Serrin-type blowup criterion of three-dimensional nonhomogeneous heat conducting

Serrin-type blowup criterion of three-dimensional nonhomogeneous heat conducting

magnetohydrodynamic flows with vacuum

Ling Zhou

Liangjiang Primary School, Yubei District, Chongqing 401120, People’s Republic of China Received 4 September 2019, appeared 4 November 2019

Communicated by Maria Alessandra Ragusa

Abstract.We consider an initial boundary value problem for the nonhomogeneous heat conducting magnetohydrodynamic flows. We show that for the initial density allowing vacuum, the strong solution exists globally if the velocity field satisfies Serrin’s condi- tion. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equations.

Keywords: heat conducting magnetohydrodynamic flows, Serrin-type criterion, vac- uum.

2010 Mathematics Subject Classification: 35Q35, 35B65, 76N10.

1 Introduction

Magnetohydrodynamics studies the dynamics of electrically conducting fluids and the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. Due to the profound physical background and important mathematical significance, a great deal of attention has been focused on studying well-posedness of solutions to the MHD system, both from a pure mathematical point of view and for concrete applications. For more background, we refer to [6] and references therein.

Let Ω⊂ _R³ be a bounded smooth domain, the motion of a viscous, incompressible, and heat conducting magnetohydrodynamic fluid in Ωcan be described by the following MHD system



















∂tρ+div(ρu) =0,

∂_t(_ρu) +_div(_ρu⊗_u)−µ∆u+∇P=_b· ∇_b,

c_v[∂_t(ρθ) +div(ρuθ)]−_κ∆θ =2µ|D(u)|²+ν|curlb|²,

∂tb−b· ∇u+u· ∇b=_ν∆b, divu=0, divb=0

(1.1)

BEmail: lzhou1204@sina.com

with the initial condition

(_ρ,u,θ,b)(_0,x) = (ρ₀,u0,θ₀,b0)(x)_, x ∈_Ω, (1.2) and the boundary condition

u=0, ∂θ

∂n =0, b=0, on∂Ω, (1.3)

where n is the unit outward normal to ∂Ω. Here ρ,u,θ,P,b are the fluid density, velocity, absolute temperature, pressure, and the magnetic field, respectively. D(u)denotes the defor- mation tensor given by

D(u) = ¹

2(∇u+ (∇u)^tr).

The constantµ>0 is the viscosity coefficient. Positive constantsc_v andκ are respectively the heat capacity, the ratio of the heat conductivity coefficient over the heat capacity, andν>_{0 is} the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field.

When b = 0, the system (1.1) reduces to the nonhomogeneous heat conducting Navier–

Stokes equations and there are a lot of results on the existence in the literature. For the initial density containing vacuum states, Lions [14, Chapter 2] established the global existence of weak solutions in any space dimensions. Later on, Cho–Kim [5] proposed a compati- bility condition on the initial data and investigated the local existence of strong solutions.

By delicate energy estimates, Zhong [19] showed the global existence of strong solutions on three-dimensional bounded domains under some smallness assumption. There are also very interesting investigations about the existence of strong solutions to the three-dimensional non- homogeneous heat conducting Navier–Stokes equations, please refer to [15,17,18,21].

Recently, the local and global existence of strong solutions to the multi-dimensional vis- cous heat conducting magnetohydrodynamic flows with non-negative density were estab- lished. Inspired by [5], Wu [16] proved the local existence of strong solutions. By using the techniques in [19], the author [20] studied the global strong solutions for small initial data. At the same time, he also obtained a blowup criterion of strong solutions. By a critical Sobolev inequality of logarithmic type, Fan–Li–Nakamura [7] showed the global strong solutions with no restrictions on the initial data in two-dimensional bounded domains. Very recently, Zhu–

Ou [22] obtained the global existence and algebraic decay of strong solutions to the non- homogeneous heat-conducting magnetohydrodynamic equations with density-temperature- dependent viscosity and resistivity coefficients. At the same time, many authors studied blowup criteria and regularity criteria of incompressible magnetohydrodynamic equations and related system, please refer to [1,2,4,10–12]. In this paper, motivated by [19], we aim at giving a Serrin-type blowup criterion of strong solutions of the system (1.1).

