Strong solutions to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection

Strong solutions to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection

without thermal diffusion

Xin Zhong

School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China Received 15 March 2020, appeared 8 April 2020

Communicated by Maria Alessandra Ragusa

Abstract. We are concerned with the Cauchy problem of nonhomogeneous Boussinesq equations for magnetohydrodynamics convection in R². We show that there exists a unique local strong solution provided the initial density, the magnetic field, and the initial temperature decrease at infinity sufficiently quickly. In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states.

Keywords:nonhomogeneous Boussinesq-MHD system, strong solutions, Cauchy prob- lem.

2020 Mathematics Subject Classification: 35Q35, 76D03.

1 Introduction

Consider the following nonhomogeneous Boussinesq system for magnetohydrodynamic con- vection (Boussinesq-MHD) inR²:

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ρ_t+div(ρu) =0,

(ρu)_t+div(ρu⊗u)−_µ∆u+∇P=b· ∇b+ρθe2, θ_t+u· ∇θ =0,

b_t−_ν∆b+u· ∇b−b· ∇u=0, divu=_divb=_0,

(1.1)

where t ≥ _{0 is time,} x = (x₁,x₂) ∈ _R² is the spatial coordinate, and ρ = ρ(x,t)_{, u} = (u¹,u²)(x,t), b = (b¹,b²)(x,t), θ = θ(x,t), and P = P(x,t) denote the density, velocity, magnetic field, temperature, and pressure of the fluid, respectively. The coefficients µandν are positive constants. e2= (_{0, 1})^T_{, where} Tis the transpose.

We consider the Cauchy problem for (1.1) with the far field behavior

(ρ,u,θ,b)→(0,0, 0,0), as|x| →_∞, (1.2)

BEmail: xzhong1014@amss.ac.cn

and the initial condition

ρ(x, 0) =ρ0(x), ρu(x, 0) =ρ0u0(x), θ(x, 0) =θ0(x), b(x, 0) =b0(x), x ∈_R², (1.3) for given initial dataρ₀,u₀,θ₀, andb₀.

The system (1.1) is a combination of the nonhomogeneous Boussinesq equations of fluid dynamics and Maxwell’s equations of electromagnetism, where the displacement current can be neglected. The Boussinesq-MHD system models the convection of an incompressible flow driven by the buoyant effect of a thermal or density field, and the Lorenz force, generated by the magnetic field of the fluid and the Lorentz force. Specifically, it closely relates to a natural type of the Rayleigh-Bénard convection, which occurs in a horizontal layer of conductive fluid heated from below, with the presence of a magnetic field. For more physics background, one may refer to [7,14,16] and references therein.

Whenρis constant, the system (1.1) reduces to the homogeneous Boussinesq-MHD system.

Recently, the well-posedness issue of solutions has attracted much attention. Bian [3] studied the initial boundary value problem of two-dimensional (2D) viscous Boussinesq-MHD system and obtained a unique classical solution forH³ initial data. Without smallness assumption on the initial data, Bian and Gui [4] proved the global unique solvability of 2D Boussinesq-MHD system with the temperature-dependent viscosity, thermal diffusivity, and electrical conduc- tivity. Later on, the authors [5] established the global existence of weak solutions with H¹ initial data. By imposing a higher regularity assumption on the initial data, they also ob- tained a unique global strong solution. In [10], Larios and Pei proved the local well-posedness of solutions to the fully dissipative 3D Boussinesq-MHD system, and also the fully inviscid, irresistive, non-diffusive Boussinesq-MHD system. Moreover, they also provided a Prodi–

Serrin-type global regularity condition for the 3D Boussinesq-MHD system without thermal diffusion, in terms of only two velocity and two magnetic components. By Fourier localiza- tion techniques, Zhai and Chen [20] investigated well-posedness to the Cauchy problem of the Boussinesq-MHD system with the temperature-dependent viscosity in Besov spaces. Very re- cently, Liu et al. [13] showed the global existence and uniqueness of strong and smooth large solutions to the 3D Boussinesq-MHD system with a damping term. Meanwhile, Bian and Pu [6] proved global axisymmetric smooth solutions for the 3D Boussinesq-MHD equations without magnetic diffusion and heat convection.

