This article established the global existence and uniqueness of solutions for the 3D compressible magneto-micropolar fluid system with vacuum. The remarkable thing is that in the context of small initial energy, we got a new result with a lower regularity than we ever have before.
Citation: Mingyu Zhang. Regularity and uniqueness of 3D compressible magneto-micropolar fluids[J]. AIMS Mathematics, 2024, 9(6): 14658-14680. doi: 10.3934/math.2024713
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This article established the global existence and uniqueness of solutions for the 3D compressible magneto-micropolar fluid system with vacuum. The remarkable thing is that in the context of small initial energy, we got a new result with a lower regularity than we ever have before.
This paper focuses on the three-dimensional compressible magneto-micropolar fluid system in R3 [17,18,24]:
{ρt+div(ρu)=0,(ρu)t+div(ρu⊗u)−(η+β)Δu−(η+κ−β)∇divu+∇P(ρ)=2βrotw+(∇×b)×b,(ρw)t+div(ρu⊗w)−μ′Δw−(η′+κ′)∇divw+4βw=2βrotu,bt+∇×(α∇×b)=∇×(u×b),divb=0. | (1.1) |
Here, ρ⩾0 is the density, u is the velocity, w is the micro-rotational velocity, b is the magnetic field and P(ρ)=Aργ(A>0,γ>1) is the pressure. The parameters η,κ,β,η′,κ′ and α are constants satisfying
η,β,η′,α>0,2η+3κ−4β⩾0,2η′+3κ′⩾0. |
For the completeness of equations (1.1), we supplement the conditions
(ρ,u,w,b)(x,t)|t=0=(ρ0,u0,w0,b0) | (1.2) |
and
(ρ,u,w,b)(x,t)||x|→∞→(1,0,0,0)ast⩾0. | (1.3) |
When we only consider the effects of the velocity field and the magnetic field in (1.1) (i.e., w=0,β=0), the system (1.1) becomes the compressible magnetohydrodynamic (MHD) system, which has been widely discussed; see [4,5,9,11,12,25,26] and references therein. When we lose sight of the effects of magnetic fields, the system (1.1) becomes a micropolar model whose theory was introduced by Eringen [10] and Lukaszewicz [16]. Chen-Xu-Zhang in [6] proved the existence of global weak and smooth solutions in cases of small energy. In bounded or unbounded domain Ω⊂R3, Chen [8] also got the local existence of strong solutions in the context of large initial data.
For system (1.1), there are still many papers discussing this model [3,7,20]. Wei, Guo, and Li [18] found the global existence and decay properties of smooth solutions with small initial perturbation [23]. Later, Tong and Tan [17] improved the results of [18], and obtained the same decay property between the linearized equations and nonlinear system. Later, Xu, Tan, and Wang in [21] considered the system in a bounded region and derived the global well-posedness strong solutions. Zhang-Cai in [24] considered the system (1.1) in a periodic region and derived the well-posedness of solutions. Moreover, in the context of small conditions and symmetrical external forces, the periodic solution was attained. Utilizing Hoff's method [13], Xu, Tan, and Wang in [22] derived the global weak solutions in context of discontinuous initial data. Very recently, Xu and Zhong in [20] proved the local well-posedness of strong solutions to (1.1) with vacuum. Chen, Sun, and Zhong in [3] deduced the well-posedness of global classical solutions in case of vacuum states. However, the results obtained in [6,7,8] still require some compatibility conditions
{−(η+β)Δu0−(η+κ−β)∇divu0+∇P(ρ0)−2βrotw0=ρ1/20g1,−η′Δw0−(η′+κ′)∇divw0+4βw0−2βrotu0=ρ1/20g2, | (1.4) |
for some (g1,g2)∈L2. In case of density including vacuum, it is known from (1.4) that
limt→0+(ρ1/2˙u)(x,t)∈L2,limt→0+(ρ1/2˙w)(x,t)∈L2, | (1.5) |
where, ˙g=gt+u⋅∇g stands for the material derivative. If ρ0>0, then (1.4) becomes
limt→0+(˙u(x,t),˙w(x,t))∈L2. | (1.6) |
To make sure (1.4)/(1.5) or (1.6) are true, the initial velocity and micro-rotational velocity satisfy
(u0,w0)∈H2. | (1.7) |
Thus,
supt∈[0,T]‖(˙u,˙w)‖L2+∫T0‖(∇˙u,∇˙w)‖2L2dt⩽C(T),∀T∈(0,∞), | (1.8) |
where, ‖(g,h)‖Lp≜‖g‖Lp+‖h‖Lp. Higher-order estimates can be deduced from (1.8).
{F1≜(2η+κ)divu−(P(ρ)−P(1)),G1≜rotu,F2≜(2η′+κ′)divw,G2≜rotw, | (1.9) |
where, the quantities F1 and F2 represent the "effective viscous flux", and G1 and G2 stand for the vorticity [6]. One can deduce from (1.9) that
{ΔF1=div(ρ˙u)−divdiv(b⊗b),ΔF2=div(ρ˙w)−4βdivw,(η+β)ΔG1=∇×(ρ˙u)−2β∇×G2,η′ΔG2−4βG2=∇×(ρ˙w)−2β∇×G1. | (1.10) |
Our purpose in this article is to modify the method of [19] and set up the following theorem.
Theorem 1.1. For s∈[9/2,6), assume that (ρ0,u0,w0,b0) satisfies
infρ0(x)>0,ρ0−1∈H1∩W1,s,(u0,w0)∈H1∩W1,3,b0∈H1. | (1.11) |
There exists a constant ε>0, depending on η,κ,β,η′,κ′,γ,α,A,infρ0, supρ0, ‖∇u0‖L2, ‖∇w0‖L2, and ‖∇b0‖L2, such that if
S0≜‖(ρ0−1,u0,w0,b0)‖2L2⩽ε, | (1.12) |
the systems (1.1)–(1.3) possess a global uniqueness solution (ρ,u,w,b) in R3×(0,∞) satisfying
{ρ−1∈C([0,T];H1∩W1,s),infρ(x,t)>0,(u,w,b)∈C([0,T];L2∩La)2⩽a<6,(u,w)∈L∞(0,T;H1∩W1,3)∩L2(0,T;H2)∩Lℓ(0,T;W1,∞),b∈L∞([0,T];H1)∩L2(0,T;H2),(t1/2˙u,t1/2˙w)∈L∞(0,T;L2),(t1/2∇˙u,t1/2∇˙w)∈L2(0,T;L2),(t1/2bt,t1/2∇2b)∈L∞([0,T];L2),(t1/2∇bt,t1/2∇3b)∈L2([0,T];L2), | (1.13) |
with 1<ℓ<(4s)/(5s−6).
In this section, we begin to hammer at the derivations to obtain the global a priori estimates. Let (ρ,u,b,w) stand for a smooth solution of (1.1)–(1.4) on R3×[0,T] for some 0<T<∞, then the system (1.1) becomes
{ρt+div(ρu)=0,ρ(ut+u⋅∇u)+∇P(ρ)−(η+β)Δu−(η+κ−β)∇divu=2βrotw+b⋅∇b−12∇|b|2,ρ(wt+u⋅∇w)−μ′Δw−(η′+κ′)∇divw=4βw+2βrotu,bt+u⋅∇b−b⋅∇u+bdivu−αΔb=0,divb=0. | (2.1) |
Thus, we have the following lemma, which can be found in [6,15].
