Global well-posedness of the full compressible Hall-MHD equations

Qiang Tao; Canze Zhu

doi:10.1515/anona-2020-0178

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Global well-posedness of the full compressible Hall-MHD equations

Qiang Tao und Canze Zhu

Veröffentlicht/Copyright: 24. April 2021

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Aus der Zeitschrift Advances in Nonlinear Analysis Band 10 Heft 1

Abstract

This paper deals with a Cauchy problem of the full compressible Hall-magnetohydrodynamic flows. We establish the existence and uniqueness of global solution, provided that the initial energy is suitably small but the initial temperature allows large oscillations. In addition, the large time behavior of the global solution is obtained.

Keywords: full compressible Hall-MHD equations; global existence; large time behavior

MSC 2010: 35A01; 35Q35; 76W05

1 Introduction

In this paper, we study the three-dimensional full compressible Hall-magnetohydrodynamic (for short, Hall-MHD) system, which is governed by the following equations (see, e.g. [4, 11]):

ρt+div(ρu)=0,(ρu)t+div(ρu⊗u)−μΔu−(μ+λ)∇divu+∇P=(curlB)×B,Cv((ρθ)t+div(ρuθ))−κΔθ+θ∂θPdivu=2μ|D(u)|2+λ|divu|2+v|curlB|2,Bt−vΔB+ϵcurl(curlB×Bρ)=curl(u×B),divB=0, (1.1)

with t ≥ 0 and x ∈ ℝ³. Here ρ, u, P, θ and B represent the fluid density, velocity, pressure, absolute temperature and magnetic field, respectively. Deformation tensor D(u) := 12 [∇u+ (∇u)^tr]. The constant viscosity coefficients μ and λ satisfy the physical restrictions:

μ>0,2μ+3λ≥0.

Here, we investigate the ideal polytropic fluids so that the pressure P and the physical constant C_v satisfy

P(ρ,θ):=Rρθ, Cv=Rγ−1,

where γ > 1 is the adiabatic constant, and for simplicity, we assume C_v = R = 1. κ and v are positive constants. ϵ > 0 is the Hall coefficient.

The Hall-MHD system can be derived from fluid mechanics with appropriate modifications to account for electrical forces and Hall effects. This compressible system(1.1) describes the dynamics of plasma flows with strong shear of magnetic fields such as in the solar flares, neutron stars and geo-dynamo, we refer to [1, 3, 10, 12, 23, 26, 30] for the physical background of this system. The Hall term curl((curlB)×B)ρ) in (1.1)₄ has been put forward by Ghosh et al. [10] to restores the influence of the electric current in the Lorentz force occurring in Ohms law. And the Hall coefficient ϵ is defined by the quotient of Alfven frequency of the lowest wave number ω_A and the ion cyclotron frequency Ω_i, it means ϵ:=ωAΩi. When the Hall effect term is neglected (ϵ = 0), the equations (1.1) is reduced to the well-known compressible full MHD system, whose applications cover a broad range of physical fields from liquid metals to cosmic plasmas. The mathematical results on this compressible heat conducting MHD system can refer for example to [2, 6, 7,8, 15, 16, 20, 25]. Fan and Yu [7] gave the local strong solution and Huang and Li established the blowup criterion. The global weak solutions was proved in [2, 6, 15, 16]. The long time behavior was discussed in [8, 25]. While, Jiang and his collaborators solved the low Mach number limit problem in [19, 20].

The compressible Hall-MHD equations are also mathematically significant. The solvability and stability of the equations has attracted considerable attention recently. For the isentropic case, Fan et al. [5] studied the global existence of strong solution and established the optimal time decay rates under the small initial perturbation condition. Gao and Yao [9] improved their work and established optimal decay rates for higher-order spatial derivatives of classical solutions. In [29], Tao, Yang and Yao established the global existence, uniqueness and exponential stability of strong solutions with large initial data for the one-dimensional case. Xiang [31] established the uniform estimates and optimal decay rates to global solution with respect to the Hall coefficient ϵ under the condition that H²-norm of initial data is small enough. If the temperature is taken into account, Fan et al. [4] first proved the local well-posedness for the full compressible Hall-MHD equations, and obtained a blow-up criterion of strong solution. The boundedness and time decay of the higher-order spatial derivatives of the smooth solution under the condition that H^k-norm (k ≥ 3) of initial data is small and bounded in Ḣ^s (0 < s < 32 ) are established by He, Samet and Zhou in [21]. Recently, Lai, Xu and Zhang [11] generalized Xiang' results [31] into the non-isentropic case. For other works on the compressible Hall-MHD system, we refer to [27, 28] and references therein. However, to our knowledge, all the results on the global smooth solutions for the three dimensional compressible Hall-MHD equation need the initial data has at least H² small norm.

In this paper, we consider an initial value problem of the Hall-MHD compressible flows (1.1) supplied with initial data

(ρ,u,θ,B)(x,0)=(ρ0,u0,θ0,B0)(x), for x∈R3 (1.2)

and the far field behavior

(ρ0−1,u0,θ0−1,B0)(x)→0, as~|x|→∞, t≥0. (1.3)

Motivated by the works for compressible Navier-Stokes equation [14, 18] and the compressible MHD equation [13, 22], we will first established global existence and uniqueness of solution with smooth initial data which is of small energy. It is worth mentioning that H²-norm of the initial data are not necessarily small. Then, the large time behavior of the solution will be given as well. Compared with the works in [13, 14, 18, 22], we also have to deal with the essential difficulties caused by Hall term in the present paper. This term includes the strong coupling between the density and the magnetic field, which together with the second-order derivative structure make the derivations of estimates more difficult. Therefore, the methods used in the MHD equation to show the bounds for the magnetic field are no longer applicable here. In order to overcome the difficulties from the Hall term, we introduce two kinds of estimates for derivatives of the density, which play an important role to establish the time-independent lower-order estimates. In addition, the temperature is considered in this paper, which brings us more nonlinear term, for instance, |curlB|² in temperature equation (1.1)₃ and makes the system more complex.

Throughout this paper, we use H^s(ℝ³)(s ∈ ℕ) to denote the usual Sobolev spaces with norm ∥⋅∥_H^s and L^p(ℝ³)(1 ≤ p ≤ ∞) to denote the L^p spaces with norm ∥⋅∥_L^p. For given initial data (ρ₀, u₀, θ₀, B₀), we define the initial energy E₀,

E0:=∫(12ρ0|u0|2+12|B0|2+(1+ρ0logρ0−ρ0)+ρ0(θ0−logθ0−1))dx. (1.4)

Now, the main result in this paper is stated as follows.