If the fluid is not affected by the Lorentz force (i.e.,b= 0), then the system (1.1) becomes the nonhomogeneous Boussinesq system. The authors [9,21] studied regularity criteria for 3D nonhomogeneous incompressible Boussinesq equations, while Qiu and Yao [17] showed the local existence and uniqueness of strong solutions of multi-dimensional nonhomogeneous incompressible Boussinesq equations in Besov spaces. A blow-up criterion was also obtained in [17]. We should point out here that the results in [9,17,21] always require the initial density is bounded away from zero. For the initial density allowing vacuum states, Zhong [22] recently showed local existence of strong solutions of the Cauchy problem in R² by making use of weighted energy estimate techniques. In this paper, we will investigate the local existence of strong solutions to the problem (1.1)–(1.3) with zero density at infinity. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support.

Before stating our main result, we first explain the notations and conventions used throughout this paper. Forr>0, set

B_r,^x∈ _R²| |x|<r .

For 1≤ p≤_∞and integerk≥0, the standard Sobolev spaces are denoted by:

L^p= L^p(_R²), W^k,p =W^k,p(_R²), H^k = H^k,2(_R²), D^k,p= {u ∈L¹_loc| ∇^ku∈ L^p}. Our main result can be stated as follows:

Theorem 1.1. Letη₀be a positive constant and

¯

x, ³+|x|²¹² log^1+η0 3+|x|². (1.4) For constants q>2and a>1, we assume that the initial data(ρ₀≥0,u₀,θ₀ ≥0,b₀)satisfy

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ρ₀x¯^a ∈ L¹∩H¹∩W^1,q, θ₀∈ H¹∩W^1,q,

√ρ₀u₀∈ L², ∇u₀ ∈ L², divu₀ =0, b₀x¯^2a ∈ L², ∇b₀ ∈ L², divb₀ =0.

(1.5)

Then there exists a positive time T0 > ₀ such that the problem (1.1)–(1.3) has a strong solution (ρ≥0,u,θ ≥0,b)onR²×(0,T₀]satisfying

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ρ∈C([0,T₀];L¹∩H¹∩W^1,q), ρx¯^a∈ L^∞(_0,T₀;L¹∩H¹∩W^1,q)_,

√ρu, ∇u, √ t√

ρu_t, √

t∇²u∈ L^∞(0,T₀;L²), θ ∈C([0,T0];H¹∩W^1,q),

b,bx¯^a2,∇b,√ tb_t,√

t∇²b∈ L^∞(0,T₀;L²),

∇u∈ L²(0,T₀;H¹)∩L^q

+1

q (0,T₀;W^1,q),

∇_b∈ L²(0,T₀;H¹), bt, ∇_bx¯^a2 ∈ L²(0,T₀;L²),

√t∇u∈ L²(0,T₀;W^1,q),

√ρut, √

t∇bx¯^a2, √

t∇ut, √

t∇bt ∈ L²(_R²×(0,T0)),

(1.6)

and

0≤inft≤T0

Z

BN1

ρ(x,t)dx≥ ¹ 4

Z

R²ρ0(x)dx, (1.7)

for some positive constant N₁. Moreover, ifθ₀x¯^a ∈ H¹∩W^1,q, then the strong solution just established is unique.

Remark 1.2. When there is no electromagnetic field effect, that isb = 0, (1.1) turns to be the nonhomogeneous Boussinesq equations, and Theorem1.1is the same as that of in [22]. Hence we generalize the main result of [22] to the nonhomogeneous Boussinesq-MHD system (1.1).

However, compared with [22], for the system (1.1) treated here, the strong coupling between the velocity field and the magnetic field, such asu· ∇b, as well as strong nonlinearityb· ∇b, will bring out some new difficulties. To this end, we requireb0x¯^a2 ∈ L²and∇_b0∈ L² beyond the typical hypothesis ofb₀ ∈ H¹. This additional hypothesis is needed in order to obtain the estimate (3.10), which plays a crucial role in dealing with coupling between the velocity field and the magnetic field.

The rest of the paper is organized as follows. In Section 2, we collect some elementary facts and inequalities which will be needed in later analysis. Sections 3 is devoted to the a priori estimates which are needed to obtain the local existence of strong solutions. The main result Theorem1.1is proved in Section4.