Lemma 2.1. Given positive numbers N (not necessarily small) and ˆρ>2, assume that (ρ0,u0, w0,b0) satisfies
0⩽infρ0⩽supρ0⩽ˆρ,‖(∇u0,∇w0,∇b0)‖L2⩽N, | (2.2) |
then there exist constants L>0 and ε>0, depending on η,κ,β,η′,κ′,γ,N,α,A,andˆρ, such that if
S0≜∫(J(ρ0)+12ρ0|w0|2+12ρ0|u0|2+12|b0|2)dx⩽ε, | (2.3) |
where J(⋅) stands for the potential energy density
J(ρ)≜ρ∫ρ1P(ς)−P(1)ς2dς, |
then
0⩽ρ(x,t)⩽2ˆρ,∀x∈R3,t∈[0,T], | (2.4) |
supt∈[0,T]‖b‖3L3+∫T0‖b‖3L9dx⩽S1/90, | (2.5) |
supt∈[0,T]‖(∇u,∇w,∇b)‖2L2+∫T0‖(ρ1/2˙u,ρ1/2˙w,∇2b,bt)‖2L2dt⩽L. | (2.6) |
supt∈[0,T]‖(ρ−1,b,ρ1/2u,ρ1/2w)‖2L2+∫T0‖(∇u,∇w,rotu−2w,∇b)‖2L2⩽CS0. | (2.7) |
Remark 2.1. The estimations given by (2.4)–(2.7) do not depend on T and infρ0. In addition, if infρ0>0, then J(⋅) implies
E0∼‖(ρ0−1,u0,b0,w0)‖2L2. | (2.8) |
Remark 2.2. We infer from (2.7) that
∫T0‖w‖2L2dt⩽C∫T0(‖∇u‖2L2+‖rotu−2w‖2L2)dt⩽CS0. | (2.9) |
Lemma 2.2. Under the circumstance of (2.2) and (2.3), then there exists positive constant ε1, such that
supt∈[0,T]‖t1/2(ρ1/2˙u,ρ1/2˙w,∇2b,bt)‖2L2+∫T0t(‖(∇˙u,∇˙w,∇bt)‖2L2+‖∇b‖2H2)dt⩽C(T), | (2.10) |
provided S0⩽ε1.
Proof. First, operate t˙uj[ddt+div(u⋅)] and t˙wj[ddt+div(u⋅)] to (2.1)j2 and (2.1)j3, respectively. Integration by parts gives
[t2‖(ρ1/2˙u,ρ1/2˙w)‖2L2]t−12‖(ρ1/2˙u,ρ1/2˙w)‖2L2=(η+β)∫t˙uj[Δujt+div(Δuju)]dx+η′∫t˙wj[Δwjt+div(Δwju)]dx+(κ+η−β)∫t˙uj[∂t∂j(divu)+div(u∂j(divu))]dx+(κ′+η′)∫t˙wj[div[u∂j(divw)]+∂t∂j(divw)]dx−∫[div(u∂jP)+∂jPt]t˙ujdx+2β∫[∂i(uirotw)+rotwt]⋅t˙udx+2β∫[∂i(uirotu)+rotut]⋅t˙wdx−4β∫[div(wju)+wjt]t˙wdx+∫[div(ub⋅∇bj)+∂t(b⋅∇bj)−12div(u∂j(|b|2))−∂t∂j(|b|2)]t˙ujdx≜10∑i=1Ii. | (2.11) |
Based upon the integration by parts, one has
I1=−(η+β)∫t(∂l˙ui∂luit−∂m∂l˙ujum∂luj−∂m˙uj∂lum∂luj)dx=−(η+β)∫t(|∇˙u|2−∂m˙uj∂mul∂luj+∂m˙uj∂lul∂muj−∂m˙uj∂lum∂luj)dx⩽−3η+4β4(t‖∇˙u‖2L2)+Ct‖∇u‖4L4, | (2.12) |
and
I2=−η′∫t(∂m˙wj∂mwjt−∂i∂m˙wjui∂mwj−∂i˙wj∂mui∂muj)dx=−η′∫t(|∇˙w|2−∂m˙wj∂mul∂lwj−∂m˙wj∂lum∂lwj+∂m˙wj∂lul∂mwj)dx⩽−3η′4(t‖∇˙w‖2L2)+Ct‖∇u‖4L4+Ct‖∇w‖4L4. | (2.13) |
Similarly
I3⩽−μ+λ−ζ2(t‖div˙u‖2L2)+C(t‖∇u‖4L4), | (2.14) |
and
I4⩽−μ′+λ′2(t‖div˙w‖2L2)+Ct‖(∇u,∇w)‖4L4. | (2.15) |
Taking notice of (1.1)1
[P(ρ)−P(1)]t+u⋅∇[P(ρ)−P(1)]=−γdivuP(ρ). | (2.16) |
The inequality (2.4), together with the integration by parts, yields
I5=∫[−∂k(∂i˙ukui)P−div˙u[γdivuP+u⋅∇P]]tdx=∫tP[∂k˙uk∂iui−∂i˙uk∂kui−γ(div˙u)(divu)]dx⩽η8(t‖∇˙u‖2L2)+Ct‖∇u‖2L2. | (2.17) |
Next, for I6 and I7, we see that
I6+I7=4β∫rot(˙w˙u)tdx−2β∫(u⋅t∇w)⋅rot˙udx−2β∫(u⋅t∇u)⋅rot˙wdx−2β∫∂i˙u⋅rotwtuidx−2β∫∂i˙w⋅rotutuidx⩽βt‖∇˙u‖2L2+η′8t‖∇˙w‖2L2+Ct‖ρ˙w‖2L2+C1tS2/30‖∇˙w‖2L2+Ct‖∇u‖6L2+Ct‖(∇u,∇w)‖3L3. | (2.18) |
I8=−4β‖t1/2˙w‖2L2+4β∫[(u⋅t∇˙w)⋅w+(u⋅t∇w)⋅˙w]dx=−4β‖t1/2˙w‖2L2−4β∫div(utw)⋅˙wdx⩽−2β‖t1/2˙w‖2L2+Ct(‖∇w‖6L2+‖∇u‖3L3). | (2.19) |
Next, for I9, the integration by parts gives
I9=∫t(∂j˙ujbibit+∂i˙ujuibk∂jbk)dx⩽Ct‖∇˙u‖L2(‖bt‖L6‖b‖L3+‖u‖L6‖∇b‖L3‖b‖L∞)⩽η8(t‖∇˙u‖2L2)+Ct‖b‖2L3‖∇bt‖2L2+Ct‖∇2b‖2L2(‖∇b‖4L2+‖∇u‖4L2), | (2.20) |
where, we have used the Gagliardo-Nirenberg's inequality:
‖g‖Lβ⩽C‖g‖6−β2βL2‖∇g‖3β−62βL2,∀g∈H1andβ∈[2,6]. | (2.21) |
Integration by parts, together with (2.11), yields
I10=∫t(˙uj∂ibjbit+˙uj∂ibjtbi−∂k˙ujukbi∂ibj)dx⩽Ct‖∇˙u‖L2‖∇bt‖L2‖b‖L3+Ct‖∇˙u‖L2‖∇u‖L2‖b‖L∞‖∇b‖L3⩽η8(t‖∇˙u‖2L2)+Ct‖∇bt‖2L2‖b‖2L3+Ct(‖∇u‖4L2+‖∇b‖4L2)‖∇2b‖2L2. | (2.22) |
Substituting (2.12)–(2.15), (2.17)–(2.20), and (2.22) into (2.11), we obtain
[t2‖(ρ1/2˙u,ρ1/2˙w)‖2L2]t+η2t‖∇˙u‖2L2+η′2t‖∇˙w‖2L2+2βt‖˙w‖2L2⩽(12+Ct)∫(ρ|˙u|2+ρ|˙w|2)dx+Ct‖∇bt‖2L2‖b‖2L3+Ct(‖(∇u,∇w)‖4L4+‖(∇u,∇w)‖3L3+‖∇u‖2L2+‖(∇u,∇w)‖4L2‖∇2b‖2L2), | (2.23) |
provided
S0⩽ε1≜(η′8C1)3/2. |
Taking note of ‖∇bt‖L2, then
btt−αΔbt=(b⋅∇u−u⋅∇b−bdivu)t. |
Thanks to ut=˙u−u⋅∇u, we get
12ddt(t‖bt‖2L2)+αt‖∇bt‖2L2−12‖bt‖2L2=∫t(bt⋅∇u−u⋅∇bt−btdivu+b⋅∇˙u−˙u⋅∇b−bdiv˙u)⋅btdx+∫t[(u⋅∇u)⋅∇b−b⋅∇(u⋅∇u)+bdiv(u⋅∇u)]⋅btdx=3∑i=1Ji. | (2.24) |
We utilize (2.21) to get that
J1⩽Ct‖bt‖L3‖bt‖L6‖∇u‖L2+Ct‖u‖L6‖∇bt‖L2‖bt‖L3⩽Ct‖∇bt‖32L2‖bt‖12L2‖∇u‖L2⩽α8(t‖∇bt‖2L2)+Ct‖bt‖2L2‖∇u‖4L2, | (2.25) |
J2=∫tbt⋅(−˙u⋅∇b+b⋅∇˙u−b⋅div˙u)dx⩽Ct‖b‖L3‖∇˙u‖L2‖∇bt‖L2⩽α8(t‖∇bt‖2L2)+Ct‖b‖2L3‖∇˙u‖2L2, | (2.26) |
and