Before stating our main results, we first explain the notations and conventions used throughout this paper. We use the notation

Z

·dx=

Z

Ω·dx.

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

(L^p = L^p(_Ω), W^k,p =W^k,p(_Ω), H^k = H^k,2(_Ω),

H₀¹={u∈ H¹|u=_{0 on}∂Ω}_, H_n² ={u∈ H²| ∇u·_n=_{0 on}∂Ω}_. Now we define precisely what we mean by strong solutions to the problem (1.1)–(1.3).

Definition 1.1. (ρ,u,θ,b) is called a strong solution to (1.1)–(1.3) in Ω×(0,T), if for some q0 >3,















ρ≥0, ρ∈ C([0,T];W^1,q0), ρ_t ∈C([0,T];L^q0), (u,b)∈ C([0,T];H₀¹∩H²)∩L²(0,T;W^2,q0), θ ≥0, θ∈ C([0,T];H²)∩L²(0,T;W^2,q0), (_bt,ut,θ_t)∈ L²(0,T;H¹), (_bt,√

ρut,√

ρθ_t)∈ L^∞(0,T;L²). Furthermore, both (1.1) and (1.2) hold almost everywhere inΩ×(0,T).

Our main results read as follows.

Theorem 1.2. For constant q ∈(3, 6], assume that the initial data(ρ₀≥0,u0,θ₀ ≥0,b0)satisfy ρ₀ ∈W^1,q(_Ω), (u₀,b₀)∈ H₀¹(_Ω)∩H²(_Ω), θ₀∈ H_n²(_Ω), divu₀=divb₀ =0, (1.4) and the compatibility conditions

(−_µ∆u0+∇P₀−b₀· ∇b₀= √ ρ₀g₁, κ∆θ₀+_2µ|D(_u0)|²+ν|curlb0|² =√

ρ₀g2, (1.5)

for some P₀ ∈ H¹(_Ω)andg1,g2 ∈ L²(_Ω). Let (_ρ,u,θ,b)be a strong solution to the problem(1.1)–

(1.3). If T^∗ <_∞is the maximal time of existence for that solution, then we have

Tlim→T^∗kuk_Ls(0,T;L^r)= _∞, (1.6)

where r and s satisfy

2 s +³

r ≤1, s>1, 3<r ≤_∞. (1.7)

Remark 1.3. The local existence of a strong solution with initial data as in Theorem 1.2 has been established in [16]. Hence, the maximal timeT^∗ is well-defined.

Remark 1.4. The conclusion in Theorem 1.2 is somewhat surprising since the criterion (1.6) is independent of the magnetic field. The result indicates that the magnetic field acts no significant roles on the mechanism of blowup of nonhomogeneous heat conducting magne- tohydrodynamic flows. Thus we generalize [19, Theorem 1.1] to the heat conducting MHD flows.

Remark 1.5. Due to the Sobolev inequality kuk_L6 ≤ Ck∇uk_L2, we thus improve the blowup criterion obtained in [20, Theorem 1.1].

We will prove Theorem1.2by contradiction in Section3. In fact, the proof of the theorem is based on a priori estimates under the assumption that kuk_Ls(0,T;L^r)is bounded independent of any T ∈ (0,T^∗). The a priori estimates are then sufficient for us to apply the local exis- tence result repeatedly to extend a local solution beyond the maximal time of existence T^∗, consequently, contradicting the maximality of T^∗.

The rest of this paper is organized as follows. In Section 2, we collect some elementary facts and inequalities that will be used later. Section3is devoted to the proof of Theorem1.2.

2 Preliminaries

In this section, we will recall some known facts and elementary inequalities that will be used frequently later.

First, the following Gagliardo–Nirenberg inequality (see [9]) will be useful in the next section.