2 Preliminaries

In this section, we will recall some known facts and elementary inequalities which will be used frequently later. First of all, if the initial density is strictly away from vacuum, the following local existence theorem on bounded balls can be shown by similar arguments as in [19].

Lemma 2.1. For R>0and B_R ={x∈ _R²| |x|<R}, assume that(ρ₀,u₀,θ₀ ≥0,b₀)satisfies (ρ₀,u₀,θ₀,b₀)∈ H²(B_R), inf

x∈BR

ρ₀(x)>0, divu₀ =divb₀=0. (2.1) Then there exists a small time T_R > 0 and a unique classical solution(ρ,u,P,θ,b)to the following initial-boundary-value problem

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ρ_t+div(ρu) =0,

(ρu)_t+div(ρu⊗u)−µ∆u+∇P=b· ∇b+ρθe2, θ_t+u· ∇θ =0,

bt−_ν∆b+u· ∇b−b· ∇u=0, divu=divb=0,

(ρ,u,θ,b)(x,t=0) = (ρ₀,u₀,θ₀,b₀), x∈ B_R,

u(x,t) =b(x,t) =0, x∈ ∂BR,t >0,

(2.2)

on B_R×(0,T_R]such that

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(ρ,θ)∈C [0,T_R];H² , (u,b)∈C [0,T_R];H²

∩L² 0,T_R;H³ , P∈C [0,T_R];H¹

∩L² 0,T_R;H² ,

(2.3)

where we denote H^k = H^k(B_R)for positive integer k.

Next, for Ω ⊂ _R², the following weighted L^m-bounds for elements of the Hilbert space D˜^1,2(_Ω),{v ∈ H_loc¹ (_Ω)|∇v∈ L²(_Ω)}can be found in [12, Theorem B.1].

Lemma 2.2. For m∈[2,∞)and s∈(1+^m₂,∞),there exists a positive constant C such that for either Ω=_R²orΩ=B_Rwith R≥1and for any v∈ D^˜1,2(_Ω),

_Z

Ω

|v|^m

3+|x|²(log(3+|x|²))^−sdx _m¹

≤Ckvk_L2(B1)+Ck∇vk_L2(_Ω). (2.4) A useful consequence of Lemma2.2is the following crucial weighted bounds for elements of ˜D^1,2(_Ω), which have been proved in [11, Lemma 2.3].

Lemma 2.3. Letx and¯ η₀be as in(1.4)andΩbe as in Lemma2.2. Assume thatρ∈ L¹(_Ω)∩L^∞(_Ω) is a non-negative function such that

Z

BN1

ρdx≥ M₁, kρk_L1(_Ω)∩L^∞(_Ω)≤ M2, (2.5) for positive constants M₁,M₂, and N₁ ≥ 1 with B_N1 ⊂ Ω. Then for ε > 0 and η > 0, there is a positive constant C depending only onε,η,M₁,M₂,N₁, andη₀such that every v∈ D^˜1,2(_Ω)satisfies

kvx¯^−ηk_L(2+ε)/ ˜η(_Ω) ≤Ck√

ρvk_L2(_Ω)+Ck∇vk_L2(_Ω) (2.6) withη˜ =_min{_1,η}.

Next, the following L^p-bound for elliptic systems, whose proof is similar to that of [8, Lemma 12], is a direct result of the combination of the well-known elliptic theory [1,2] and a standard scaling procedure.

Lemma 2.4. For p>1and k≥0, there exists a positive constant C depending only on p and k such that

k∇^k+2vk_Lp(BR)≤Ck_∆vk_Wk,p(BR), (2.7) for every v∈W^k+2,p(B_R)satisfying

v =0 on B_R.

3 A priori estimates

Throughout this section, forr∈ [_1,∞]_andk≥0, we denote Z

·dx=

Z

BR

·dx, L^r= L^r(B_R), W^k,r=W^k,r(B_R), H^k =W^k,2.

Moreover, for R > 4N₀ ≥ 4 withN₀ fixed, assume that(ρ₀,u₀,θ₀,b₀)satisfies, in addition to (2.1), that

1 2 ≤

Z

BN0

ρ0(x)dx ≤

Z

BR

ρ0(x)dx≤1. (3.1)

Thus Lemma 2.1 yields that there exists some T_R > 0 such that the initial-boundary-value problem (1.1) and (2.2) has a unique classical solution (ρ,u,P,θ,b)on B_R×[0,T_R] satisfying (2.3).