J3=∫t(bium∂muk∂ibkt+uk∂kui∂ibmbmt−∂mbiuk∂kumbit−biuk∂kum∂mbit)dx⩽‖∇bt‖L2‖∇b‖L2‖∇u‖L2‖∇u‖L6⩽α8(t‖∇bt‖2L2)+Ct‖∇b‖2L2‖∇u‖2L2‖∇u‖2L6. | (2.27) |
Putting (2.25)–(2.27) into (2.24), one has
ddt(t‖bt‖2L2)+t‖∇bt‖2L2⩽Ct‖b‖2L3‖∇˙u‖2L2+C‖bt‖2L2+Ct‖bt‖2L2‖∇u‖4L2+Ct‖∇b‖2L2‖∇u‖2L2‖∇u‖2L6. | (2.28) |
The inequalities (2.23), (2.28), (2.5), and (2.6) yield
\begin{equation} \begin{split} &\sup\limits_{t\in [0, T]}\left(t\| (\rho^{1/2} \dot{ \mathbf{u}}, \rho^{1/2} \dot{ \mathbf{w}}, \mathbf{b}_t)\|_{L^2}^2\right)+\int_0^Tt \|(\nabla\dot{ \mathbf{u}}, \nabla\dot{ \mathbf{w}}, \nabla \mathbf{b}_t)\|_{L^2}^2 dt\\ &\quad \leqslant C(T)+C\int_0^T t \Big(\|(\nabla \mathbf{u}, \nabla \mathbf{w})\|_{L^4}^4+\|(\nabla \mathbf{u}, \nabla \mathbf{w})\|_{L^3}^3+\|\nabla \mathbf{u}\|_{L^6}^2\Big)dt. \end{split} \end{equation} | (2.29) |
We deduce from (1.9) and (1.10) that
\begin{equation} \|(\nabla \mathbf{ u}, \nabla \mathbf{w})\|_{L^6}\leqslant C \|(\rho \dot{\mathbf{ u}}, \rho \dot{\mathbf{ w}})\|_{L^2}+C\| \mathbf{b}\cdot \nabla \mathbf{b}\|_{L^2}+C\|P(\rho)-P(1)\|_{L^6}, \end{equation} | (2.30) |
which can be found in [3]. The inequalities (2.4)–(2.8), (2.21), and (2.30) give
\begin{align} &C\int_0^T t \Big(\|(\nabla \mathbf{u}, \nabla \mathbf{w})\|_{L^4}^4+\|(\nabla \mathbf{u}, \nabla \mathbf{w})\|_{L^3}^3+\|\nabla \mathbf{u}\|_{L^6}^2 \Big)dt\\ &\leqslant C\int_0^T t \Big(\|\nabla \mathbf{u}\|_{L^2}\|\nabla \mathbf{u}\|_{L^6}^3+\|\nabla \mathbf{w}\|_{L^2}\|\nabla \mathbf{w}\|_{L^6}^3+\|\nabla \mathbf{u}\|_{L^6}^2 \Big)dt\\ &\quad+C\int_0^T t \Big(\|\nabla \mathbf{u}\|_{L^2}^{3/2}\|\nabla \mathbf{u}\|_{L^6}^{3/2}+\|\nabla \mathbf{w}\|_{L^2}^{3/2}\|\nabla \mathbf{w}\|_{L^6}^{3/2}\Big)dt\\ &\leqslant C(T)+C\int_0^T t \Big(\|\nabla \mathbf{u}\|_{L^6}^{3}+\|\nabla \mathbf{w}\|_{L^6}^{3}\Big)dt\\ &\leqslant C(T)+C \int_0^T t\left(\| \rho^{1/2} \dot{ \mathbf{u}}\|_{L^2}^3+\| \rho^{1/2} \dot{ \mathbf{w}}\|_{L^2}^3\right)+C \int_0^T t\| \mathbf{b}\|_{L^\infty}^3\|\nabla \mathbf{b}\|_{L^2}^3dt\\ &\leqslant C(T)+C \sup\limits_{0\leqslant t \leqslant T} t^{1/2}\left(\| \rho^{1/2} \dot{ \mathbf{u}}\|_{L^2}+\| \rho^{1/2} \dot{ \mathbf{w}}\|_{L^2}\right)\int_0^Tt^{1/2}\left(\| \rho^{1/2} \dot{ \mathbf{u}}\|_{L^2}^2+\| \rho^{1/2} \dot{ \mathbf{w}}\|_{L^2}^2\right) dt\\ &\quad + C \sup\limits_{0\leqslant t \leqslant T} t^{1/2}\| \nabla^2 \mathbf{b}\|_{L^2}\int_0^T t^{1/2}\|\nabla \mathbf{b}\|_{L^2}^2dt\\ &\leqslant C(T)+\frac{1}{2} \sup\limits_{0\leqslant t \leqslant T} t\left(\| \rho^{1/2} \dot{ \mathbf{u}}\|_{L^2}^2+\| \rho^{1/2} \dot{ \mathbf{w}}\|_{L^2}^2+\| \nabla^2 \mathbf{b}\|_{L^2}^2\right). \end{align} | (2.31) |
The inequality (2.1) _4 gives
\begin{equation*} \|\nabla^2 \mathbf{b}\|_{L^2} \leqslant C\left(\| \mathbf{b}_t\|_{L^2}+\|\nabla \mathbf{b}\|_{L^2}\|\nabla \mathbf{u}\|_{L^2}^2\right), \end{equation*} |
thus
\begin{equation} \begin{split} \sup\limits_{0\leqslant t \leqslant T}\left(t \|\nabla^2 \mathbf{b}\|_{L^2}^2\right) \leqslant C \sup\limits_{0\leqslant t \leqslant T} t \left(\| \mathbf{b}_t\|_{L^2}^2+\|\nabla \mathbf{b}\|_{L^2}^2\|\nabla \mathbf{u}\|_{L^2}^4\right) \leqslant C+ C \int_0^T t \|\nabla \mathbf{u}\|_{L^4}^4 dt. \end{split} \end{equation} | (2.32) |
Based upon (2.1) _4 , it is easy to get that
\begin{equation*} \|\nabla \mathbf{b}\|_{H^2} \leqslant C+C\left(\|\nabla \mathbf{b}_t\|_{L^2}+\|\nabla^2 \mathbf{u}\|_{L^2}\|\nabla^2 \mathbf{b}\|_{L^2}^{1/2}+\|\nabla^2 \mathbf{b}\|_{L^2}+\|\nabla^2 \mathbf{u}\|_{L^2}\right), \end{equation*} |
thus
\begin{equation} \begin{split} \int_0^T t\|\nabla \mathbf{b}\|_{H^2}^2 dt &\leqslant C+C\int_0^T t\|\nabla \mathbf{b}_t\|_{L^2}^2 dt+C\sup\limits_{0\leqslant t \leqslant T} \left(t\|\nabla^2 \mathbf{u}\|_{L^2}^2\right)\int_0^T\|\nabla^2 \mathbf{b}\|_{L^2} dt\\ &\quad+C\int_0^T t \left(\|\nabla^2 \mathbf{b}\|_{L^2}^2+\|\nabla^2 \mathbf{u}\|_{L^2}^2\right)\leqslant C(T). \end{split} \end{equation} | (2.33) |
Therefore, putting (2.31)–(2.33) into (2.29), we obtain (2.10).