Lemma 2.1 (Gagliardo–Nirenberg). LetΩ ⊂ _R³ be a bounded smooth domain. Assume that1 ≤ q,r ≤ _∞, and j,m are arbitrary integers satisfying 0 ≤ j < m. If v ∈ W^m,r(_Ω)∩L^q(_Ω), then we have

kD^jvk_L^p ≤Ckvk¹_L⁻q^akvk^a_Wm,r, where

−j+ ³

p = (1−a)³ q+a

−m+ ³ r

, and

a∈ ( _j

m, 1

, if m−j− ³_r is an nonnegative integer, _j

m, 1

, otherwise.

The constant C depends only on m,j,q,r,a, andΩ.

Next, we give some regularity results for the following Stokes system











−µ∆U+∇P=_F, x ∈_Ω, divU=0, x∈_Ω,

U=0, x∈ _∂Ω.

(2.1)

Lemma 2.2. Let m ≥ 2 be an integer, r any real number with 1 < r < _∞and let Ωbe a bounded domain ofR³of class C^m−1,1. LetF∈W^m−2,r(_Ω)be given. Then the Stokes system(2.1)has a unique solutionU∈W^m,r(_Ω)and P∈W^m−1,r(_Ω)/R. In addition, there exists a constant C>0depending only on m,r,andΩsuch that

kUk_Wm,r +kPk_Wm−1,r/R≤ CkFk_Wm−2,r. Proof. See [3, Theorem 4.8].

3 Proof of Theorem 1.2

Let (_ρ,u,θ,b) be a strong solution described in Theorem 1.2. Suppose that (1.6) were false, that is, there exists a constant M₀ >0 such that

Tlim→T^∗kuk_Ls(0,T;L^r)≤ M₀<_∞. (3.1) Rewrite the system (1.1) as



















ρ_t+_u· ∇ρ=0,

ρu_t+ρu· ∇u−_µ∆u+∇P=b· ∇b,

cv[ρθ_t+ρu· ∇θ]−κ∆θ =2µ|D(u)|²+ν|curlb|², b_t−b· ∇u+u· ∇b−_ν∆b=0,

divu=0, divb=0.

(3.2)

In what follows,Cstands for a generic positive constant which may depend onM₀,µ,c_v,κ,ν,T^∗, and the initial data.

First of all, we have the following basic energy estimates and the upper bound of the density.

Lemma 3.1. It holds that for any T∈(0,T^∗), sup

0≤t≤T

kρk_L∞+c_vkρθk_L1 +k√

ρuk²_L2+kbk²_L2+

Z _T

0 µk∇uk²_L2+νk∇bk²_L2dt

≤ kρ₀k_L∞+c_vkρ₀θ₀k_L1+k√

ρ₀u₀k²_L2+kb₀k²_L2. (3.3) Proof. Since divu=0, we then derive from [14, Theorem 2.1] that for anyt ∈(0,T^∗),

kρ(t)k_L∞ = kρ₀k_L∞. (3.4) Applying standard maximum principle to (3.2)₃along withθ₀ ≥0 shows (see [8, p. 43])

Ω×[inf0,T]θ ≥0. (3.5)

Multiplying (3.2)2byuand integrating (by parts) over Ω, we derive that 1

2 d dt

Z

ρ|u|²dx+µ Z

|∇u|²dx=

Z

b· ∇b·udx. (3.6)

Multiplying (3.2)₄byband integrating (by parts) over Ω, we get after using (1.3) that 1

2 d dt

Z

|b|²dx+ν Z

|∇b|²dx=

Z

b· ∇u·bdx−

Z

u· ∇b·bdx. (3.7) Due to divb=0 andu|_∂Ω=0, we have

Z

b· ∇b·udx=

Z

bⁱ∂_ib^ju^jdx=−

Z

b· ∇u·bdx. (3.8)

Similarly, one obtains

−

Z

u· ∇b·bdx =−

Z

uⁱ∂_ib^jb^jdx=

Z

u· ∇b·bdx,

and thus _Z

u· ∇b·bdx =0. (3.9)