Let ¯x,η₀,a, and q be as in Theorem 1.1, the main aim of this section is to derive the following key a priori estimate onψdefined by

ψ(t),¹+k√

ρuk_L2+k∇_uk_L2+kθk_H1∩W^1,q+k∇_bk_L2+kx¯^a2bk_L2 +kx¯^aρk_L1∩H¹∩W^1,q. (3.2) Proposition 3.1. Assume that(ρ₀,u0,θ₀,b0)satisfies(2.1)and(3.1). Let(ρ,u,P,θ,b)be the solution to the initial-boundary-value problem (1.1) and(2.2) on B_R×(0,T_R]obtained by Lemma 2.1. Then there exist positive constants T₀and M both depending only onµ,ν,η₀,q, a, N₀,and E₀ such that

sup

0≤t≤T₀

h

ψ(t) +√ t

k√

ρutk_L2+k∇²uk_L2+kbtk_L2+k∇²bk_L2+k∇bx¯^2ak_L2

i

+

Z _T0

0

k√

ρu_tk²_L2+k∇²uk²_L2+k∇²bk²_L2+kb_tk²_L2+k∇bx¯^2ak²_L2dt +

Z _T0

0

k∇²uk

q+1 q

L^q +k∇Pk

q+1 q

L^q +tk∇²uk²_Lq+tk∇Pk²_Lq

dt

+

Z _T0

0

tk∇_utk²_L2 +tk∇_btk²_L2 +tk∇²_bx¯^a2k²_L2dt≤ M, (3.3) where

E0,k√

ρ0u0k_L2 +k∇u0k_L2 +kθ0k_H1∩W^1,q+k∇b0k_L2 +kx¯^a2b0k_L2 +kx¯^aρ0k_L1∩H¹∩W^1,q. To show Proposition3.1, whose proof will be postponed to the end of this subsection, we begin with the following standard energy estimate for (ρ,u,P,θ,b) and the estimate on the L^p-norm of the density.

Lemma 3.2. Under the conditions of Proposition 3.1, let (ρ,u,P,θ,b) be a smooth solution to the initial-boundary-value problem(1.1)and(2.2). Then for any t ∈(0,T₁],

sup

0≤s≤t

kρk_L1∩L^∞+kθk_L2∩L^∞+k√

ρuk²_L2+kbk²_L2+

Z _t

0

k∇uk²_L2 +k∇bk²_L2ds≤ C, (3.4) where (and in what follows) C denotes a generic positive constant depending only onµ,ν,q,a, N₀,η₀ and E0. T₁is as that of Lemma3.3.

Proof. 1. Since divu=0, we deduce from (1.1)₁ that

ρt+u· ∇ρ=0. (3.5)

Define particle path

(d

dtX(x,t) =u(X(x,t),t), X(x, 0) =x.

Thus, along particle path, we obtain from (3.5) that d

dtρ(X(x,t),t) =0, which implies

ρ(X(x,t),t) =ρ₀. (3.6)

Similarly, one derives from (1.1)₃that

θ(X(x,t),t) =θ0. (3.7)

2. Multiplying (1.1)₂byuand then integrating the resulting equation overB_R, we have 1

2 d dt

Z

ρ|u|²dx+µ Z

|∇u|²dx=

Z

b· ∇b·udx+

Z

ρθe₂·udx. (3.8) Multiplying (1.1)₄byband integrating by parts, we arrive at

1 2

d dt

Z

|b|²dx+ν Z

|∇b|²dx+

Z

b· ∇b·udx=0, which combined with (3.8) and (3.7) implies that

1 2

d dt k√

ρuk²_L2 +kbk²_L2

+ µk∇uk²_L2+νk∇bk²_L2

=

Z

ρθu·e₂dx

≤ kρk_L¹²_∞k√

ρuk_L2kθk_L2

≤Ck√

ρuk²_L2 +C. (3.9) Thus, Gronwall’s inequality leads to

sup

0≤s≤t

k√

ρuk²_L2+kbk²_L2

+

Z _t

0

k∇uk²_L2+k∇bk²_L2

ds≤C,

which together with (3.6) and (3.7) yields (3.4) and completes the proof of Lemma3.2.

