.70)

(3.71)

(3.72)

(3.73)

(3.74)

For 0 ≤ k ≤ s − 3, it follows from (3.65), (3.67), (3.69), (3.70), and (3.71) that ( 1 ) ||(𝜀𝜕t )k (pxx , uxx , wxx , bxx )|| ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)),

(3.75)

( 1 ) ||(p, u, w, b)|| s−1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)),

(3.76)

and

which, together with 105 and 22, yields ( 1 ) (t) ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)).

(3.77)

Combining (3.64), (3.65), (3.68), (3.72), (3.73), (3.74), (3.76), and (3.77), we conclude (3.66). Lemma 3.7. It holds for s ≤ 4 that t

∫0

( 1 ) ||(px , ux , wx , bx )(𝜏)||2 s d𝜏 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)).

(3.78)

Proof. Firstly, we infer from (3.64) and (3.77) that t

∫0

( 1 ) ||(𝜀𝜕t )s ux (𝜏)||2 d𝜏 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)),

(3.79)

LIU AND WANG

17

which, together with (3.65)1 , gives ( 1 ) ||(𝜕t )s ux (𝜏)||2 d𝜏C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)) ∫0 ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)).

t

∫0

t

||(𝜀𝜕t )s+1 p(𝜏)||2 d𝜏 ≤ C

(3.80)

On the other hand, it follows from (3.65)2 that [ ] (𝜀𝜕t )s px = −𝜀b(−𝜃)(𝜀𝜕t )s ut + (𝜀𝜕t )s −𝜀b(−𝜃)uux + 𝜆𝜀a(𝜀p)uxx − 𝜀a(𝜀p)b · bx .

(3.81)

Multiplying (3.81) by (𝜀𝜕 t )s px and integrating the resulting equation, we have ||(𝜀𝜕t )s px ||2 = −𝜀

∫

∫

b(−𝜃)(𝜀𝜕t )s px (𝜀𝜕t )s (−𝜀b(−𝜃)uux + 𝜆𝜀a(𝜀p)uxx − 𝜀a(𝜀p)b · bx )

b(−𝜃)(𝜀𝜕t )s ut (𝜀𝜕t )s px dx

d b(−𝜃)(𝜀𝜕t )s u(𝜀𝜕t )s px dx + b(−𝜃)(𝜀𝜕t )s u(𝜀𝜕t )s+1 px dx ∫ dt ∫ 1 + 𝜕 b(−𝜃)(𝜀𝜕t )s u(𝜀𝜕t )s (𝜀p)x dx + ||(𝜀𝜕t )s px ||2 ∫ t 8 [ ]|2 | + |(𝜀𝜕t )s −𝜀b(−𝜃)uux + 𝜆𝜀a(𝜀p)uxx − 𝜀a(𝜀p)bbx | dx. | ∫ | ≤−

(3.82)

Integrating (3.82) with respect to t, using (3.79) and (3.80) and Lemmas 3.1, 3.2, 3.3, and 3.6, we obtain t

∫0

( 1 ) ||(𝜀𝜕t )s px (𝜏)||2 d𝜏 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)),

(3.83)

which, together with (3.65) and (3.80), yields ( 1 ) ||(𝜀𝜕t )s px (𝜏)||2 d𝜏 + C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)) ∫0 ) ( 1 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)).

t

∫0

t

||(𝜀𝜕t )s−1 pxx (𝜏)||2 d𝜏 ≤ C

(3.84)

Combining (3.64) and (3.84), we arrive at ( 1 ) ||px (𝜏)||2 s d𝜏 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)).

(3.85)

( 1 ) ||(ux , wx , bx )(𝜏)||2 s d𝜏 ≤ C (1 + Λ((0))) + C t 2 + 𝜀 Λ( (t)).

(3.86)

t

∫0 In the same way, we have t

∫0

Now, we have complete the proof of Lemma 3.7. Proof of Proposition 3.1. Proposition 3.1 follows immediately from Lemmas 3.1, 3.2, 3.3, 3.6, and 3.7. Proof of Theorem 2.1. Using Proposition 3.1, one can prove Theorem 2.1 by combining the local existence theorem and the bootstrap arguments; we omitted the details.