Theorem 1.1

Assume that for all given M₁ > 0 (not necessarily small), the initial data (ρ₀, u₀, θ₀, B₀) satisfies

infρ0>0, infθ0>0, (ρ0−1,u0,θ0−1,B0)∈H3, (1.5)

∥∇ρ0∥H12+∥∇u0∥H12+∥∇B0∥H12+∥∇θ0∥L22≤M1. (1.6)

Then, there is a positive constant δ depending only on μ, λ, κ, v, ϵ and M₁, such that if

E0≤δ, (1.7)

the Cauchy problem (1.1)–(1.3) has a unique global solution (ρ, u, θ, B) in ℝ³ × [0, ∞) satisfying

ρ−1,u,θ−1,B∈L∞([0,T];H3),∇ρ∈L2([0,T];H2),∇u,∇θ,∇B∈L2([0,T];H3), (1.8)

and the large time behavior:

limt→∞(∥ρ−1∥Lq2+∥u∥Lq2+∥θ−1∥Lq2+∥B∥Lq2)=0,for any q∈(2,∞]. (1.9)

Remark 1.1

From (1.6) and the small initial energy, we can find that the initial data in Theorem 1.1 have small H¹-norm for (ρ₀, u₀, B₀), which is weaker than that in [21, 31]. Indeed, by Gagliardo-Nirenberg inequality, the smallness of H²-norm of the initial data is required in [21, 31]. Moreover, the absolute temperature is considered in this paper and the initial temperature allows large oscillations.

Remark 1.2

Since the Hall term involves second-order derivative of magnetic field and first-order derivative of density, in order to establish the global existence of the solution, we need to get the bound of ∥∇ ρ∥_H¹, which leads (1.6) including H²-norm of initial data.

The rest of this paper is organized as follow. In section 2, we derive the time-independent lower-order estimates and the higher-order estimates depending on time of the solutions. In section 3, the proof of Theorem 1.1 will be showed.

2 Global existence

In this section, a known inequality and some facts are first collected, and then we will establish some suitable a priori estimates by the energy method.

2.1 Preliminaries

For the convenience of the proof below, let us rewrite the system (1.1) as follows,

ρt+div(ρu)=0,ρut−μΔu−(μ+λ)∇divu+θ∇ρ+ρ∇θ=−ρu⋅∇u+(curlB)×B,ρθt−κΔθ=−ρu⋅∇θ−ρθdivu+2μ|D(u)|2+λ|divu|2+v|curlB|2,Bt−vΔB=−ϵcurl(curlB×Bρ)+curl(u×B),divB=0, (2.1)

here we use the equalities

(curlB)×B=(B⋅∇)B−12∇(|B|2)

and

curl(u×B)=u(divB)−(u⋅∇)B+(B⋅∇)u−B(divu).

In addition, the initial data satisfies

(ρ,u,θ,B)(x,t)|t=0=(ρ0(x),u0(x),θ0(x),B0(x)) (2.2)

and

(ρ−1,u,θ−1,B)(x,t)=(ρ0−1,u0,θ0−1,B0)(x)→(0,0,0,0), as |x|→∞. (2.3)

The following Gagliardo-Nirenberg inequality are well-known (see for example [24]).

Lemma 2.1

Let 0 ≤ m, α ≤ l and the function f ∈ C0∞ (ℝ³), then we have

∥∇αf∥Lp≤C∥∇mf∥L21−θ∥∇lf∥L2θ,

where 0 ≤ θ ≤ 1 and α satisfy

1p−α3=(12−m3)(1−θ)+(12−l3)θ.

Now, we are ready to define some functions which will be frequently used later. First of all, let σ = σ(t) = min{1, t} and σ′=ddtσ(t). Then, we set:

A1(T)=sup0≤t≤T∫((ρ−1)2+(θ−1)2)dx+∫0T∫(|∇u|2+|∇B|2+|∇θ|2)dxdt,A2(T)=sup0≤t≤T∫(|∇u|2+|∇B|2+|∇ρ|2)dx+∫0T∫(|∇2u|2+|∇2B|2)dxdt,A3(T)=sup0≤t≤T∫(|∇2u|2+|∇2B|2+|∇2ρ|2)dx,A4(T)=sup0≤t≤T(σ4∥∇2θ∥L22).

In what follows, we denote the generic constant and suitably small constant by C > 0 and δ₁ > 0 depending only on some known constants μ, λ, k, v and ϵ but independent of time t, respectively. Particularly, we will use C(M) to emphasize that C may depend on M = max{(1 + C1−1 )M₁, (1 + C₃) M₁}, where the given constants C₁ and C₃ are defined in Lemma 2.8 and 2.10.

2.2 Time-independent lower-order estimates

The aim of this subsection is to derive the lower-order estimates on the solutions which are independent of time. Now, let (ρ, u, θ, B) be a solution to system (2.1)–(2.3) on ℝ³ × (0, T) for some positive time T > 0 and without loss of generality, let E₀ ≤ 1.

Proposition 2.1

Assume that the solution (ρ, u, θ, B) satisfies

A1(T)≤4E016,A2(T)≤2M,A3(T)≤2M,A4(T)≤2E0132, (2.4)

for all (x, t) ∈ ℝ³ × (0, T), and the initial data satisfies (1.5)–(1.6), then it holds that

A1(T)≤2E016,A2(T)≤32M,A3(T)≤32M,A4(T)≤E0132, (2.5)

provided E₀ ≤ δ, where δ is a positive constant depending on μ, λ, κ, v, ϵ and M₁ but independent of T.

The proof of Proposition 2.1 consists of Lemma 2.2-2.12 and is to be completed by the end of this subsection.

Lemma 2.2

Under all the assumption of Proposition 2.1, it holds that

12<ρ<32 (2.6)

in ℝ³ × (0, T), provided E₀ ≤ (4(C(M))⁻²⁴.

Proof

By (2.4) and Sobolev inequality, we obtain

∥ρ−1∥L∞≤C∥ρ−1∥L214∥∇2ρ∥L234≤C(M)E0124,

provided E₀ ≤ (4C(M))⁻²⁴. Thus, (2.6) holds. □

Lemma 2.3

Under the conditions of Proposition 2.1, it holds that

sup0≤t≤T∫(12ρ|u|2+12|B|2+(1+ρlogρ−ρ)+ρ(θ−logθ−1))dx+∫0T∫θ−1(μ|∇u|2+(λ+μ)|divu|2+v2|∇B|2)+κθ−2|∇θ|2dxdt≤E0. (2.7)

Proof

It follows from [18] and maximum principle, we have θ > 0 for all (x, t) ∈ ℝ³ × (0, T). Multiplying (2.1)₂–(2.1)₄ by u, 1 − θ⁻¹ and B, respectively, then adding them up and integrating by parts over ℝ³, using (2.1)₁ and the equality

∫curl(curlB×Bρ)⋅Bdx=0,

we have

ddt∫(12ρ|u|2+12|B|2+(1+ρlogρ−ρ)+ρ(θ−logθ−1))dx=−∫(θ−1(μ|∇u|2+(λ+μ)|divu|2+v2|∇B|2)+κθ−2|∇θ|2)dx.

We thus derive (2.7) directly by integrating the above equality over (0, T) and finish the proof of Lemma 2.3. □

Lemma 2.4

Under the assumptions of Proposition 2.1, it holds that

sup0≤t≤T(∥u∥L22+∥B∥L22+∥ρ−1∥L22)≤CE0 (2.8)

and

∥θ−1∥L2≤CE012+CE013∥∇θ∥L2. (2.9)

The proof of Lemma 2.4 is the same as Lemma 3.1 in [18], so we omit it for brevity.

The following lemma is given to estimate A₁(T).

Lemma 2.5

Under the assumptions of Proposition 2.1, it holds that

sup0≤t≤T∫(θ−1)2dx+∫0T∫(|∇u|2+|∇B|2+|∇θ|2)dxdt≤E016, (2.10)

provided E0≤min{(2C)−124,(κC(M))12}.