Next, we will give some spatial weighted estimates on the density and the magnetic.

Lemma 3.3. Under the conditions of Proposition 3.1, let (ρ,u,P,θ,b) be a smooth solution to the initial-boundary-value problem(1.1)and(2.2). Then there exists a T₁ =T₁(N0,E0)> 0such that for all t∈(0,T₁],

sup

0≤s≤t

kρx¯^ak_L1+kbx¯^2ak²_L2+

Z _t

0

k∇bx¯^2ak²_L2ds≤C. (3.10) Proof. 1. ForN>1, let ϕ_N ∈C^∞₀ (B_N)satisfy

0≤ ϕ_N ≤1, ϕ_N(x) =1, if |x| ≤ ^N

2, |∇ϕ_N| ≤CN⁻¹. (3.11) It follows from (1.1)₁and (3.4) that

d dt

Z

ρϕ_2N0dx =

Z

ρu· ∇ϕ_2N0dx

≥ −CN₀⁻¹ _Z

ρdx ¹_{2 Z}

ρ|_u|²dx ¹₂

≥ −C^˜(E₀). (3.12) Integrating (3.12) and using (3.1) give rise to

0≤inft≤T1

Z

B2N0

ρdx≥ inf

0≤t≤T1

Z

ρϕ_2N0dx≥

Z

ρ₀ϕ_2N0dx−CT^˜ ₁ ≥ ¹

4. (3.13)

Here, T₁ ,^min{1,(4 ˜C)⁻¹}. From now on, we will always assume that t ≤ T₁. The combina- tion of (3.13), (3.4), and (2.6) implies that forε>_{0 and}η>_{0, every}v∈D^˜1,2(B_R)_satisfies

kvx¯^−ηk²

L

2+ε

˜ η

≤ C(ε,η)k√

ρvk²_L2+C(ε,η)k∇vk²_L2, (3.14) with ˜η=min{1,η}.

2. Noting that

|∇x¯| ≤(3+2η₀)log^1+η0(3+|x|²)≤ C(a,η₀)x¯^8+4a, multiplying (1.1)₁by ¯x^a and integrating by parts imply that

d

dtkρx¯^ak_L1 =

Z

ρ(_u· ∇)xa¯ x¯^a−1dx

≤C Z

ρ|u|x¯^a−1+8+4adx

≤Ckρx¯^a−1+8+8ak

L^87++aakux¯^−8+4ak_L8+a

≤Ckρk_L⁸⁺_∞^1akρx¯^ak⁷

+a 8+a

L¹ (k√

ρuk_L2+k∇uk_L2)

≤C(1+kρx¯^ak_L1) 1+k∇uk²_L2

due to (3.4) and (3.14). This combined with Gronwall’s inequality and (3.4) leads to sup

0≤s≤t

kρx¯^ak_L1 ≤ Cexp

C Z _t

0 1+k∇uk²_L2

ds

≤C. (3.15)

3. Multiplying (1.1)₃ bybx¯^a and integrating by parts yield 1

2 d

dtkbx¯^a/2k²_L2 +νk∇bx¯^a/2k²_L2 = ^ν 2

Z

|b|²_∆x¯^adx+

Z

b· ∇u·bx¯^adx+¹ 2

Z

|b|²u· ∇x¯^adx

,^I¯1+I^¯₂+I^¯₃, (3.16)

where

|I^¯₁| ≤C Z

|b|²x¯^ax¯⁻²log^2(1−η0)(₃+|x|²)dx≤C Z

|b|²x¯^adx,

|I^¯2| ≤Ck∇uk_L2kbx¯^2ak²_L4

≤Ck∇uk_L2kbx¯^2ak_L2(k∇bx¯^2ak_L2+kb∇x¯^a2k_L2)

≤C(k∇uk²_L2 +1)kbx¯^a2k²_L2+ ^ν

4k∇bx¯^a2k²_L2,

|I^¯₃| ≤Ckbx¯^2ak_L4kbx¯^a2k_L2kux¯⁻³⁴k_L4

≤Ckbx¯^2ak²_L4+Ckbx¯^a2k²_L2 k√

ρuk²_L2+k∇uk²_L2

≤C 1+k∇uk²_L2

kbx¯^a2k²_L2 +^ν

4k∇bx¯^a2k²_L2, (3.17) due to Gagliardo–Nirenberg inequality, (3.4), and (3.14). Putting (3.17) into (3.16), we get after using Gronwall’s inequality and (3.4) that

sup

0≤s≤t

k_bx¯^a2k²_L2+

Z _t

0

k∇_bx¯^a2k²_L2ds≤Cexp

C Z _t

0 1+k∇_uk²_L2ds

≤ C, (3.18) which together with (3.15) gives (3.10) and finishes the proof of Lemma3.3.