Combining (3.6)–(3.9), we deduce that 1

2 d dt

Z

ρ|u|²+|b|²dx+

Z

µ|∇u|²+ν|∇b|²dx=0. (3.10) Integrating (3.2)₃ with respect to the spatial variable gives rise to

c_v d dt

Z

ρθdx−2µ Z

|D(u)|²dx−ν Z

|∇b|²dx=0. (3.11) Substituting (3.11) into (3.10) and noting that

−2µ Z

|D(u)|²dx= −^µ 2 Z

(∂_iu^j+∂_juⁱ)²dx

= −µ Z

|∇u|²dx−µ Z

∂_iu^j∂_juⁱdx

= −µ Z

|∇_u|²dx,

we derive that

d dt

Z

cvρθ+ ¹

2ρ|u|²+ ¹ 2|b|²

dx=0, (3.12)

which combined with (3.10) leads to d

dt Z

cvρθ+ρ|u|²+|b|²dx+

Z

µ|∇u|²+ν|∇b|²dx=0.

Integrating the above equality over(0,T)yields sup

0≤t≤T

c_vkρθk_L1+k√

ρuk²_L2 +k_bk²_L2+

Z _T

0

µk∇_uk²_L2+νk∇_bk²_L2dt

≤c_vkρ₀θ₀k_L1+k√

ρ₀u₀k²_L2+kb₀k²_L2.

This along with (3.4) implies the desired (3.3) and consequently completes the proof.

The following lemma was deduced in [13], we sketch it here for completeness.

Lemma 3.2. Under the condition(3.1), it holds that for p∈[2, 12]and T∈[0,T^∗), sup

0≤t≤T

kbk_L^p +

Z _T

0

Z

|b|^p−2|∇b|²dxdt≤C. (3.13) Proof. Multiplying (3.2)₄byp|b|^p−2band integrating the resulting equation overΩ, we deduce

d dt

Z

|b|^pdx+νp(p−1)

Z

|b|^p−2|∇b|²dx ≤ p Z

(b· ∇u−u· ∇b)· |b|^p−2bdx. (3.14) Integration by parts together with (3.2)₅ yields

−p Z

(_u· ∇)_b· |_b|^p−2bdx=

Z

divu|_b|^pdx=0. (3.15) We derive from integration by parts, Hölder’s inequality, and Gagliardo–Nirenberg inequality that forr andssatisfying (1.7),

p Z

(b· ∇)u· |b|^p−2bdx

≤ ^νp(p−1) 4

Z

|b|^p−2|∇b|²dx+C(ν,p)

Z

|u|²|b|^pdx

≤ ^νp(p−1) 4

Z

|b|^p−2|∇b|²dx+Ckuk²_Lrk|b|^p2k²

(r−3) r

L² k|b|^p2k^6r

L⁶

≤ ^νp(p−1) 4

Z

|b|^p−2|∇b|²dx+δk∇|b|^2pk²_L2+C(δ)(1+kuk^s_Lr)kbk_L^pp. (3.16) Substituting (3.15) and (3.16) into (3.14) and choosingδsuitably small give that

d dt

Z

|b|^pdx+^νp(p−1) 2

Z

|b|^p−2|∇b|²dx≤C(1+kuk^s_Lr)

Z

|b|^pdx.

We thus obtain (3.13) directly after using Gronwall’s inequality and (3.1). This finishes the proof of Lemma3.2.

Next, the following lemma concerns the key time-independent estimates on the L^∞(_0,T;L²)-norm of the gradients of the velocity and the magnetic field.

Lemma 3.3. Under the condition(3.1), it holds that for any T ∈(0,T^∗), sup

0≤t≤T

k∇uk²_L2 +k∇bk²_L2+

Z _T

0

k√

ρu_tk²_L2+kb_tk²_L2 +kuk²_H2+k∇²bk²_L2dt≤C. (3.17) Proof. Multiplying (3.2)2byut and integrating the resulting equations overΩ, we derive from Cauchy–Schwarz inequality that