Proof

The lemma can be established by a similar way in [14]. For completeness, we prove it here. Multiplying (2.1)₂ by u, integrating the result over ℝ³ and using integration by parts, we have

12ddt∫(ρ|u|2+|B|2)dx+∫(μ|∇u|2+(λ+μ)|divu|2+v|∇B|2)dx≤−∫∇(ρθ)⋅udx. (2.11)

By Hölder inequality and using (2.8) and (2.9), the right-hand side of (2.11) is estimated as follows,

−∫∇(ρθ)⋅udx=∫(ρ(θ−1)+ρ−1)divudx≤C(∥θ−1∥L2+∥ρ−1∥L2)∥∇u∥L2≤μ2∥∇u∥L22+CE0+CE023∥∇θ∥L22. (2.12)

Taking (2.12) into (2.11), we have

12ddt∫(ρ|u|2+|B|2)dx+∫μ2|∇u|2+v|∇B|2dx≤CE0+CE023∥∇θ∥L22. (2.13)

On the order hand, multiplying (2.1)₃ by θ − 1, integrating the resulting inequality over ℝ³, we obtain

12ddt∫ρ(θ−1)2dx+κ∥∇θ∥L22≤C∫ρθ|θ−1||divu|dx+C∫|θ−1|(|∇u|2+|∇B|2)dx:=E1+E2. (2.14)

By (2.4), (2.8) and (2.9), using Hölder, Sobolev and Young inequalities, we can bound E₁ as follows,

E1≤C∫|θ−1|2|divu|dx+∫|θ−1||divu|dx≤C∥θ−1∥L3∥θ−1∥L6∥∇u∥L2+C∥θ−1∥L2∥∇u∥L2≤C∥θ−1∥L212∥∇θ∥L232∥∇u∥L2+C∥θ−1∥L2∥∇u∥L2≤C(E014+E016∥∇θ∥L212)∥∇θ∥L232∥∇u∥L2+C(E012+E013∥∇θ∥L2)∥∇u∥L2≤CE016∥∇θ∥L22∥∇u∥L2+CE012∥∇u∥L2+CE013(∥∇θ∥L22+∥∇u∥L22)≤C(M)E014+C(M)E016∥∇θ∥L22.

Similarly, we have following estimate for E₂,

E2≤C∥θ−1∥L3(∥∇u∥L2∥∇u∥L6+∥∇B∥L2∥∇B∥L6)≤C∥θ−1∥L212∥∇θ∥L212(∥∇u∥L2∥∇u∥L6+∥∇B∥L2∥∇B∥L6)≤C(E014∥∇θ∥L212+E016∥∇θ∥L2)(∥∇u∥L2∥∇u∥L6+∥∇B∥L2∥∇B∥L6)≤C(M)E014+CE0112∥∇θ∥L22+C(M)E014(∥∇2u∥L22+∥∇2B∥L22)).

Inserting the estimates of E₁ and E₂ into (2.14), we get

ddt∫ρ(θ−1)2dx+κ∥∇θ∥L22≤C(M)(E014+E014(∥∇2u∥L22+∥∇2B∥L22)), (2.15)

provided that E0≤(κC(M))12. Summing up (2.13) and (2.15), we obtain

ddt∫(ρ|u|2+|B|2+ρ(θ−1)2)dx+∫(μ|∇u|2+v|∇B|2+κ|∇θ|2)dx≤C(M)(E013+E014(∥∇2u∥L22+∥∇2B∥L22)). (2.16)

Integrating (2.16) over (0, σ(T)), it follows from (2.4) that

sup0≤t≤σ(T)12∫(ρ|u|2+|B|2+ρ(θ−1)2)dx+∫0σ(T)∫(|∇u|2+|∇B|2+|∇θ|2)dxdt≤C(M)E014. (2.17)

For t ∈ (σ(T), T), by (2.4), we have the following estimate

supσ(T)≤t≤T∥θ−1∥L∞≤sup0≤t≤Tσ2(∥θ−1∥L214∥∇2θ∥L234)≤CE0124≤12,

provided E₀ ≤ (2C)⁻²⁴. Thus

12≤θ≤32,t∈(σ(T),T). (2.18)

Applying (2.7) and (2.18), we have

supσ(T)≤t≤T12∫(ρ|u|2+|B|2+ρ(θ−1)2)dx+∫σ(T)T∫(|∇u|2+|∇B|2+|∇θ|2)dxdt≤CE0. (2.19)

Then, the combination of (2.17) with (2.19) yields

sup0≤t≤T∫(|u|2+|B|2+(θ−1)2)dx+∫0T∫(|∇u|2+|∇B|2+|∇θ|2)dxdt≤E016,

provided E0≤min{(2C)−124,(κC(M))12}. Thus, we complete the proof of Lemma 2.5. □

The estimates of ∥∇B∥_L² and ∥∇^2B∥_L² will be given by following Lemma.

Lemma 2.6

Under the assumptions of Proposition 2.1, it holds that

sup0≤t≤T∥∇B∥L22+v∫0T∥∇2B∥L22dt≤C(M)E016+∥∇B0∥L22, (2.20)

sup0≤t≤Tσ2∥∇B∥L22+∫0Tσ2∥∇2B∥L22dt≤C(M)E016, (2.21)

sup0≤t≤T∥∇2B∥L22+v∫0T∥∇3B∥L22dt≤C(M)E0112+∥∇2B0∥L22, (2.22)

sup0≤t≤Tσ4∥∇2B∥L22+∫0Tσ4∥∇3B∥L22dt≤C(M)E0112. (2.23)

Proof

Applying ∇ to (2.1)₄ and multiplying by ∇B, then integrating it over ℝ³, we obtain

12ddt∥∇B∥L22+v∥∇2B∥L22=∫∇B⋅∇(−u⋅∇B+B⋅∇u−Bdivu)dx−ϵ∫∇B⋅∇curl(curlB)×Bρdx:=I1+I2. (2.24)

Using the integration by parts and with the help of Hölder, Sobolev and Young inequalities, I₁ and I₂ can be bounded as

I1≤C∥∇2B∥L2(∥u∥L6∥∇B∥L3+2∥∇u∥L2∥B∥L∞)≤C∥∇u∥L2∥∇2B∥L2(∥∇2B∥L212∥∇B∥L212+∥∇2B∥L234∥∇B∥L214)≤v4∥∇2B∥L22+C(∥∇u∥L24+C∥∇u∥L28)∥∇B∥L22≤v4∥∇2B∥L22+C(M)∥∇B∥L22

and

I2=ϵ∫∇(curlB)⋅∇((curlB)×Bρ)dx=ϵ∫∇(curlB)⋅(curlB)×∇(Bρ)dx≤C∥∇2B∥L2(∥ρ−1∥L∞∥∇B∥L42+∥ρ−2∥L∞∥B∥L∞∥∇B∥L3∥∇ρ∥L6)≤C∥∇2B∥L252∥∇B∥L212+C∥∇2B∥L2∥∇B∥L212∥∇2B∥L212∥B∥L234∥∇2B∥L214∥∇2ρ∥L2≤v4∥∇2B∥L22+C(M)(∥∇2B∥L24∥∇B∥L22+∥∇B∥L24∥B∥L26∥∇2ρ∥L28)≤v4∥∇2B∥L22+C(M)∥∇B∥L22.