Lemma 3.4. Let T₁be as in Lemma 3.3. Then there exists a positive constantα> 1such that for all t∈ (0,T₁],

sup

0≤s≤t

k∇uk²_L2+k∇bk²_L2

+

Z _t

0

k√

ρusk²_L2 +k∇²uk²_L2 +kbsk²_L2 +k∇²bk²_L2

ds

≤C+C Z _t

0 ψ^α(s)ds. (3.19)

Proof. 1. It follows from (3.4), (3.10), and (3.14) that for anyε>0 and any η>0, kρ^ηvk

L² +ε

˜ η

≤Ckρ^ηx¯^{4(3 ˜2ηa+ε)}k

L

4(2+ε) 3 ˜η

kvx¯^{−4(3 ˜2ηa+ε)}k

L

4(2+ε)

˜ η

≤C _Z

ρ

4(2+ε)η 3 ˜η −1

ρx¯^adx ₄₍₂^{3 ˜}₊^η

ε)

kvx¯^{−4(3 ˜2ηa+ε)}k

L

4(2+ε) η˜

≤Ckρk

4(2+ε)η−3 ˜η 4(2+ε)

L^∞ kρx¯^ak

3 ˜η 4(2+ε) L¹ (k√

ρvk_L2 +k∇vk_L2)

≤Ck√

ρvk_L2+Ck∇vk_L2, (3.20)

where ˜η=min{1,η}andv∈ D^˜1,2(B_R). In particular, this together with (3.4) and (3.14) yields kρ^ηuk

L² +ε

˜ η

+kux¯^−ηk

L² +ε

˜ η

≤C(1+k∇uk_L2), (3.21) kρ^ηθk

L² +ε η˜

+kθx¯^−ηk

L² +ε η˜

≤C(1+k∇θk_L2). (3.22) 2. Multiplying (1.1)₂byu_t and integrating by parts, one has

µd dt

Z

|∇u|²dx+

Z

ρ|u_t|²dx≤C Z

ρ|u|²|∇u|²dx+

Z

b· ∇b·u_tdx+

Z

ρθ|u_t|dx. (3.23)

We derive from (3.21), Hölder’s inequality, and Gagliardo–Nirenberg inequality that Z

ρ|u|²|∇u|²dx≤Ck√

ρuk²_L8k∇uk²

L⁸³

≤Ck√

ρuk²_L8k∇uk_L³²₂k∇uk_H¹²₁

≤Cψ^α+εk∇²uk²_L2, (3.24)

where (and in what follows) we useα>1 to denote a genetic constant, which may be different from line to line. For the second term on the right-hand side of (3.23), integration by parts together with (1.1)₅and Gagliardo–Nirenberg inequality indicates that for any ε>0,

Z

b· ∇b·utdx=− ^d dt

Z

b· ∇u·bdx+

Z

bt· ∇u·bdx+

Z

b· ∇u·btdx

≤ − ^d dt

Z

b· ∇_u·bdx+ ^ν

−1

2 k_btk²_L2 +Ck_bk²_L4k∇_uk²_L4

≤ − ^d dt

Z

b· ∇u·bdx+ ^ν

−1

2 kb_tk²_L2 +Ckbk_L2k∇bk_L2k∇uk_L2k∇uk_H1

≤ − ^d dt

Z

b· ∇u·bdx+ ^ν

−1

2 kbtk²_L2 +εk∇²uk²_L2+Cψ^α. (3.25) From Cauchy–Schwarz inequality and (3.4), we have

Z

ρθ|_ut|dx≤ ¹ 2k√

ρutk²_L2+ ¹

2kρk_L∞kθk²_L2 ≤ ¹ 2

Z

ρ|_ut|²dx+C. (3.26) Thus, inserting (3.24)–(3.26) into (3.23) gives

d

dtB(t) + ¹ 2k√

ρu_tk²_L2 ≤εk∇²uk²_L2+ ^ν

−1

2 kb_tk²_L2 +Cψ^α, (3.27) where

B(t),^µk∇uk²_L2+

Z

b· ∇u·bdx satisfies

µ

2k∇_uk²_L2−C₁k∇_bk²_L2 ≤ B(t)≤Ck∇_uk²_L2 +Ck∇_bk²_L2, (3.28) owing to Hölder’s inequality, Gagliardo–Nirenberg inequality, and (3.4).