Lemma 2.3. Under the circumstance of (2.2) and (2.3),
\begin{equation} \int_0^T \Big(\|(\rho^{1/2}\mathbf{ \dot{u}}, \rho^{1/2}\mathbf{ \dot{w}}, \mathbf{ b}\cdot \nabla \mathbf{ b})\|_{L^r}^s+\| (\rm{div} \mathbf{ u}, \rm{div} \mathbf{ w}, \rm{rot} \mathbf{ u}, \rm{rot} \mathbf{ w})\|_{L^\infty}^s\Big) dt \leqslant C(T), \end{equation} | (2.34) |
where (r, s) satisfies
\begin{equation} r\in (3, 6)\quad{and}\quad 1 < s < \frac{4r}{5r-6} < \frac{4}{3}. \end{equation} | (2.35) |
Proof. The inequality (2.4) together with (2.21) leads to
\begin{equation*} \begin{split} &\| \mathbf{b} \cdot \nabla \mathbf{b}\|_{L^r}^s+\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^r}^s+\|\rho^{1/2}\dot{ \mathbf{w}}\|_{L^r}^s\\ &\leqslant C \|\nabla \mathbf{b}\|_{L^2}^{\tfrac{3s}{r}} \|\nabla^2 \mathbf{b}\|_{L^2}^{\tfrac{s(2r-3)}{r}}+C\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^2}^{\tfrac{s(6-r)}{2r}}\|\nabla \dot{ \mathbf{u}}\|_{L^2}^{\tfrac{s(3r-6)}{2r}}+C\|\rho^{1/2}\dot{ \mathbf{w}}\|_{L^2}^{\tfrac{s(6-r)}{2r}}\|\nabla \dot{ \mathbf{w}}\|_{L^2}^{\tfrac{s(3r-6)}{2r}}, \end{split} \end{equation*} |
thus, by using Hölder's inequality and the inequalities (2.6) and (2.10) gives
\begin{equation} \begin{split} &\int_{0}^T\left(\| \mathbf{b} \cdot \nabla \mathbf{b}\|_{L^r}^s+\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^r}^s+\|\rho^{1/2}\dot{ \mathbf{w}}\|_{L^r}^s\right)dt \leqslant C \sup\limits_{0\leqslant t \leqslant T}\left(t \|\nabla^2 \mathbf{b}\|_{L^2}^2\right)^{\tfrac{s(2r-3)}{2r}}\int_0^T t^{-\tfrac{s(2r-3)}{2r}}dt\\ &\quad+ C \sup\limits_{0\leqslant t \leqslant T}\left[ t\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^2}^2+ t\|\rho^{1/2}\dot{ \mathbf{w}}\|_{L^2}^2\right]^{\tfrac{s(6-r)}{4r}}\int_0^T t^{-\tfrac{s}{2}}\left(t \|\nabla \dot{ \mathbf{u}}\|_{L^2}^2+t \|\nabla \dot{ \mathbf{w}}\|_{L^2}^2\right)^{\tfrac{s(3r-6)}{4r}} dt\\ & \leqslant C \left(\int_0^T t^{-\tfrac{2rs}{4r-3sr+6s}} dt \right)^{\tfrac{4r-3rs+6s}{4r}} \left(\int_0^T (t \|\nabla \dot{ \mathbf{u}}\|_{L^2}^2+ t \|\nabla \dot{ \mathbf{w}}\|_{L^2}^2 ) dt \right)^{\tfrac{s(3r-6)}{4r}}\\ &\quad+C \int_0^T t^{-\tfrac{s(2r-3)}{2r}} dt \leqslant C(T). \end{split} \end{equation} | (2.36) |
Thanks to r\in (3, 6) and 1 < s < \frac{4r}{5r-6} < 2 , one can deduce
0 < \frac{s(2r-3)}{2r} < 1, \quad \quad 0 < \frac{2rs}{4r-3rs+6s} < 1, \quad \quad 0 < \frac{s(3r-6)}{4r} < 1. |
With the help of (1.9) and (1.10), we get
\begin{equation} \|(F_1, G_1, F_2, G_2)\|_{L^2} \leqslant C \|(\nabla \mathbf{ u}, \nabla \mathbf{w}, P(\rho)-P(1))\|_{L^2}, \end{equation} | (2.37) |
and for any m \in [2, 6] ,
\begin{equation} \|(\nabla F_1, \nabla G_1, \nabla F_2, \nabla G_2)\|_{L^m}+\|\nabla G_1\|_{L^m}\leqslant C\|(\rho \dot{\mathbf{ u}}, \rho \dot{ \mathbf{ w}}, \mathbf{b} \cdot \nabla \mathbf{b}, \nabla \mathbf{u}, \nabla \mathbf{w}, \mathbf{w})\|_{L^m}, \end{equation} | (2.38) |
thus, we have from (2.4)–(2.7), (2.37), (2.30), (2.38) that