Putting the bounds of I₁ and I₂ into (2.24), and then applying (2.4) and (2.8), it leads to

ddt∥∇B∥L22+v∥∇2B∥L22≤C(M)∥∇B∥L22. (2.25)

Integrating (2.25) over (0, T) and by (2.10), we obtain

sup0≤t≤T∥∇B∥L22+v∫0T∥∇2B∥L22dt≤C(M)E016+∥∇B0∥L22. (2.26)

Multiplying (2.25) by σ², then, integrating it over (0, T) and by (2.10), we have

sup0≤t≤Tσ2∥∇B∥L22+v∫0Tσ2∥∇2B∥L22dt≤(C(M)+2σ⋅σ′)∫0T∥∇B∥L22dt≤C(M)E016. (2.27)

By (2.26) and (2.27), we complete the proof of (2.20) and (2.21).

Applying ∇² to (2.1)₄ and multiplying it by ∇²B, then integrating it over ℝ³, we get

12ddt∥∇2B∥L22+v∥∇3B∥L22=∫∇2B⋅∇2(−u⋅∇B+B⋅∇u−Bdivu)dx−ϵ∫∇2B⋅∇2curl(curlB)×Bρdx:=II1+II2. (2.28)

Using the integration by parts and with the help of Hölder, Sobolev and Young inequalities, by (2.4) and (2.10), II₁ and II₂ can be bounded as

II1≤C∥∇3B∥L2(∥u∥L6∥∇2B∥L3+∥∇u∥L3∥∇B∥L6+∥∇2u∥L2∥B∥L∞)≤v4∥∇2B∥L22+C(M)(E0−112∥∇u∥L22+E014∥∇2u∥L22+E0112∥∇2B∥L22)≤v4∥∇2B∥L22+C(M)∥∇B∥L22

and

II2=ϵ∫∇2(curlB)⋅∇2((curlB)×Bρ)dx=ϵ∫∇2(curlB)⋅(curlB)×∇2(Bρ)dx≤C∫|∇3B⋅∇B|(|∇2B|+|∇B∥∇ρ|+|B∥∇2ρ|)≤C∥∇3B∥L2(∥ρ−1∥L∞∥∇B∥L42+∥ρ−2∥L∞∥B∥L∞∥∇B∥L3∥∇ρ∥L6)dx≤C∥∇3B∥L2(∥∇B∥L4∥∇2B∥L4+∥∇B∥L∞∥∇B∥L3∥∇ρ∥L6+∥B∥L∞∥∇2ρ∥L2)≤C(M)∥∇3B∥L274(∥∇B∥L2∥∇2B∥L214+∥∇B∥L234∥∇2B∥L212∥∇2ρ∥L2)≤v4∥∇2B∥L22+C(M)∥∇B∥L22.

Inserting the estimates of II₁ and II₂ into (2.28), and then applying (2.4) and (2.8), it implies

ddt∥∇2B∥L22+v∥∇3B∥L22≤C(M)∥∇B∥L22. (2.29)

Integrating (2.29) over (0, T) and by (2.10), we obtain

sup0≤t≤T∥∇2B∥L22+v∫0T∥∇3B∥L22dt≤C(M)E0112+∥∇B0∥L22. (2.30)

Multiplying (2.29) by σ⁴, then, integrating it over (0, T) and using (2.10) and (2.21), we have

sup0≤t≤Tσ4∥∇2B∥L22+v∫0Tσ4∥∇3B∥L22dt≤4σ3⋅σ′∫0T∥∇2B∥L22dt+C(M)∫0T∥∇B∥L22dt≤C(M)E0112. (2.31)

Thus, by (2.26), (2.27), (2.30) and (2.31), we complete the proof of this lemma. □

For ∥∇ρ∥_L², we want to establish two different estimates in the following lemma.

Lemma 2.7

Under the assumptions of Proposition 2.1, it holds that

∫0T∥∇ρ∥L22dt≤C(M)E0112(∫0T∥∇2ρ∥L22dt+1), (2.32)

∫0T∥∇ρ∥L22dt≤C(M)E016+C(M)∫0T∥∇2u∥L22dt. (2.33)

Proof

Multiplying (2.1)₂ by ∇ρρ and integrating the resulting equality over ℝ³, one has

∫ρ−1|∇ρ|2dx=−∫∇ρ⋅ut−∇ρ⋅[(u⋅∇)u]+1ρ∇ρ⋅(μΔu+(μ+λ)∇divu)+1ρ∇ρ⋅((curlB)×B)dx−∫θ−1ρ|∇ρ|2+∇θ⋅∇ρdx=−ddt∫u⋅∇ρdx+∫divu⋅div(ρu)dx−∫∇ρ⋅u⋅∇udx+∫1ρ∇ρ⋅(μΔu+(μ+λ)∇divu)dx+∫1ρ∇ρ⋅((curlB)×B)dx−∫θ−1ρ|∇ρ|2+∇θ⋅∇ρdx=−ddt∫u⋅∇ρdx+∑i=15Ji. (2.34)

By (2.4) and (2.8), with the help of Hölder, Young and Sobolev inequalities, we obtain

J1+J4≤C(M)∥∇u∥L22+C∥∇ρ∥L3∥∇B∥L2∥B∥L6≤C(M)(∥∇u∥L22+∥∇B∥L22).

Similarly, we give the estimates for J₅ as follows,

J5≤∥θ−1∥L6∥∇ρ∥L2∥∇ρ∥L3+C∥∇θ∥L2∥∇ρ∥L2≤δ1∥∇ρ∥L22+C(M)∥∇θ∥L22.

Particularly, we established the different estimates for J₂ + J₃,

J2+J3≤C(∥ρ−1∇2ρ∥L2+∥|∇ρ|2ρ−2∥L2)∥∇u∥L2+C∥∇u∥L2∥u∥L3∥∇ρ∥L6≤C(∥∇2ρ∥L2∥∇u∥L2+∥∇ρ∥L4∥∇ρ∥L4∥∇u∥L2+∥u∥L212∥∇u∥L232∥∇2ρ∥L2)≤C(M)(E0112∥∇2ρ∥L22+E0−112∥∇u∥L22+E0112∥∇ρ∥L22)

J2+J3≤C∥ρ−1∇ρ∥L2∥∇2u∥L2+C∥∇u∥L6∥u∥L3∥∇ρ∥L2≤δ1∥∇ρ∥L22+C(M)(∥∇u∥L22+∥∇2u∥L22).

Substituting the estimates of J_i(i = 1, ⋯, 5) into (2.30), it is clear that

∥∇ρ∥L22≤−Cddt∫u⋅∇ρdx+C(M)E0−112(∥∇u∥L22+∥∇B∥L22+∥∇θ∥L22)+E0112∥∇2ρ∥L22 (2.35)

∥∇ρ∥L22≤−Cddt∫u⋅∇ρdx+C(M)(∥∇u∥L22+∥∇B∥L22+∥∇θ∥L22)+C(M)∥∇2u∥L22. (2.36)

Integrating (2.35) and (2.36) from 0 to T, and then applying (2.8) and (2.10), we obtain (2.32) and (2.33), respectively. □

The following lemma is established to estimate A₂(T).