µ 2

d dt

Z

|∇u|²dx+

Z

ρ|u_t|²dx

=

Z

b· ∇b·utdx−

Z

ρu· ∇u·utdx

=−^d dt

Z

b· ∇u·bdx+

Z

[b_t· ∇u·b+b· ∇u·b_t−ρu· ∇u·u_t]dx

≤ −^d dt

Z

b· ∇u·bdx+¹ 2

Z

ρ|u_t|²+|b_t|²dx+

Z

2ρ|u|²|∇u|²+8|b|²|∇u|²dx, and thus

d dt

Z

µ|∇u|²+2b· ∇u·b dx+

Z

ρ|ut|²dx

≤

Z

|b_t|²dx+

Z

4ρ|u|²|∇u|²+16|b|²|∇u|²dx. (3.18) Multiplying (3.2)₄bybtand integrating by parts yield

νd dt

Z

|∇b|²dx+2 Z

|b_t|²dx=2 Z

(b· ∇u−u· ∇b)·b_tdx

≤ ¹ 2

Z

|bt|²dx+8 Z

|b|²|∇u|²+|u|²|∇b|²dx, (3.19) which combined with (3.18) implies

d dt

Z

µ|∇u|²+ν|∇b|²+2b· ∇u·b dx+

Z

ρ|u_t|²+|b_t|²dx

≤ C Z

ρ|u|²|∇u|²+|b|²|∇u|²+|u|²|∇b|²dx. (3.20) Recall that(u,P)satisfies the following Stokes system











−µ∆u+∇P=−ρut−ρu· ∇u+_b· ∇b, x∈_Ω,

divu=0, x∈_Ω,

u=0, x∈_∂Ω.

Applying Lemma2.2 withF,−_ρut−_ρu· ∇_u+_b· ∇b, we obtain from (3.4) that kuk²_H2 ≤C kρutk²_L2+kρu· ∇uk²_L2+kb· ∇bk²_L2

≤C k√

ρutk²_L2 +k√

ρu· ∇_uk²_L2+k_b· ∇_bk²_{L2. (3.21)} It follows from the standard L²-estimates of elliptic system and (3.2)₄ that

k∇²bk²_L2 ≤C kb_tk²_L2+ku· ∇bk²_L2+kb· ∇uk²_L2, (3.22)

which together with (3.21) leads to for some K>0, kuk²_H2+k∇²bk²_L2 ≤K k√

ρu_tk²_L2 +kb_tk²_L2+C k√

ρu· ∇uk²_L2+kb· ∇bk²_L2 +C ku· ∇bk²_L2 +kb· ∇uk²_L2

. (3.23)

Adding (3.23) multiplied by _2K¹ to (3.20), we get from Hölder’s inequality, (3.13), and the Gagliardo–Nirenberg inequality that

A⁰(t) +¹ 2 k√

ρutk²_L2 +k_btk²_L2+ ¹

2K k_uk²_H2+k∇²_bk²_L2

≤C Z

ρ|u|²|∇u|²+|b|²|∇u|²+|u|²|∇b|²+|b|²|∇b|²dx

≤Ckρk_L∞k_uk²_Lrk∇_uk²

L^r2r−2

+Ck_bk²_L6k∇_uk_L2k∇_uk_L6

+Ck_uk²_Lrk∇_bk²

L^r2r−2

+Ck_bk²_L6k∇_bk_L2k∇_bk_L6

≤Ckuk²_Lrk∇uk^2−6r

L² k∇uk^6r

H¹ +Ck∇uk_L2k∇uk_H1

+Ckuk²_Lrk∇bk²_L⁻₂ ^6rk∇bk_H^6r₁ +Ck∇bk_L2k∇bk_H1

≤ ¹

2 kuk²_H2+k∇²bk²_L2

+C 1+kuk^s_Lr+k∇uk²_L2+k∇bk²_L2

k∇uk²_L2+k∇bk²_L2

, and thus

A⁰(t) + ¹ 2 k√

ρu_tk²_L2+kb_tk²_L2

+ ¹

2K kuk²_H2 +k∇²bk²_L2

≤ 1+kuk^s_Lr +k∇uk²_L2 +k∇bk²_L2 k∇uk²_L2 +k∇bk²_L2. (3.24) Here

A(t),

Z

µ|∇_u|²+ν|∇_b|²+_2b· ∇_u·_bdx satisfies

µ

2k∇uk²_L2+νk∇bk²_L2−C≤ A(t)≤ ^3µ

2 k∇uk²_L2 +νk∇bk²_L2+C (3.25)

due to

Z

2b· ∇u·bdx

≤ ^µ

2k∇uk²_L2 + ⁸

µkbk⁴_L4 ≤ ^µ

2k∇uk²_L2+C.