\begin{equation} \begin{split} &\|({\rm rot} \mathbf{u}, {\text{div}} \mathbf{u}, {\rm rot} \mathbf{w}, {\text{div}} \mathbf{w})\|_{L^\infty}\\ & \leqslant C\left(\|F_1\|_{L^\infty}+\|G_1\|_{L^\infty}+\|P(\rho)-P(1)\|_{L^\infty}+\| \mathbf{b}^2\|_{L^\infty}+\|F_2\|_{L^\infty}+\|G_2\|_{L^\infty}\right)\\ &\leqslant C+C\left(\|F_1\|_{L^2}+\|G_1\|_{L^2}+ \| F_2\|_{L^2}+\| G_2\|_{L^2}+\|\nabla \mathbf{b}\|_{L^2}\|\nabla^2 \mathbf{b}\|_{L^2}\right)\\ &\quad+C\left(\|\nabla F_1\|_{L^r}+\|\nabla G_1\|_{L^r}+ \|\nabla F_2\|_{L^r}+\|\nabla G_2\|_{L^r}\right)\\ &\leqslant C+C\left(\|\nabla \mathbf{u}\|_{L^2}+\|P(\rho)-P(1)\|_{L^2}+\|\nabla \mathbf{w}\|_{L^2}+\|\nabla^2 \mathbf{b}\|_{L^2}\right)\\ &\quad+C\left(\|\rho \dot{\mathbf{ u}}\|_{L^r}+\|\rho \dot{ \mathbf{ w}}\|_{L^r}+\| \mathbf{b} \cdot \nabla \mathbf{b}\|_{L^r}+\|\nabla \mathbf{ u}\|_{L^r}+\|\nabla \mathbf{ w}\|_{L^r}+\|\mathbf{ w}\|_{L^r}\right)\\ &\leqslant C+ C\left(\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^r}+\|\rho^{1/2} \dot{ \mathbf{w}}\|_{L^r}\right)+C\left(\|\nabla \mathbf{u}\|_{L^2}+\|\nabla \mathbf{w}\|_{L^2}+\| \mathbf{w}\|_{L^2}\right)\\ &\quad+C\left(\|\nabla \mathbf{u}\|_{L^6}+\|\nabla \mathbf{w}\|_{L^6}+\|\nabla^2 \mathbf{b}\|_{L^2}\right)+C \| \mathbf{b} \cdot \nabla \mathbf{b}\|_{L^r}\\ &\leqslant C+ C\left(\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^r}+\|\rho^{1/2} \dot{ \mathbf{w}}\|_{L^r}\right)+C\left(\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^2}+\|\rho^{1/2} \dot{ \mathbf{w}}\|_{L^2}\right)\\ &\quad+C\left(\| \mathbf{b} \cdot \nabla \mathbf{b}\|_{L^2}+\|P(\rho)-P(1)\|_{L^6}\right)+C \| \mathbf{b} \cdot \nabla \mathbf{b}\|_{L^p}+C\|\nabla^2 \mathbf{b}\|_{L^2}\\ &\leqslant C+ C\left(\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^r}+\|\rho^{1/2} \dot{ \mathbf{w}}\|_{L^r}\right)+C\left(\|\rho^{1/2}\dot{ \mathbf{u}}\|_{L^2}+\|\rho^{1/2} \dot{ \mathbf{w}}\|_{L^2}\right)\\ &\quad+C \| \mathbf{b} \cdot \nabla \mathbf{b}\|_{L^r}+C\|\nabla^2 \mathbf{b}\|_{L^2}. \end{split} \end{equation} | (2.39) |
Due to s\in (1, 4/3) , one has
\int_0^T \|\nabla^2 \mathbf{b}\|_{L^2}^s dt \leqslant \left( t \|\nabla^2 \mathbf{b}\|_{L^2}^2\right)^{\tfrac{s}{2}}\int_0^T t^{-\tfrac{s}{2}} dt \leqslant C(T), |
which, together with (2.36) and (2.39), gives (2.34).
In the next step, we mainly focus on estimating that \|\nabla \mathbf{ u} \|_{L^p} holds for p\in (1, \infty) and p = \infty , respectively. Due to -\Delta \mathbf{ u} = \nabla {\rm div}\mathbf{ u}-\nabla \times {\rm rot}\mathbf{ u} , for 0 < p < \infty , \|\nabla \mathbf{ u} \|_{L^p}\leqslant C\left(\|{\rm div} \mathbf{ u}\|_{L^p}+\| {\rm rot} \mathbf{ u}\|_{L^p}\right) . However, for p = \infty , we have the following lemma.
Lemma 2.4. (Beale-Kato-Majda type inequality (cf. [1,14])) For k \in \mathbb{Z}^+ and s\in(1, +\infty) , let D^{k, s}\triangleq \Big\{\mathbf{ g}\in L^1_{loc} | \partial^k \mathbf{ g}\in L^s\} and D^{1}\triangleq D^{1, 2} be the homogeneous Sobolev spaces. For \mathbf{ g}\in D^1\cap D^{2, s} with s \in (3, +\infty) , there exists a positive constant C(s) > 0 , such that for all \nabla \mathbf{ g} \in L^2 \cap D^{1, s} ,
\begin{equation} \|\nabla \mathbf{ g}\|_{L^{\infty}}\leqslant C\left(1+\|\nabla \mathbf{ g}\|_{L^2}\right)+C(\|{\rm div} \mathbf{ g}\|_{L^{\infty}}+\|{\rm rot} \mathbf{ g}\|_{L^{\infty}})\ln \left(e+\|\nabla^2 \mathbf{ g}\|_{L^s}\right). \end{equation} | (2.40) |
Lemma 2.5. Let (1.11) and (1.12) be in force, then
\begin{equation} \sup\limits_{t\in [0, T]}\left(\|\rho_t \|_{L^2}+\|\nabla \rho \|_{{L^2}{\cap} {L^r}}\right) +\int_0^T\left(\|\nabla \mathbf{ u}\|_{L^\infty}^s+\|\nabla^2 \mathbf{ u}\|_{L^r}^s \right) dt\leqslant C(T), \end{equation} | (2.41) |
where r\in (3, 6) and s\in (1, \infty) are the same ones as in (2.35).