Lemma 2.8

Under the assumptions of Proposition 2.1, it holds that

sup0≤t≤T(∥∇u∥L22+∥∇ρ∥L22)+C1∫0T∥∇2u∥L22dt≤C(M)E016+(∥∇u0∥L22+∥∇ρ0∥L22), (2.37)

sup0≤t≤Tσ2(∥∇u∥L22+∥∇ρ∥L22)+∫0Tσ2∥∇2u∥L22dt≤C(M)E0112(∫0T∥∇2ρ∥L22dt+1), (2.38)

where C₁ is a positive constant depending on μ and λ.

Proof

Using ∇ to (2.1)₁ and (2.1)₂, multiplying the resulting equations by ∇ρ and ∇u, respectively, then summing up and integrating it over ℝ³, we obtain

12ddt(∥∇u∥L22+∥∇ρ∥L22)+∫1ρ∇2u⋅(μΔu+(μ+λ)∇divu)dx=∫∇ρ⋅∇div((ρ−1)u)dx+∫∇u⋅∇1ρ∇(ρθ)−∇ρdx+∫(∇2u⋅(u⋅∇u)−1ρ∇2u⋅((curlB)×B))dx=∑i=13Ki. (2.39)

By (2.4) and (2.8), using Hölder, Young and Sobolev inequalities, let us show the estimates of K_i (i = 1, 2, 3), respectively.

K1=∫12|∇ρ|2∇u+(ρ−1)∇ρ⋅divudx≤C∥∇ρ∥L2∥∇ρ∥L3∥∇u∥L6+∥∇ρ∥L2∥ρ−1∥L3∥∇u∥L6≤C(∥ρ−1∥L212∥∇ρ∥L22∥∇2ρ∥L232+∥ρ−1∥L2∥∇ρ∥L23)+δ1∥∇2u∥L22≤δ1∥∇2u∥L22+C(M)E014∥∇ρ∥L22, (2.40)

similarly,

K2=∫∇2u⋅1ρ∇(ρθ)−∇ρdx≤C∥∇2u∥L2(∥θ−1∥L6∥∇ρ∥L3+∥ρ−1∥L4∥∇ρ∥L4+∥∇θ∥L2)≤δ1∥∇2u∥L22+C(M)∥∇θ∥L22+C(M)E014∥∇ρ∥L22 (2.41)

and

K3≤C∥∇2u∥L2∥∇u∥L4∥u∥L4+C∥∇2u∥L2∥∇B∥L4∥B∥L4≤C∥∇2u∥L274∥∇u∥L2∥u∥L214+C∥∇2u∥L2∥∇2B∥L234∥∇B∥L2∥B∥L214≤δ1∥∇2u∥L22+C(∥∇u∥L28∥u∥L22+∥∇B∥L22∥B∥L212)≤δ1∥∇2u∥L22+C(M)E014(∥∇u∥L22+∥∇B∥L22). (2.42)

Substituting (2.40)-(2.42) into (2.39), one has

ddt(∥∇u∥L22+∥∇ρ∥L22)+2C1∥∇2u∥L22≤C(M)E014(∥∇u∥L22+∥∇B∥L22+∥∇ρ∥L22)+C(M)∥∇θ∥L22. (2.43)

Integrating (2.43) from 0 to T, applying (2.10) and (2.33), we arrive at (2.37).

Next, multiplying (2.43) by σ² and integrating the resulting inequality from 0 to T, applying (2.10) and (2.32), we can obtain (2.38) as follows,

σ2(∥∇u∥L22+∥∇ρ∥L22)+∫0Tσ2∥∇2u∥L22dt≤C∫0Tσσ′(∥∇u∥L22+∥∇ρ∥L22)dt+C(M)∫0Tσ2∥∇θ∥L22dt.C(M)E014∫0Tσ2(∥∇u∥L22+∥∇B∥L22+∥∇ρ∥L22)dt≤C(M)E0112(∫0T∥∇2ρ∥L22dt+1).

This completes the proof of this lemma. □

Lemma 2.9

Under the assumptions of Proposition 2.1, it holds that

sup0≤t≤T∥∇θ∥L22+C2∫0T(∥θt∥L22+∥∇2θ∥L22)dt≤C(M)E0112+∥∇θ0∥L22, (2.44)

sup0≤t≤Tσ2∥∇θ∥L22+∫0Tσ2∫(|θt|2+|∇2θ|2)dxdt≤C(M)E0112, (2.45)

where C₂ is a positive constant depending on κ.

Proof

It follows from (2.1)₃, (2.4), Hölder and Sobolev inequalities that

κddt∫|∇θ|2dx+∫ρ|θt|2+κ2ρ|Δθ|2dx=∫(ρθt−κΔθ)θt−κρΔθdx≤C∫(|u|2|∇θ|2+|θ|2|divu|2+|∇u|4+|∇B|4)dx≤(∥u∥L∞2∥∇θ∥L22+∥θ−1∥L62∥divu∥L6∥divu∥L2+2∥θ−1∥L6∥divu∥L3∥divu∥L2+∥divu∥L22+∥∇u∥L6∥∇u∥L6∥∇u∥L6∥∇u∥L2+∥∇B∥L6∥∇B∥L6∥∇B∥L6∥∇B∥L2)≤C(M)(∥∇θ∥L22+∥∇θ∥L2∥∇u∥L2+∥∇u∥L22+∥∇2u∥L2∥∇u∥L2+∥∇2B∥L2∥∇B∥L2).

By above inequality and Young inequality, we obtain

ddt∫|∇θ|2dx+C2∫(|θt|2+|∇2θ|2)dx≤C(M)E0−112(∥∇θ∥L22+∥∇u∥L22+∥∇B∥L22)+E0112(∥∇2u∥L22+∥∇2B∥L22) (2.46)

and

ddt∫|∇θ|2dx+C2∫(|θt|2+|∇2θ|2)dx≤C(M)E0−124(∥∇θ∥L22+∥∇u∥L22+∥∇B∥L22)+E0124(∥∇2u∥L22+∥∇2B∥L22). (2.47)

Integrating (2.46) from 0 to T, applying (2.10) and (2.37), (2.44) holds.

Multiplying (2.47) by σ² and integrating the resulting inequality from 0 to T, applying (2.10) and (2.33), we obtain (2.41) as follows,

σ2∫|∇θ|2dx+∫0Tσ2∫(|θt|2+|∇2θ|2)dx≤C∫0Tσσ′∫|∇θ|2dxdt+C(M)E0−124∫0T(∥∇θ∥L22+∥∇u∥L22+∥∇B∥L22)dt+E0124∫0T(∥∇2u∥L22+∥∇2B∥L22)dt≤C(M)E0112.

Thus, we complete the proof of this lemma. □

By the bound of ∥∇θ∥L22+∫0T∥∇2θ∥L22dt, we give the following lemma to estimate A₃(T).