18

4

LIU AND WANG

LOW MAC H NUMBER LIMIT

In this section, we shall prove the Theorem 2.2 by modifying the arguments developed by Métivier and Schochet6 ; see also some extensions in Alazard and Levermore et al.7-9 As per Theorem 2.1, we have ̃  s+1 < ∞. sup ||(p𝜀 , u𝜀 , w𝜀 , b𝜀 )|| s + sup ||(𝜃 𝜀 − 𝜃)||

0≤𝜏≤T0

(4.1)

0≤𝜏≤T0

Thus, after a subsequence is extracted, as 𝜀 → 0+ , we obtain {

(p𝜀 , u𝜀 , w𝜀 , b𝜀 ) ⇀ (p, u, w, b), weak − ⋆ in L∞ (0, T0 ;  s (R)) , ) ( ̃ weak − ⋆ in L∞ 0, T0 ;  s+1 (R) . 𝜃 𝜀 − 𝜃̃ ⇀ 𝜃 − 𝜃,

(4.2)

It follows from the equations of w𝜀 , b𝜀 , and 𝜃 𝜀 and (4.1) that ( ) wt𝜀 , b𝜀t , 𝜃t𝜀 ∈ L∞ 0, T0 ;  s−2 (R) ,

(4.3)

which, together with Aubin-Lions lemma, yields that for s < s ′

( ) ⎧ w𝜀 → w, strong in C 0, T0 ;  s′ (R) , ( ) ⎪ 𝜀 s′ ⎨ b → b, strong in C ( 0, T0 ;  (R) ,) ⎪ 𝜃 𝜀 → 𝜃, strong in C 0, T0 ;  s′ +1 (R) . ⎩

(4.4)

) ( ′ ′ To obtain limiting system (1.11), (we need to show the strong convergence of (p𝜀 , u𝜀 ) in L2 0, T0 ;  s (R) for s < s. To ) 𝜀 +𝜃 𝜀 𝜀 𝜀 𝜀 −𝜀p + 𝜃x x converge strongly to 0 as 𝜀 → 0 . In fact, (1.10)1 and (1.10)2 can this end, we will show that p and 2u − 𝜅e be rewritten as { ( ) 𝜀 𝜀 𝜀p𝜀t 2u𝜀 − 𝜅e−𝜀p +𝜃 𝜃x𝜀 x = 𝜀𝑓 𝜀 , (4.5) 𝜀 𝜀e−𝜃 u𝜀t + p𝜀x = 𝜀g𝜀 . It follows from (2.3) that f𝜀 and g𝜀 are bounded in C(0, T0 ;  s−1 (R)). p𝜀 is uniformly bounded in L∞ (0, T0 ; L∞ (R)). Passing the weak limit in (4.5), we have px = 0 and (2u − 𝜅e𝜃 𝜃 x )x = 0. Since p ∈ L∞ (0, T0 ;  s (R)), we infer that p = 0. In addition, similarly, in Alazard and Huang et al,8,14 we can obtain that 𝜃 satisfies the following estimates: |𝜃 − 𝜃+ | ≤ Cx−1−𝜎 ,

x ∈ [1, +∞).

(4.6)

To obtain the strong convergence of u𝜀 and p𝜀 , we need the following Proposition 4.1, which will be proved in the end of this section. Proposition 4.1. Let (2.3) and (4.6) hold. Then, for s < s ′

{

( ) ′ p𝜀 → 0, strong in L)2 0, T0 ;  s (R) , ( ) 𝜀 𝜀 ′ (2u𝜀 − 𝜅e−𝜀p +𝜃 𝜃x𝜀 x → 0, strong in L2 0, T0 ;  s −1 (R) .