Proof. We operate \nabla to both sides of (2.1) _1 and multiply the result equation by |\nabla \rho|^{r-2}\nabla\rho for r \in [2, 6] . One can deduce from (2.4) and integration by parts that
\begin{equation} \frac{d}{dt}\|\nabla\rho\|_{L^r}\leqslant C \left(\|\nabla \mathbf{u}\|_{L^\infty}\|\nabla\rho\|_{L^r}+\|\nabla^2 \mathbf{u}\|_{L^r}\right). \end{equation} | (2.42) |
Recalling that \mathcal{G}\triangleq-(\eta+\beta)\Delta-(\eta+\kappa-\beta)\nabla {\text{div}} is a strong elliptic operator ([2]), the inequalities (2.1) _2 and (2.4) give
\begin{equation} \|\nabla^2 \mathbf{u}\|_{L^r} \leqslant C\|(\nabla \rho, \rho^{1/2}\dot{ \mathbf{u}}, \mathbf{b} \cdot \nabla \mathbf{b}, {\rm rot} \mathbf{w})\|_{L^r} \quad \forall\; r\in(3, 6), \end{equation} | (2.43) |
which, combined with (2.42), gives
\begin{equation} \frac{d}{dt}\|\nabla\rho\|_{L^r}\leqslant C \left(\|\nabla \mathbf{u}\|_{L^\infty}+1\right)\|\nabla\rho\|_{L^r}+C\left( \| \rho^{1/2} \dot{ \mathbf{u}}\|_{L^r}+\|{\rm rot} \mathbf{w}\|_{L^r}+ \| \mathbf{b} \cdot \nabla \mathbf{b}\|_{L^r}\right). \end{equation} | (2.44) |
The combination of (2.40) and (2.43) gives
\begin{equation} \begin{split} \|\nabla \mathbf{u}\|_{L^{\infty}} &\leqslant C+C\|({\rm rot} \mathbf{u}, {\text{div}} \mathbf{u})\|_{L^{\infty}}\ln \left(\|\nabla \rho\|_{L^r}+e\right) \\ &\quad+C(\|({\rm rot} \mathbf{u} , {\text{div}} \mathbf{u})\|_{L^{\infty}}\ln \left(\|(\rho^{1/2} \dot{ \mathbf{u}}, {\rm rot} \mathbf{w}, \mathbf{b} \cdot \nabla \mathbf{b})\|_{L^r}+e\right), \end{split} \end{equation} | (2.45) |
for any 3 < r < 6 . Taking (2.45) into (2.44), one gets
\begin{equation} \frac{d}{dt}\left(\|\nabla\rho\|_{L^r}+e\right)\leqslant C \Im(t) \ln \left(\|\nabla\rho\|_{L^r}+e\right), \end{equation} | (2.46) |
where
\begin{equation*} \begin{split} \Im(t) &\triangleq C\left(1+\|( {\text{div}} \mathbf{u}, {\rm rot} \mathbf{u})\|_{L^\infty}+\|(\rho^{1/2} \dot{ \mathbf{u}}, {\rm rot} \mathbf{w}, \mathbf{b} \cdot \nabla \mathbf{b})\|_{L^r}\right)\\ &\quad + C\|( {\text{div}} \mathbf{u}, {\rm rot} \mathbf{u})\|_{L^\infty} \ln \left(\|(\rho^{1/2} \dot{ \mathbf{u}}, {\rm rot} \mathbf{w}, \mathbf{b} \cdot \nabla \mathbf{b})\|_{L^r}+e\right). \end{split} \end{equation*} |
Noting that the relationship \ln(e+z) \leqslant (e+z)^\tau holds for any z \geqslant 0 and \tau > 0 , the inequality (2.33) gives
\begin{equation*} \int_0^T \Im(t)dt \leqslant C(T), \end{equation*} |
which, combined with (2.46), leads to
\begin{equation} \sup\limits_{t\in [0, T]} \|\nabla \rho(t)\|_{L^r} \leqslant C(T), \quad \forall\ r\in(3, 6). \end{equation} | (2.47) |
The inequalities (2.7) and (2.34) give
\begin{equation} \begin{split} &\int_0^T\left(\|\nabla \mathbf{u}\|_{L^\infty}^s +\|\nabla^2 \mathbf{u}\|_{L^r}^s\right) dt \leqslant C \int_0^T\left(\|\nabla^2 \mathbf{u}\|_{L^r}^s+1\right) dt\\ &\quad \leqslant C \int_0^T\left(\|(\rho^{1/2} \dot{ \mathbf{u}}, \nabla P, \mathbf{b} \cdot \nabla \mathbf{b}, {\rm rot} \mathbf{w})\|_{L^r}^s+1\right) dt\leqslant C(T), \end{split} \end{equation} | (2.48) |
for any r, s being as the ones in (2.31). Next, taking r = 2 in (2.42), one has
\begin{equation*} \begin{split} \frac{d}{dt}\|\nabla\rho\|_{L^2} &\leqslant C \left(\|\nabla \mathbf{u}\|_{L^\infty}\|\nabla\rho\|_{L^2}+\|\nabla^2 \mathbf{u}\|_{L^2}\right)\\ &\leqslant C \left(\|\nabla \mathbf{u}\|_{L^\infty}+1 \right)\|\nabla\rho\|_{L^2}+C \left(1+\|(\rho^{1/2} \dot{ \mathbf{u}}, \nabla \mathbf{w})\|_{L^2}\right), \end{split} \end{equation*} |
thus
\begin{equation} \sup\limits_{t\in [0, T]} \|\nabla \rho\|_{L^2} \leqslant C(T). \end{equation} | (2.49) |
Note that the Eq (2.1) _1 gives
\begin{equation*} \begin{split} \|\rho_t\|_{L^2} &\leqslant C\left(\|\nabla \mathbf{u}\|_{L^2}+\|\nabla \rho\|_{L^3}\right)\leqslant C\left(\|\nabla \rho\|_{L^3}+1\right)\\ &\leqslant C\left(\|\nabla \rho\|_{L^r}+1+\|\nabla \rho\|_{L^2}\right)\leqslant C(T), \quad \forall r\in (3, 6), \end{split} \end{equation*} |
which, combined with (2.47)–(2.49), gives (2.41).
Lemma 2.6. Assume that \rho_0 satisfies \inf_{x \in \mathbb{R}^3} \rho_0(x) \geqslant \check{\rho} > 0 , then there exists a constant c > 0 , depending on \check{\rho} and T , such that
\begin{equation} \rho(x, t)\geqslant c, \quad \forall\; x\in \mathbb{R}^3, \quad t\in [0, T], \end{equation} | (2.50) |
and
\begin{equation} \sup\limits_{t\in [0, T]}\left(t\|(\nabla^2 \mathbf{ u}, \mathbf{ u}_t)\|_{L^2}^2\right)+\int_0^T \|(\nabla^2 \mathbf{ u}, \nabla^2 \mathbf{ w}, \mathbf{ u}_t, t^{1/2}\nabla \mathbf{ u}_t)\|_{L^2}^2dt \leqslant C(T). \end{equation} | (2.51) |
Proof. The inequality (2.34) gives
\begin{equation*} \rho(x, t)\geqslant \inf\limits_{x \in \mathbb{R}^3} \rho_0(x)e^{\Big\{-\int_0^t\| {\text{div}} \mathbf{u}\|_{L^\infty} ds \Big\}} \geqslant c(\check{\rho}, T), \end{equation*} |
which yields (2.50).