Lemma 2.10

Under the assumptions of Proposition 2.1, it holds that

∫0T∥∇2ρ∥L22dt≤C(M), (2.48)

∫0T∥∇ρ∥L22dt≤C(M)E0112, (2.49)

sup0≤t≤Tσ2(∥∇u∥L22+∥∇ρ∥L22)+∫0Tσ2∥∇2u∥L22dτ≤C(M)E0112, (2.50)

sup0≤t≤T(C3∥∇θ∥L22+∥∇2u∥L22+∥∇2ρ∥L22)+C4∫0T(∥∇3u∥L22+∥∇2θ∥L22)dt≤C(M)E0130+C3∥∇θ0∥L22+∥∇2u0∥L22+∥∇2ρ0∥L22, (2.51)

where C₃ and C₄ are positive constants depending on some known constants μ, λ, k, v.

Proof

Multiplying (2.1)₂ by ∇3ρρ and integrating the resulting equality over ℝ³, then using integration by parts, one has

∫|∇2ρ|2dx=−∫∇2ρ⋅∇ut−∫∇2ρ⋅∇(1ρ∇(ρθ))−∇2ρdx−∇2ρ⋅∇(u⋅∇u)+∇2ρ⋅∇(1ρ(μΔu+(μ+λ)∇divu))+∇2ρ⋅∇((curlB)×B)dx=−ddt∫∇u⋅∇2ρdx+∫div∇u⋅∇div(ρu)dx+∫∇2ρ⋅∇(u⋅∇u)dx+∫∇2ρ⋅∇(1ρ(μΔu+(μ+λ)∇divu))dx+∫∇2ρ⋅∇(1ρ(curlB)×B)dx−∫(∇2ρ⋅∇(1ρ∇(ρθ))−|∇2ρ|2)dx=−ddt∫∇u⋅∇2ρdx+∑i=15Ji. (2.52)

By (2.4) and using Hölder, Sobolev and Young inequalities, we can bound ∑1≤i≤4Ji as

∑1≤i≤4Ji≤14∥∇2ρ∥L22+C(M)(∥∇3u∥L22+∥∇2u∥L22+∥∇2B∥L22+∥∇B∥L22).

For J₅, it follows from (2.4), (2.44), Hölder, Sobolev and Young inequalities that

J5≤C∥∇2ρ∥L2(∥∇2θ∥L2+∥θ−1∥L∞∥∇2ρ∥L2+∥1−ρ∥L∞∥|∇2ρ∥L2+∥∇ρ∥L3∥∇θ∥L6+∥∇ρ∥L3∥∇ρ∥L6)≤14∥∇2ρ∥L22+C(M)(∥∇2θ∥L22+∥∇ρ∥L22).

Substituting the estimates of J_i (i = 1, ⋯, 5) into (2.48), we get

∥∇2ρ∥L22≤−Cddt∫∇u⋅∇2ρdx+C(M)(∥∇ρ∥L22+∥∇2θ∥L22+∥∇2u∥L22+∥∇3u∥L22+∥∇B∥L22+∥∇2B∥L22). (2.53)

Applying ∇² to (2.1)₁ and (2.1)₂, multiplying the resulting equations by ∇²ρ and ∇²u, then summing up and integrating it over ℝ³, we have

12ddt(∥∇2u∥L22+∥∇2ρ∥L22)+∫1ρ∇3u⋅∇(μΔu+(μ+λ)∇divu)dx=∫∇2ρ⋅∇2div((ρ−1)u)dx−∫∇2u⋅∇2(1ρ∇(ρθ)−∇ρ)dx+∫(∇3u⋅∇(u⋅∇u)−∇3u⋅∇(1ρ(curlB×B)))dx+∫∇ρρ2⋅∇3u⋅(μΔu+(μ+λ)∇divu)dx=∑i=14Fi. (2.54)

Now, let us bound F_i(i = 1, 2, 3, 4) by (2.4), (2.8), (2.10) and Hölder, Sobolev and Young inequalities.

F1=∫∇2ρ⋅∇3ρ⋅u+|∇2ρ|2∇u+∇2ρ⋅∇2((ρ−1)divu)dx≤C(∥∇2ρ∥L2∥∇2ρ∥L2∥∇u∥L∞+∥∇2ρ∥L2∥∇ρ∥L3∥∇2u∥L6+∥∇2ρ∥L2∥ρ−1∥L∞∥∇3u∥L2)≤C(∥∇2ρ∥L22∥∇u∥L214∥∇3u∥L234+∥∇ρ∥L212∥∇2ρ∥L232∥∇3u∥L2+∥∇2ρ∥L2∥ρ−1∥L214∥∇2ρ∥L234∥∇3u∥L2)≤δ1∥∇3u∥L2+C(∥∇2ρ∥L2165∥∇u∥L225+∥∇ρ∥L2∥∇2ρ∥L23+∥ρ−1∥L212∥∇2ρ∥L272)≤δ1∥∇3u∥L2+E0130∥∇2ρ∥L22+C(M)(E0−124∥∇ρ∥L22+E0−215∥∇u∥L22).

Similarly, we obtain

F2≤∥∇3u∥L22∥∇2θ∥L22+∥ρ−1∥L∞∥∇3u∥L22(∥θ−1∥L∞∥∇ρ∥L2+∥ρ−1∥L∞∥∇2ρ∥L2+∥∇2ρ∥L2∥∇θ∥L3+∥θ−1∥L2∥∇ρ∥L62)≤2δ1∥∇3u∥L22+14δ1∥∇2θ∥L22+C(∥θ−1∥L212∥∇2θ∥L232∥∇ρ∥L22+∥ρ−1∥L212∥∇2ρ∥L272+∥∇2ρ∥L22∥∇θ∥L2∥∇2θ∥L2+∥∇θ∥L2∥∇2ρ∥L22)≤2δ1∥∇3u∥L22+12δ1∥∇2θ∥L22+C(M)(E0124∥∇ρ∥L22+E0112∥∇2ρ∥L22+E0−112∥∇θ∥L22)

and

F3+F4≤C∥∇3u∥L2(∥∇2u∥L4∥u∥L4+∥∇u∥L4∥∇u∥L4+∥∇ρ∥L6∥∇B∥L6∥B∥L6+∥∇2B∥L4∥B∥L4+∥∇B∥L4∥∇B∥L4+∥∇ρ∥L4∥∇2u∥L4)≤δ1∥∇3u∥L22+C(M)E0−112(∥∇u∥L22+∥∇B∥L22)+C(M)∥∇ρ∥L22+C(M)E0112(∥∇2u∥L22+∥∇2B∥L22).

Substituting the estimates of F_i (i = 1, ⋯, 4) into (2.54), yields

ddt(∥∇2u∥L22+∥∇2ρ∥L22)+C5∥∇3u∥L22≤C(M)E0−215(∥∇u∥L22+∥∇B∥L22+∥∇θ∥L22)+C(M)E0−124∥∇ρ∥L22+E0130(∥∇2u∥L22+∥∇2B∥L22+∥∇2ρ∥L22)+12δ1∥∇2θ∥L22. (2.55)

Multiplying (2.46) by (C₂δ₁)⁻¹, then adding up it and (2.55), substituting (2.35) and (2.53) into the resulting inequality, we obtain

ddt(1C2δ1∥∇θ∥L22+∥∇2u∥L22+∥∇2ρ∥L22)+C5∥∇3u∥L22+12δ1∥∇2θ∥L22≤−CE0−124ddt∫u⋅∇ρdx+C(M)E0−215(∥∇u∥L22+∥∇B∥L22+∥∇θ∥L22)−CE0130(ddt∫∇u⋅∇2ρdx+∥∇2u∥L22+∥∇2B∥L22).