3. It follows from (1.1)₃that ν d

dtk∇bk²_L2+kb_tk²_L2+ν²k_∆bk²_L2

≤Ck|b||∇u|k²_L2 +Ck|u||∇b|k²_L2

≤Ck_bk_L2k∇²_bk_L2k∇_uk²_L2+Ckx¯^−a4uk²_L8kx¯^2a∇_bk_L2k∇_bk_L4

≤ ^ν

2

2 k_∆bk²_L2+Cψ^α+Ckx¯^a2∇bk²_L2 (3.29) due to (2.7), (3.21), and Gagliardo–Nirenberg inequality. Multiplying (3.29) by ν⁻¹(C₁+1) and adding the resulting inequality to (3.27) imply

d

dt B(t) + (C₁+1)k∇bk²_L2

+¹ 2k√

ρu_tk²_L2+ ^ν

−1

2 kb_tk²_L2+ ^ν

2k_∆bk²_L2

≤Cψ^α+Ckx¯^2a∇bk²_L2+εk∇²uk²_L2. (3.30)

Since(ρ,u,P,θ,b)satisfies the following Stokes system











−_µ∆u+∇P=−ρu_t−ρu· ∇u+b· ∇b+ρθe₂, x∈ B_R,

divu=0, x∈ BR,

u(x) =0, x∈ ∂B_R,

(3.31)

applying regularity theory of Stokes system to (3.31) (see [18]) yields that for any p ∈[2,∞), k∇²uk_L^p+k∇Pk_L^p ≤Ckρu_tk_L^p +Ckρu· ∇uk_L^p +Ck|b||∇b|k_L^p +Ckρθk_L^p. (3.32) Hence, we infer from (3.32), (3.4), (3.21), and Gagliardo–Nirenberg inequality that

k∇²uk²_L2 +k∇Pk²_L2

≤Ckρu_tk²_L2+Ckρu· ∇uk²_L2+Ck|b||∇b|k²_L2 +Ckρθk²_L2

≤Ckρk_L∞k√

ρu_tk²_L2 +Ckρuk²_L4k∇uk²_L4+Ckbk²_L4k∇bk²_L4 +Ckρk²_L∞kθk²_L2

≤Ck√

ρutk²_L2 +Ckρuk²_L4k∇uk_L2k∇uk_H1 +Ckbk_L2k∇bk²_L2k∇bk_H1+C

≤Ck√

ρutk²_L2 +¹

4k∇²bk²_L2+ ¹

2k∇²uk²_L2+C

1+k∇bk⁴_L2 +k∇uk⁶_L2

≤Ck√

ρutk²_L2 +¹

4k∇²_bk²_L2+ ¹

2k∇²_uk²_L2+Cψ^α. (3.33)

Substituting (3.33) into (3.30) and choosingεsuitably small, one gets d

dt B(t) + (C₁+1)k∇bk²_L2+¹ 4k√

ρu_tk²_L2+ ^ν

−1

2 kb_tk²_L2 +^ν

4k_∆bk²_L2 ≤Cψ^α+Ckx¯^2a∇bk²_L2. Integrating the above inequality over(0,t), then we obtain (3.19) from (2.7), (3.28), (3.10), and (3.33). The proof of Lemma3.4 is finished.