Next, the inequalities (2.1) _1 , (2.4), and (2.5) give
\begin{equation*} \|\nabla^2 \mathbf{u}\|_{L^2}\leqslant C\left(\|(\rho \dot{ \mathbf{u}}, \nabla P, {\rm rot} \mathbf{w})\|_{L^2}+\| \mathbf{b}\|_{L^3}\|\nabla^2 \mathbf{b}\|_{L^2}\right)\leqslant C\|(\dot{ \mathbf{u}}, \nabla \rho, \nabla \mathbf{w}, \nabla^2 \mathbf{b})\|_{L^2}, \end{equation*} |
thus, using (2.4), (2.5), (2.10), (2.41), and (2.50), one has
\begin{equation} \begin{split} &\sup\limits_{t\in [0, T]}\left(t\|\nabla^2 \mathbf{u}\|_{L^2}^2\right)+\int_0^T\|\nabla^2 \mathbf{u}\|_{L^2}^2 dt\\ &\quad\leqslant C \sup\limits_{t\in [0, T]}\left[t\|(\dot{ \mathbf{u}}, \nabla \rho, \nabla \mathbf{w}, \nabla^2 \mathbf{b})\|_{L^2}^2 \right]+C \int_0^T \|(\dot{ \mathbf{u}}, \nabla \rho, \nabla \mathbf{w}, \nabla^2 \mathbf{b})\|_{L^2}^2 dt\leqslant C(T). \end{split} \end{equation} | (2.52) |
Taking note of \dot{ \mathbf{u}} ,
\| \mathbf{u}_t\|_{L^2}^2\leqslant \| \dot{ \mathbf{u}}\|_{L^2}^2+\| \mathbf{u}\cdot \nabla \mathbf{u}\|_{L^2}^2 \leqslant C\left(\|\dot{ \mathbf{u}}\|_{L^2}^2+\|\nabla \mathbf{u}\|_{L^2}^2\|\nabla \mathbf{u}\|_{H^1}^2\right), |
and the inequalities (2.4), (2.6), (2.10), (2.50), and (2.52) give
\begin{equation} \sup\limits_{t\in [0, T]}\left(t\| \mathbf{u}_t \|_{L^2}^2\right)+\int_0^T\| \mathbf{u}_t\|_{L^2}^2 dt \leqslant C(T), \end{equation} | (2.53) |
and
\begin{equation} \begin{split} \int_0^Tt\|\nabla \mathbf{u}_t\|_{L^2}^2 dt &\leqslant \int_0^T t\|(\nabla \dot{ \mathbf{u}}, \nabla ( \mathbf{u} \cdot\nabla \mathbf{u} ))\|_{L^2}^2 dt \\ &\leqslant C+C\int^T_0t\|\nabla^2 \mathbf{u} \|^4_{L^2} dt+C\int^T_0 t \| \mathbf{u} \|^2_{L^\infty} \|\nabla^2 \mathbf{u} \|^2_{L^2} dt\\ &\leqslant C+C\int^T_0t\|\nabla^2 \mathbf{u} \|^4_{L^2} dt\leqslant C. \end{split} \end{equation} | (2.54) |
On the other hand, due to (2.1) _3 , (2.4)–(2.6), and (2.9), one has
\begin{equation} \int_0^T\|\nabla^2 \mathbf{w}\|_{L^2}^2 dt\leqslant C\int_0^T \Big(\|\rho^{1/2}\dot{ \mathbf{w}}\|_{L^2}^2 +\| \mathbf{w}\|_{L^2}^2+\|\nabla \mathbf{u}\|_{L^2}^2 \Big)dt\leqslant C. \end{equation} | (2.55) |
Utilizing (2.52)–(2.55), we can obtain the estimate (2.51).
Based upon the foregoing, the following estimation of \|\nabla \mathbf{ u}\|_{L^3} is very crucial for this article.
Lemma 2.7. Under the circumstance of Theorem 1.1, we have
\begin{equation} \sup\limits_{t\in [0, T]}\|\nabla \mathbf{ u}\|_{L^3}^3+\int_0^T\|(|{\rm div}\mathbf{ u}|^{1/2}\nabla {\rm div}\mathbf{ u}, |{\rm rot}\mathbf{ u}|^{1/2}\nabla {\rm rot}\mathbf{ u})\|_{L^2}^2dt\leqslant C(T). \end{equation} | (2.56) |
Proof. Utilizing the inequality \|\nabla \mathbf{ u}\|_{L^m}\leqslant C\| ({\rm div} \mathbf{ u}, {\rm rot} \mathbf{ u})\|_{L^m} for 1 < m < \infty , thus, we just have to estimate \| {\rm div} \mathbf{ u}\|_{L^m} and \|{\rm rot} \mathbf{ u}\|_{L^m} . Operating div and rot to (2.1) _2 , one gets
\begin{equation} \begin{split} &\rho ({\rm div} \mathbf{u})_t+\rho \mathbf{u}\cdot\nabla({\rm div} \mathbf{u})-(2\eta+\kappa)\Delta ({\rm div} \mathbf{u})\\ & = -(\nabla \rho)\cdot \mathbf{u}_t -\partial_k(\rho u^i)\partial_i u^k-\Delta P+\partial_k b^i \partial_i b^k-\frac{1}{2}\Delta | \mathbf{b}|^2, \end{split} \end{equation} | (2.57) |
and
\begin{equation} \begin{split} &\rho({\rm rot} \mathbf{u})_t+\rho \mathbf{u}\cdot\nabla({\rm rot} \mathbf{u})-(\eta+\beta)\Delta({\rm rot} \mathbf{u})\\ & = -(\nabla\rho)\times \mathbf{u}_t-\nabla(\rho u^k)\times(\partial_k \mathbf{u})+2\beta {\rm rot}({\rm rot} \mathbf{w})+(\nabla b^i) \times (\partial_i \mathbf{b})+ \mathbf{b} \cdot \nabla ({\rm rot} \mathbf{b}). \end{split} \end{equation} | (2.58) |
We first to estimate \|{\rm div} \mathbf{u}\|_{L^3} . Multiplying (2.57) by | {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u} , one can deduce from integration by parts that
\begin{equation} \begin{split} &\Big(\frac{1}{3}\|\rho^{1/3}{\rm div} \mathbf{u}\|_{L^3}^3\Big)_t+(2\eta+\kappa)\|(|{\rm div} \mathbf{u}|^{1/2}\nabla {\rm div} \mathbf{u}, |{\rm div} \mathbf{u}|^{1/2}\nabla |{\rm div} \mathbf{u}|)\|_{L^2}^2\\ & = -\int \left[ \mathbf{u}_t\cdot\nabla \rho +\partial_k (\rho u^i)\partial_i u^k\right] \left( | {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right) dx\\ &\quad +\int \nabla \left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right) \cdot \nabla P dx+\int \partial_j b^i \partial_i b^j \left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right) dx \\ &\quad+\int \nabla \left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right) \cdot \nabla | \mathbf{b}|^2 dx\triangleq \sum\limits_{i = 1}^5J_i. \end{split} \end{equation} | (2.59) |
With the help of (2.1) _2 and the integration by parts, we get
\begin{equation} \begin{split} J_1 & = -\int\rho^{-1}\Big[((\eta+\beta)\Delta \mathbf{u}+(\eta+\kappa-\beta)\nabla {\text{div}} \mathbf{u}-\nabla P-\rho \mathbf{u}\cdot\nabla \mathbf{u}\Big]\cdot\nabla\rho\left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right) dx\\ &\quad -\int\rho^{-1}\left(2\beta {\rm rot} \mathbf{w}+ \mathbf{b} \cdot \nabla \mathbf{b}-\frac{1}{2}\nabla | \mathbf{b}|^2\right)\cdot\nabla\rho\left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right) dx\\ & = (\eta+\beta)\int \Big[\left(\partial_m u^k\partial_k (\ln\rho)\right)\partial_m\left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right)- \left(\partial_m {\text{div}} \mathbf{u}\right)\left(\partial_{m}(\ln \rho)\right) \left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right)\Big]dx\\ &\quad-(\eta+\beta)\int \left(\partial_m u^k\partial_{m}(\ln \rho) \right)\partial_k\left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right) dx\\ &\quad-(\eta+\kappa-\beta)\int \left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right) \nabla {\text{div}} \mathbf{u} \cdot\nabla (\ln\rho) dx\\ &\quad+\int \left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right)\rho^{-1}\nabla P \cdot\nabla\rho dx +\int\left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right)\left( \mathbf{u}\cdot\nabla \mathbf{u}\right)\cdot \nabla \rho dx\\ &\quad- \int \left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right)\Big[ \Big(2\beta{\rm rot} \mathbf{w}+ \mathbf{b} \cdot \nabla \mathbf{b}-\frac{1}{2}\nabla | \mathbf{b}|^2\Big)\cdot \nabla (\ln \rho)\Big] dx\triangleq \sum\limits_{i = 1}^5J_{1, i}. \end{split} \end{equation} | (2.60) |