Consequently, the desired (3.17) follows from (3.24), Gronwall’s inequality, (3.25), (3.3), and (3.1). This completes the proof of Lemma3.3.

Finally, the following lemma will deal with the higher order estimates of the solutions which are needed to guarantee the extension of the local strong solution to be a global one.

Lemma 3.4. Under the condition(3.1), it holds that for any T∈ (0,T^∗), sup

0≤t≤T

kρk_W1,q+kuk²_H2+kθk²_H2 +kbk²_H2

≤C. (3.26)

Proof. Differentiating (3.2)₂with respect to tand using (1.1)₁, we arrive at ρu_tt+ρu· ∇u_t−µ∆u_t

=−∇P_t+div(ρu) (u_t+u· ∇u)−ρu_t· ∇u+b_t· ∇b+b· ∇b_t. (3.27)

Multiplying (3.27) byu_t and integrating (by parts) overΩyield 1

2 d dt

Z

ρ|_ut|²dx+µ Z

|∇_ut|²dx=

Z

div(_ρu)|_ut|²dx+

Z

div(_ρu)_u· ∇_u·_utdx

−

Z

ρu_t· ∇u·u_tdx+

Z

b_t· ∇b·u_tdx +

Z

b· ∇bt·utdx,

∑

J_k. (3.28)

By virtue of Hölder’s inequality, Sobolev’s inequality, (3.4), and (3.17), we find that forδ>0,

|J₁|=

−

Z

ρu· ∇|ut|²dx

≤2kρk_L¹²_∞kuk_L6k√

ρu_tk_L3k∇u_tk_L2

≤Ckρk_L¹²_∞k∇uk_L2k√ ρu_tk¹²

L²k√ ρu_tk¹²

L⁶k∇u_tk_L2

≤Ckρk_L³⁴_∞k∇_uk_L2k√ ρutk¹²

L²k∇_utk³²

L²

≤ ^µ

10k∇u_tk²_L2+Ck√

ρu_tk²_L2;

|J₂|=

−

Z

ρu· ∇(u· ∇u·ut)dx

≤

Z

ρ|u||∇u|²|ut|+ρ|u|²|∇²u||ut|+ρ|u|²|∇u||∇ut|dx

≤ kρk_L∞kuk_L6k∇uk_L2k∇uk_L6ku_tk_L6+kρk_L∞kuk²_L6k∇²uk_L2ku_tk_L6

+kρk_L∞kuk²_L6k∇uk_L6k∇utk_L2

≤Ck∇uk²_L2kuk_H2k∇utk_L2

≤ ^µ

10k∇u_tk²_L2+Ckuk²_H2;

|J3| ≤ k∇uk_L2k√

ρutk²_L4 ≤ k∇uk_L2k√

ρutk_L¹²₂k√ ρutk_L³²₆

≤Ckρk_L³⁴_∞k∇uk_L2k√ ρu_tk¹²

L²k∇u_tk³²

L²

≤ ^µ

10k∇utk²_L2+Ck√

ρutk²_L2;

|J₄| ≤ k_btk_L3k∇_bk_L2k_utk_L6 ≤Ck_btk¹²

L²k_btk¹²

L⁶k∇_utk_L2

≤ ^µ

10k∇utk²_L2+C(δ)kbtk²_L2 + ^δ

2k∇btk²_L2;

|J₅|=

−

Z

b· ∇_ut·_btdx

≤ kbk_L6k∇utk_L2kbtk_L3 ≤Ck∇bk_L2k∇utk_L2kbtk_L¹²₂kbtk_L¹²₆

≤ ^µ

10k∇u_tk²_L2+C(δ)kb_tk²_L2 + ^δ

2k∇b_tk²_L2. Substituting the above estimates into (3.28), we deduce that

d dtk√

ρu_tk²_L2 +µk∇u_tk²_L2 ≤Ck√

ρu_tk²_L2+Ckuk²_H2 +Ckb_tk²_L2+δk∇b_tk²_L2. (3.29)