Integrating above inequality over (0, T), it follows from (2.4), (2.8), (2.10), (2.26) and (2.37) that

sup0≤t≤T(C3∥∇θ∥L22+∥∇2u∥L22+∥∇2ρ∥L22)+C4∫0T(∥∇3u∥L22+∥∇2θ∥L22)dt≤C(M)E0130+C3∥∇θ0∥L22+∥∇2u0∥L22+∥∇2ρ0∥L22,

where C3=1C2δ1 and C4=min{C5,12δ1}. Thus, we complete the proof of (2.44).

Integrating (2.53) from 0 to T, by (2.10), (2.26), (2.37), (2.45) and (2.51), we obtain (2.48). Substituting (2.48) into (2.35) and (2.38), we get (2.49) and (2.50), respectively, which completes the proof of this lemma. □

Next, the following lemma is needed to bound A₄(T).

Lemma 2.11

Under the assumptions of Proposition 2.1, it holds that

∫0Tσ4(∥ut∥L22+∥Bt∥L22+∥∇ut∥L22+∥∇Bt∥L22)dt≤C(M)E0112, (2.56)

∫0T(∥ut∥L22+∥Bt∥L22+∥∇ut∥L22+∥∇Bt∥L22)dt≤C(M). (2.57)

Proof

Applying ∇ to (2.1)₂, squaring both sides of resulting equation, by(2.4) and (2.8), using Hölder, Sobolev and Young inequalities, we have

∫|∇ut−μ∇Δu−(μ+λ)∇2divu|2dx≤∫(|∇(1ρ)⋅∇2u|2+|(ρ−1ρ)⋅∇3u|2+|∇2(ρθ)|2+|∇(ρu⋅∇u)|2+|∇((curlB)×B)|2)dx≤C(M)(∥∇ρ∥L62∥∇2u∥L6∥∇2u∥L2+∥ρ−1∥L∞∥∇3u∥L22+∥θ−1∥L∞2∥∇2ρ∥L22+∥θ−1∥L∞∥∇2ρ∥L22+∥∇2ρ∥L22+∥∇ρ∥L22∥∇ρ∥L62∥∇θ∥L62∥∇θ∥L62+∥ρ∥L∞2∥∇2θ∥L22+∥ρ∥L∞2∥∇u∥L44+∥u∥L∞2∥∇ρ∥L2∥∇ρ∥L6∥∇u∥L6∥∇u∥L6+∥ρ∥L∞2∥u∥L∞2∥∇2u∥L22+∥B∥L∞2∥∇2B∥L22+∥∇B∥L44)≤δ1∥∇3u∥L22+C(M)(∥∇2ρ∥L22+∥∇2u∥L22+∥∇2B∥L22). (2.58)

Similarly, it follows from (2.1)₄, (2.4), (2.8), Hölder, Sobolev and Young inequalities that

vddt∥∇2B∥L22+∥∇Bt∥L22+v2∥∇3B∥L22≤C(M)(∥∇curl(curlB×Bρ)∥L22+∥∇curl(u×B)∥L22)≤C(M)(∥B∥L∞2∥∇3B∥L22+∥∇B∥L32∥∇2B∥L62+∥∇2B∥L62∥B∥L62∥∇ρ∥L62+∥∇B∥L62∥∇B∥L62∥∇ρ∥L62+∥∇B∥L62∥B∥L62∥∇ρ∥L62+∥∇B∥L62∥B∥L62∥∇2ρ∥L62+∥∇2B∥L22∥u∥L∞2+∥∇u∥L32∥∇B∥L62+∥∇2u∥L22∥B∥L∞2)≤C(M)(∥∇B∥L22+∥∇2B∥L22+∥∇3B∥L22+∥∇2u∥L22). (2.59)

Summing up (2.58) and (2.59), we have

C6ddt(∥∇2u∥L22+∥∇2B∥L22)+(∥∇ut∥L22+∥∇3u∥L22+∥∇Bt∥L22+∥∇3B∥L22)≤C(M)(∥∇2ρ∥L22+∥∇B∥L22+∥∇2B∥L22+∥∇3B∥L22+∥∇2u∥L22), (2.60)

where C₆ is a positive constant depending on some known constants μ, λ, v. Integrating (2.60) over (0, T), using (2.10), (2.22), and (2.37), we obtain

∫0T(∥∇ut∥L22+∥∇Bt∥L22)dt≤C(M). (2.61)

Multiplying (2.60) by σ⁴ and integrating it over (0, T), by (2.10), (2.23), and (2.49), one has

∫0Tσ4(∥∇ut∥L22+∥∇Bt∥L22)dt≤C(M)E0112. (2.62)

Similarly to (2.61) and (2.62), we can get (2.63) and (2.64).

∫0T(∥ut∥L22+∥Bt∥L22)dt≤C(M), (2.63)

∫0Tσ4(∥ut∥L22+∥Bt∥L22)dt≤C(M)E0112. (2.64)

Therefore, we complete the proof of this lemma by (2.61)-(2.64). □

For A₄(T), we have

Lemma 2.12

Under the assumptions of Proposition 2.1, it holds that

sup0≤t≤Tσ4∫(|θt|2+|∇2θ|2)dx≤E0132. (2.65)

Proof

This lemma can be proved in the similar way of Lemma 3.12 in [14]. Here, we omit the proof of them here for brevity. □

Thus, we are ready to show the proof of Proposition 2.1.

Proof of Proposition 2.1

By Lemma 2.4 and Lemma 2.5, we obtain

A1(T)≤CE0+E016<2E016,

provided E0<C−65. By Lemma 2.6 and Lemma 2.8, we obtain

A2(T)≤C(M)E016+M<32M,

provided E0<(M2C(M))6. By Lemma 2.6 and 2.10, we obtain

A3(T)≤C(M)E0130+M<32M,

provided E0<(M2C(M))30. A4(T)≤E0132 was given by using 2.12 directly. Hence, we conclude the proof of Proposition 2.1. □

In addition, we also need to establish the estimates for ∇ρ_t and ∇²θ, which are independent of the time.

Lemma 2.13

Under the assumption of Proposition 2.1, it holds that

∥∇ρt∥L22+∫0T∥∇ρt∥L22dt≤C(M), (2.66)

∥∇2θ∥L22+∫0T(∥∇3θ∥L22+∥∇θt∥L2)dt≤C(M)+∥∇2θ0∥L22. (2.67)

Proof

By (2.1)₁ and using (2.4), Hölder, Young and Sobolev inequalities, one can get

∥∇ρt∥L22≤C∫(|∇2ρ|2|u|2+|∇ρ|2|∇u|2+|ρ|2|∇2u|2)dx≤C(∥u∥L∞2∥∇2ρ∥L22+∥∇u∥L3∥∇u∥L6∥∇ρ∥L62+∥ρ∥L∞2∥∇2u∥L22)≤C(M)∥∇2ρ∥L22,

which together with (2.48) yields (2.66). Similarly, it follows from (2.1)₃ that

κddt∫|∇2θ|2dx+∫(|∇θt|2+κ2|∇Δθ|2)dx≤C∫(|ρ−1ρ|2|∇3θ|2+|∇(1ρ)|2|∇2θ|2+|∇(u⋅∇θ)|2+|∇(θ⋅divu)|2)dx+C∫(|∇u|2|∇2u|2+|∇B|2|∇2B|2)dx≤(E016+δ1)∥∇3θ∥L22+C(M)(∥∇2θ∥L22+∥∇u∥H22+∥∇B∥H22).