Lemma 3.5. Let T₁be as in Lemma 3.3. Then there exists a positive constantα> 1such that for all t∈ (0,T₁],

sup

0≤s≤t

s k√

ρusk²_L2+k_bsk²_L2+

Z _t

0

s k∇_usk²_L2 +k∇_bsk²_L2ds≤Cexp

C Z _t

0

ψ^αds

. (3.34) Proof. 1. Differentiating (1.1)₂with respect to tgives

ρutt+ρu· ∇ut−_µ∆ut =−ρt(ut+u· ∇u)−ρut· ∇u− ∇Pt+ (b· ∇b)_t+ (ρθe2)_t. (3.35) Multiplying (3.35) by ut and integrating the resulting equality by parts over B_R, we obtain after using (1.1)₁and (1.1)₅ that

1 2

d dt

Z

ρ|u_t|²dx+µ Z

|∇u_t|²dx

≤ C Z

ρ|u||ut| |∇ut|+|∇u|²+|u||∇²u|dx+C Z

ρ|u|²|∇u||∇ut|dx +C

Z

ρ|ut|²|∇u|dx+

Z

bt· ∇b·utdx+

Z

b· ∇bt·utdx

+

Z

ρ_tθe₂·u_tdx+

Z

ρθ_te₂·u_tdx,

∑

Iˆ_i. (3.36)

It follows from (3.20), (3.21), and Gagliardo–Nirenberg inequality that Iˆ₁≤Ck√

ρuk_L6k√

ρutk_L¹²₂k√

ρutk_L¹²₆ k∇utk_L2+k∇uk²_L4

+Ckρ

1

4uk²_L12k√ ρu_tk¹²

L²k√ ρu_tk¹²

L⁶k∇²uk_L2

≤C(₁+k∇_uk²_L2)k√ ρutk¹²

L²(k√

ρutk_L2 +k∇_utk_L2)¹²

× k∇u_tk_L2 +k∇uk²_L2+k∇uk_L2k∇²uk_L2+k∇²uk_L2

≤ ^µ

8k∇_utk²_L2+Cψ^αk√

ρutk²_L2+Cψ^α+C 1+k∇_uk²_L2k∇²_uk²_L2. (3.37) Hölder’s inequality combined with (3.20) and (3.21) leads to

Iˆ₂+I^ˆ₃≤ Ck√

ρuk²_L8k∇uk_L4k∇u_tk_L2+Ck∇uk_L2k√ ρu_tk³²

L⁶k√ ρu_tk¹²

L²

≤ ^µ

8k∇utk²_L2 +Cψ^αk√

ρutk²_L2+C ψ^α+k∇²uk²_L2

. (3.38)

Integration by parts together with (1.1)₅, Hölder’s and Gagliardo–Nirenberg inequalities indi- cates that

Iˆ₄+I^ˆ5 =−

Z

bt· ∇ut·bdx−

Z

b· ∇ut·btdx

≤ ^µ

8k∇utk²_L2+Ckbk²_L4kbtk²_L4

≤ ^µ

8k∇u_tk²_L2+ ^µν

4(C₂+₁)k∇b_tk²_L2 +Cψ^αkb_tk²_L2. (3.39) Integration by parts together with (1.1)₁, (1.1)5, Hölder’s inequality, Gagliardo–Nirenberg in- equality, and (3.7) indicates that

Iˆ₆=

Z

ρu· ∇(θe₂·u_t)dx

≤

Z

ρ|u||∇θ||u_t|dx+

Z

ρ|u|θ|∇u_t|dx

≤ k√

ρu_tk_L2k√ ρuk

L

2q q−2

k∇θk_L^q+k∇u_tk_L2kρuk_L4kθk_L4

≤ ^µ

6k∇u_tk²_L2+Cψ^αk√

ρu_tk²_L2 +Cψ^α. (3.40)

We get from Hölder’s inequality, (3.4), and (3.21) that Iˆ7≤

Z

ρ|u||∇θ||ut|dx

≤ k√

ρutk_L2k√ ρuk

L

2q

q−2k∇θk_L^q

≤ Cψ^αk√

ρutk²_L2+Cψ^α. (3.41)

Substituting (3.37)–(3.41) into (3.36), we obtain after using (3.33) that d

dtk√

ρutk²_L2 +µk∇_utk²_L2 ≤Cψ^α 1+k√

ρutk²_L2+k_btk²_L2 + ^µν

2(C₂+1)k∇btk²_L2+C 1+k∇uk²_L2

k∇²bk²_L2. (3.42) 2. Differentiating (1.1)₃with respect to tshows

b_tt−b_t· ∇u−b· ∇u_t+u_t· ∇b+u· ∇b_t =_ν∆bt. (3.43)