Next, differentiating (3.2)₄with respect tot and multiplying the resulting equations byb_t, we obtain after using integrating by parts and (3.17) that

1 2

d dt

Z

|bt|²dx+ν Z

|∇bt|²dx≤C(k|ut||b|k_L2+k|u||bt|k_L2)k∇btk_L2

≤C

kutk_L6kbk_L3 +kuk_L6kbtk_L¹²₂kbtk_L¹²₆

k∇btk_L2

≤C

k∇utk_L2+kbtk_L¹²₂k∇btk_L¹²₂

k∇btk_L2

≤ ^ν

2k∇b_tk²_L2 +Ck∇u_tk²_L2 +Ckb_tk²_L2, which implies that for someC₁>0,

d

dtkbtk²_L2+νk∇btk²_L2 ≤C₁k∇utk²_L2 +Ckbtk²_L2. (3.30) Adding (3.29) multiplied by ^2C_µ¹ to (3.30) and then choosingδsuitably small, we deduce that

d dt

2C₁µ⁻¹k√

ρutk²_L2+kbtk²_L2

+C₁k∇utk²_L2+^ν

2k∇btk²_L2≤Ckuk²_H2+C k√

ρutk²_L2+kbtk²_L2

. Thus we infer from the Gronwall inequality and (3.17) that

sup

0≤t≤T

k√

ρutk²_L2+k_btk²_L2+

Z _T

0

k∇_utk²_L2+k∇_btk²_L2dt≤C. (3.31) As a consequence, we derive from the regularity theory of elliptic system, (3.2)4, (3.31), and (3.17) that

kbk²_H2 ≤C kbtk²_L2+ku· ∇bk²_L2+kb· ∇uk²_L2+kbk²_H1

≤C+Ckuk²_L6k∇bk²_L3+Ckbk²_L∞k∇uk²_L2

≤C+Ck∇_uk²_L2k∇_bk_L2k∇_bk_L6 +Ck∇_bk_L2k∇_bk_H1k∇_uk²_L2

≤C+ ¹ 2k_bk²_H2,

where we have used the following Sobolev’s inequality kvk_L∞ ≤Ck∇vk_L¹²₂k∇vk¹²

H¹ forv∈ H₀¹∩H². Hence we get

sup

0≤t≤T

kbk²_H2 ≤C. (3.32)

Furthermore, it follows from Lemmas2.2 and2.1, (3.4), (3.17), (3.31), and (3.32) that kuk²_H2 ≤ C kρutk²_L2 +kρu· ∇uk²_L2+kb· ∇bk²_L2

≤ Ckρk_L∞k√

ρutk²_L2 +Ckρk²_L∞kuk²_L6k∇uk²_L3+Ckbk²_L6k∇bk²_L3

≤ C+Ck∇uk³_L2k∇uk_L6

≤ C+ ¹ 2k_uk²_H2,

which yields

sup

0≤t≤T

kuk²_H2 ≤C. (3.33)

Now we estimatek∇ρk_L^q. First of all, applying Lemma2.2once more, we have k_uk²_W2,6 ≤ C k_ρutk²_L6 +k_ρu· ∇_uk²_L6+k_b· ∇_bk²_L6

≤ Ckρk²_L∞ku_tk²_L6 +Ckρk²_L∞kuk²_L∞k∇uk²_L6+Ckbk²_L∞k∇bk²_L6

≤ Ck∇u_tk²_L2+C, which together with (3.31) implies

Z _T

0

kuk²_W2,6dt≤C. (3.34)

Then taking spatial derivative∇on the transport equation (3.2)₁ leads to

∂_t∇ρ+_u· ∇²ρ+∇_u· ∇ρ=0.