Integrating above inequalities over (0, T), and by virtue of Lemma 2.5, Lemma 2.6, Lemma 2.8 and Lemma 2.9, we complete the proof of this lemma. □

2.3 Time-dependent higher-order estimates

In this subsection, we establish the higher-order estimates for the solution (ρ₀, u, θ, B). Throughout this subsection, we always assume E₀ ≤ δ ≤ 1, and we denote positive constant C_T depending on μ, λ, κ, v, M, ϵ and T.

Lemma 2.14

Under the conditions of Theorem 1.1, it holds that

∥∇3ρ∥L22+∥∇3u∥L22+∥∇3θ∥L22+∥∇3B∥L22+∫0T(∥∇3ρ∥L22+∥∇4u∥L22+∥∇4θ∥L22+∥∇4B∥L22)dt≤CT. (2.68)

Proof

Applying ∇³ to (2.1)₁, multiplying it by ∇³ρ, and integrating the resulting equation over ℝ³, after integration by parts, we obtain

12ddt∥∇3ρ∥L22=−∫∇3div(ρu)⋅∇3ρdx≤C(M)(∥∇3ρ∥L22(∥∇3u∥L22+1)+∥∇u∥H22)+δ1∥∇4u∥L22. (2.69)

Applying ∇² to (2.1)₂, squaring both sides of resulting equation, then integrating it over ℝ³, we have

∫|∇2ut−μ∇2Δu−(μ+λ)∇3divu|2dx≤∫(|∇2(1ρ)⋅∇2u|2+|∇(1ρ)⋅∇3u|2+|(ρ−1ρ)∇4u|2)dx+∫(|∇2(1ρ)∇(ρθ)|2+|∇2(u⋅∇u)|2)dx+∫|∇2(1ρ(curlB)×B)|2dx=G1+G2+G3. (2.70)

It follows from (2.4), (2.8), Hölder, Sobolev and Young inequalities that

G1+G2≤C(M)(∥∇2ρ∥L6∥∇2ρ∥L2∥∇2u∥L62+∥∇ρ∥L64∥∇2u∥L62+∥∇ρ∥L6∥∇ρ∥L2∥∇2u∥L62+∥ρ−1∥L∞∥∇4u∥L22+∥∇θ∥H22+∥∇u∥H22+∥∇3ρ∥L22)≤C(M)(∥∇θ∥H22+∥∇u∥H22)+(δ1+E016)∥∇4u∥L22+C(M)∥∇3ρ∥L22. (2.71)

For G₃, we have

G3≤C(M)(∥∇2ρ∥L62∥∇B∥L62∥B∥L62+∥∇ρ∥L64)∥∇B∥L62∥B∥L∞2+∥∇ρ∥L62(∥∇2B∥L62∥B∥L62+∥∇B∥L64)+∥∇3B∥L22∥B∥L∞2+∥∇2B∥L62∥∇B∥L6∥∇B∥L2)≤C(M)∥∇B∥H22+C(M)∥∇3ρ∥L22. (2.72)

Substituting (2.71)-(2.72) into (2.70), together with (2.69), it yields

ddt(∥∇3ρ∥L22+∥∇3u∥L22)+C∥∇4u∥L22≤C(M)(∥∇θ∥H22+∥∇u∥H22+∥∇B∥H22)+C(M)∥∇3ρ∥L22(∥∇3u∥L22+1). (2.73)

Similarly, for ∥∇3B∥L22, it follows from (2.1)₃, (2.1)₄, (2.4), (2.8), Hölder, Sobolev and Young inequalities that

vddt∥∇3B∥L22+∥∇2Bt∥L22+v2∥∇4B∥L22≤C(M)(∥∇2curl(curlB×Bρ)∥L22+∥∇2curl(u×B)∥L22)≤v22∥∇4B∥L22+C(M)(∥∇3ρ∥L22(∥∇3B∥L22+1)+∥∇B∥H22+∥∇u∥H22). (2.74)

Adding (2.73) and (2.74) up, and then using Gronwall inequality, we obtain

∥∇3ρ∥L22+∥∇3u∥L22+∥∇3B∥L22+∫0T(∥∇4u∥L22+∥∇4B∥L22)dt≤CT. (2.75)

Finally, similar to the proof of (2.75), we can bound

∥∇3θ∥+∫0T∥∇4θ∥L22dt≤CT, (2.76)

we omit the proof of it for brevity. Thus, we complete the proof of this lemma by (2.75) and (2.76). □

3 Proof of Theorem 1.1

Applying Lemma 2.4–2.6, 2.8, 2.9 and 2.11–2.13, we obtain the following estimate,

∥ρ−1∥H32+∥u∥H32+∥θ−1∥H32+∥B∥H32+∫0T(∥∇ρ∥H22+∥∇u∥H32+∥∇θ∥H32+∥∇B∥H32)dt≤CT. (3.1)

With the help of the existence and uniqueness of local solutions which has been proved in [4] and all the a priori estimates above, using the standard continuum arguments, we extend the local solution to the global one.

Next, we investigate the large time behavior of solution. It follows from (2.10) and (2.33) that

∫0∞(∥∇ρ∥L22+∥∇u∥L22+∥∇θ∥L22+∥∇B∥L22)dt≤C(M). (3.2)

Then, by Young inequality, (3.2), (2.45), (2.58) and (2.66), we obtain

∫0∞|ddt(∥∇ρt∥L22+∥∇ut∥L22+∥∇Bt∥L22+∥∇θt∥L22)|dt≤∫0∞(∥∇ρ∥L22+∥∇u∥L22+∥∇θ∥L22+∥∇B∥L22)dt+∫0∞(∥∇ρt∥L22+∥∇ut∥L22+∥∇Bt∥L22+∥∇θt∥L22)dt≤C(M). (3.3)

Using (3.2) and (3.3), we have

limt→∞(∥∇ρ∥L22+∥∇u∥L22+∥∇θ∥L22+∥∇B∥L22)=0, (3.4)

which combining

∥ρ−1∥H22+∥u∥H22+∥θ−1∥H22+∥B∥H22≤C(M)

and using Sobolev inequality, we arrive at (1.9). Thus, the proof of Theorem 1.1 is completed. □

taoq060@126.com

Acknowledgement

This paper was done while the first author visited the Department of Mathematics, University of Pittsburgh, Pennsylvania, USA. This research was supported by the National Science Foundation of China (11971320, 11871346), Guangdong Basic and Applied Basic Research Foundation (2020A1515010530, 2018A030313024), NSF of Shenzhen City (JCYJ20180305125554234) and China Scholarship Council (201908440027). The authors are deeply grateful to the referees for the valuable comments and suggestions.

Conflict of interest: Authors state no conflict of interest

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Received: 2020-10-08

Accepted: 2021-03-11

Published Online: 2021-04-24

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https://doi.org/10.1515/anona-2020-0178

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full compressible Hall-MHD equations; global existence; large time behavior

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