(4.7)

The proof of Proposition 4.1 is based on the following dispersive estimates on the wave equation obtained by Métivier and Schochet6 and reformulated in Alazard.8 Lemma 4.1. Let T > 0, v𝜀 be a bounded sequence in C([0, T], H2 (Rd )), and 𝜀𝜕 t v𝜀 are bounded in L2 (0, T; L2 (Rd )) satisfying 𝜀2 𝜕t (a𝜀 𝜕t v𝜀 ) − ∇ · (b𝜀 ∇v𝜀 ) = c𝜀 , (4.8) where c𝜀 converges to 0 strongly in L2 (0, T; L2 (Rd )). Assume further that, for some k > 1 + d2 , the coefficients (a𝜀 , b𝜀 ) are ) ( ) ( k k (Rd ) and converge in C 0, T; Hloc (Rd ) to (a, b) satisfying for all 𝜏 ∈ R positive and uniformly bounded in C 0, T; Hloc a𝜏 2 + ∇ · (b∇) in L2 (Rd ) is reduced to 0.

(4.9)

LIU AND WANG

19

Then, the sequence v𝜀 converges to 0 strongly in L2 (0, T; L2loc (Rd )). 𝜀

Proof of Proposition 4.1. Applying 𝜀𝜕 t to (4.5)1 and 𝜕 x to e𝜃 × (4.5)2 , we obtain that 1 2 𝜀 ( 𝜃𝜀 𝜀 ) 𝜀 ptt − e px x = 𝜀F 𝜀 (p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 ), 2

(4.10)

where F𝜀 (p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 ) is a smooth function with F(0) = 0. From (2.3), 𝜀F𝜀 converges to 0 strongly in L2 (0, T; L2 (R)) as 𝜀 → 0+ . According to the strong convergence of 𝜃 𝜀 , the initial condition (2.5), and the arguments in section 8.1 in Alazard,8 we easily prove that the coefficients in (4.10) satisfy the conditions in Lemma 4.1. Therefore, we can apply Lemma 4.1 to obtain (4.11) p𝜀 → 0 strong in L2 (0, T; L2loc (R)). Since 𝜃 𝜀 is bounded uniformly in C([0, T0 ], Hs (R)), after an interpolation argument, we arrive at s (R)), p𝜀 → 0 strong in L2 (0, T; Hloc ′

𝜀

s′ < s.

(4.12)

𝜀

Similarly, we can obtain the strong convergence of (2u𝜀 − 𝜅e−𝜀p +𝜃 𝜃x𝜀 )x . This completes the proof. Proof of Theorem 2.2. If Proposition 4.1 holds, passing the limit in Equation (1.10) for (p𝜀 , u𝜀 , w𝜀 , b𝜀 , 𝜃 𝜀 ), it is easy to prove that the limit (0, u, w, b, 𝜃) solves (1.11) in the sense of distribution. On the other hand, according to the arguments in Métivier and Schochet,6 we easily obtain that (u, w, b, 𝜃) satisfies the initial condition (u, w, b, 𝜃)|t=0 = (uin , win , bin , 𝜃in ), (4.13) where uin is determined by uin = 12 𝜅e𝜃in (𝜃in )x . Moreover, we can obtain the uniqueness of solutions to the limit system (1.11) with initial data (4.13) by the energy method, and then, the above conclusions hold for the whole sequence (u, w, b, 𝜃). Thus, the proof of Theorem 2.2 is completed. □

ACKNOWLEDGEMENTS The work was supported by the National Natural Science Foundation of China ( no. 11801357, no. 11671075, and no. 11801358). The authors therefore acknowledge with thanks National Natural Science Foundation of China for technical and financial support.

CONFLICT OF INTERESTS This work does not have any conflict of interests.

ORCID Xin Liu

https://orcid.org/0000-0002-9333-0222

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How to cite this article: Liu X, Wang L. Low Mach limit to one-dimensional nonisentropic planar compressible magnetohydrodynamic equations. Math Meth Appl Sci. 2019;1–20. https://doi.org/10.1002/mma.5909