Thus standard energy methods yields for anyq∈(3, 6], d

dtk∇ρk_L^q ≤C(q)k∇uk_L∞k∇ρk_L^q ≤Ckuk_W2,6k∇ρk_L^q, which combined with Gronwall’s inequality and (3.24) gives

sup

0≤t≤T

k∇ρk_L^q ≤C.

This along with (3.4) yields

sup

0≤t≤T

kρk_W1,q ≤ C. (3.35)

Finally, we need to estimatekθk_H2. Motivated by [19], denote by ¯θ, _|Ω¹_| R

θdx, the average of θ, then we obtain from (3.4), (3.3), and the Poincaré inequality that

|θ^¯|

Z

ρdx ≤ Z

ρθdx

+ Z

ρ(θ−θ^¯)dx

≤ C+Ck∇θk_L2, which together with the fact that

R vdx

+k∇vk_L2 is an equivalent norm to the usual one in H¹(_Ω)implies that

kθk_H1 ≤C+Ck∇θk_L2. (3.36)

Similarly, one deduces

kθtk_H1 ≤Ck√

ρθtk_L2+Ck∇θtk_L2. (3.37) Multiplying (3.2)₃byθ_t and integrating the resulting equation overΩyield that

κ 2

d dt

Z

|∇θ|²dx+cv

Z

ρ|θ_t|²dx = −cv

Z

ρ(u· ∇θ)θ_tdx+2µ Z

|D(u)|²θ_tdx +ν

Z

|∇b|²θ_tdx,

∑

I_i. (3.38)

By Hölder’s inequality, (3.4), and (3.33), we get

|I₁| ≤c_vkρk_L¹²_∞k√

ρθ_tk_L2kuk_L∞k∇θk_L2 ≤ ^cv 2k√

ρθ_tk²_L2 +Ck∇θk²_L2. (3.39) From (3.33) and (3.36), one has

I₂=2µd dt

Z

|D(u)|²θdx−2µ Z

(|D(u)|²)_tθdx

≤2µd dt

Z

|D(u)|²θdx+C Z

θ|∇u||∇u_t|dx

≤2µd dt

Z

|D(_u)|²θdx+Ckθk_L6k∇_uk_L3k∇_utk_L2

≤2µd dt

Z

|D(_u)|²θdx+_Ckθk_H1kuk_H2k∇utk_L2

≤2µd dt

Z

|D(u)|²θdx+Ck∇utk²_L2+Ck∇θk²_L2 +C. (3.40) Moreover, one infers

I₃ =νd dt

Z

|∇b|²θdx−ν Z

(|∇b|²)_tθdx

≤νd dt

Z

|∇b|²θdx+C Z

θ|∇b||∇bt|dx

≤νd dt

Z

|∇b|²θdx+Ckθk_L6k∇bk_L3k∇b_tk_L2

≤νd dt

Z

|∇b|²θdx+Ckθk_H1kbk_H2k∇b_tk_L2

≤νd dt

Z

|∇_b|²θdx+Ck∇_btk²_L2+Ck∇θk²_L2+C. (3.41) Inserting (3.39)–(3.41) into (3.38), we get

d dt

Z

κ|∇θ|²−4µ|D(u)|²θ−2ν|∇b|²θ

dx+c_vk√ ρθ_tk²_L2

≤Ck∇_utk²_L2+Ck∇_btk²_L2+Ck∇θk²_L2 +C. (3.42) Noting that

4µ Z

|D(u)|²θdx≤Ckθk_L6k∇uk²

L¹²⁵ ≤Ckθk_H1kuk²_H2 ≤ ^κ

4k∇θk²_L2+C, and

2ν Z

|∇b|²θdx ≤Ckθk_L6k∇bk²

L¹²⁵ ≤Ckθk_H1kbk²_H2 ≤ ^κ

4k∇θk²_L2+C, which combined with (3.42), Gronwall’s inequality, and (3.31) leads to

sup

0≤t≤T

k∇θk²_L2+

Z _T

0

k√

ρθ_tk²_L2dt≤C.

This together with (3.36) yields sup

0≤t≤T

kθk²_H1+

Z _T

0

k√

ρθtk²_L2dt≤C. (3.43)