The inequality (2.6), together with (2.41) gives
\begin{equation} \begin{split} J_{1, 1} &\leqslant C \|\nabla \rho\|_{L^3} \|\nabla \mathbf{w}\|_{L^3} \| {\text{div}} \mathbf{u}\|_{L^6}^2\\ &\leqslant C \|\nabla \mathbf{w}\|_{L^2}^{1/2}\|\nabla^2 \mathbf{w}\|_{L^2}^{1/2} \| {\text{div}} \mathbf{u}\|_{L^3}^{1/2} \| {\text{div}} \mathbf{u}\|_{L^9}^{3/2}\\ &\leqslant C \|\nabla^2 \mathbf{w}\|_{L^2}^{1/2} \| {\text{div}} \mathbf{u}\|_{L^3}^{1/2} \|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}\\ &\leqslant \frac{2\mu+\lambda}{16} \|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C \|\nabla^2 \mathbf{w}\|_{L^2}^2 \| {\text{div}} \mathbf{u}\|_{L^3}, \end{split} \end{equation} | (2.61) |
where, we used the following facts:
\begin{equation} \begin{cases} \| {\text{div}} \mathbf{v}\|_{L^6} \leqslant C\| {\text{div}} \mathbf{v}\|_{L^3}^{1/4}\| {\text{div}} \mathbf{v}\|_{L^9}^{3/4}, \\ \| {\text{div}} \mathbf{v}\|_{L^9} = \|| {\text{div}} \mathbf{v}|^{3/2}\|_{L^6}^{2/3}\leqslant C\|| {\text{div}} \mathbf{v}|^{1/2}\nabla {\text{div}} \mathbf{v}\|_{L^2}^{2/3}. \end{cases} \end{equation} | (2.62) |
In terms of J_{1, 2} and the inequality (2.41), together with (2.62), gives
\begin{equation} \begin{split} J_{1, 2} &\leqslant C \|\nabla \rho\|_{L^p} \Big\|\nabla \mathbf{u}\Big\|_{L^{\tfrac{9r}{4r-9}}} \| {\text{div}} \mathbf{u}\|_{L^9}^{1/2}\|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}\\ &\leqslant \frac{2\mu+\lambda}{32} \|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C \Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^2 \| {\text{div}} \mathbf{u}\|_{L^9}\\ &\leqslant \frac{2\mu+\lambda}{16} \|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C \Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3, \end{split} \end{equation} | (2.63) |
\begin{equation} \begin{split} J_{1, 3}+J_{1, 4} &\leqslant C \|\nabla \rho\|_{L^3}^2 \| {\text{div}} \mathbf{u}\|_{L^6}^2+C \|\nabla \rho\|_{L^3} \|\nabla^2 \mathbf{u}\|_{L^2}\|\nabla \mathbf{u}\|_{L^2} \| {\text{div}} \mathbf{u}\|_{L^6}^2\\ &\leqslant C \|\nabla^2 \mathbf{u}\|_{L^2}^2+C \|\nabla^2 \mathbf{u}\|_{L^2}\| {\text{div}} \mathbf{u}\|_{L^6}^2\\ &\leqslant C\|\nabla^2 \mathbf{u}\|_{L^2}^2+C\| {\text{div}} \mathbf{u}\|_{L^3}^{1/2}\|\nabla^2 \mathbf{u}\|_{L^2} \|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}\\ &\leqslant \frac{2\eta+\kappa}{16} \|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C \| {\text{div}} \mathbf{u}\|_{L^3}\|\nabla^2 \mathbf{u}\|_{L^2}^2+C\|\nabla^2 \mathbf{u}\|_{L^2}^2, \end{split} \end{equation} | (2.64) |
and
\begin{equation} \begin{split} J_{1, 5} &\leqslant C \|\nabla \rho\|_{L^3} \| \mathbf{b}\|_{L^6} \|\nabla \mathbf{b}\|_{L^6} \| {\text{div}} \mathbf{u}\|_{L^6}^2\\ &\leqslant C \| {\text{div}} \mathbf{u}\|_{L^3}^{1/2}\|\nabla^2 \mathbf{b}\|_{L^2} \| {\text{div}} \mathbf{u}\|_{L^9}^{3/2}\\ &\leqslant C \| {\text{div}} \mathbf{u}\|_{L^3}^{1/2} \|\nabla^2 \mathbf{b}\|_{L^2}\|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}\\ &\leqslant \frac{2\eta+\kappa}{16} \|| {\text{div}} \mathbf{u}|^{1/2} \nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C \|\nabla^2 \mathbf{b}\|_{L^2}^2 \| {\text{div}} \mathbf{u}\|_{L^3}. \end{split} \end{equation} | (2.65) |
Putting (2.61), (2.63)–(2.65) into (2.60), one has
\begin{equation} \begin{split} J_{1} &\leqslant \frac{2\eta+\kappa}{4}\|| {\text{div}} \mathbf{u}|^{1/2}\nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C\left(\|\nabla^2 \mathbf{u}\|_{L^2}^2+ \Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3\right)\\ &\quad+C \left(\|(\nabla^2 \mathbf{w}, \nabla^2 \mathbf{b}, \nabla^2 \mathbf{u})\|_{L^2}^2+1\right)\left(1+\| {\text{div}} \mathbf{u}\|_{L^3}^2\right), \quad 3 < r < 6. \end{split} \end{equation} | (2.66) |
Next, by virtue of (2.4), (2.6), (2.41), and (2.62),
\begin{equation} \begin{split} J_2 &\leqslant C\|\nabla \mathbf{u}\|_{L^2}\|\nabla \rho\|_{L^3}\|\nabla^2 \mathbf{u}\|_{L^2}\| {\text{div}} \mathbf{u}\|_{L^6}^2+C\|\nabla^2 \mathbf{u}\|_{L^2}^2\| {\text{div}} \mathbf{u}\|_{L^3}^2\\ &\leqslant \frac{2\eta+\kappa}{8}\|| {\text{div}} \mathbf{u}|^{1/2}\nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C\|\nabla^2 \mathbf{u}\|_{L^2}^2\left(\| {\text{div}} \mathbf{u}\|_{L^3}^2+1\right), \end{split} \end{equation} | (2.67) |
and
\begin{equation} \begin{split} J_3 &\leqslant \| {\text{div}} \mathbf{u}\|_{L^3}^{1/2}\|\nabla \rho\|_{L^3}\|| {\text{div}} \mathbf{u}|^{1/2}\nabla {\text{div}} \mathbf{u}\|_{L^2}\\ &\leqslant \frac{2\eta+\kappa}{8}\|| {\text{div}} \mathbf{u}|^{1/2}\nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C\left(1+\| {\text{div}} \mathbf{u}\|_{L^3}^2\right). \end{split} \end{equation} | (2.68) |
Noticing that {\rm div}\mathbf{ b} = 0 , the inequality of (2.6) gives
\begin{equation} \begin{split} J_4+J_5 & = C \int [\partial_jb^i b^i-b^i \partial_i b^j ]\partial_j \left(| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}\right)dx\\ &\leqslant \| \mathbf{b}\|_{L^3}\| {\text{div}} \mathbf{u}\|_{L^3}^{1/2}\|\nabla \mathbf{b}\|_{L^6}\|| {\text{div}} \mathbf{u}|^{1/2}\nabla {\text{div}} \mathbf{u}\|_{L^2}\\ &\leqslant \frac{2\eta+\kappa}{8}\|| {\text{div}} \mathbf{u}|^{1/2}\nabla {\text{div}} \mathbf{u}\|_{L^2}^2+C\|\nabla^2 \mathbf{b}\|_{L^2}^2\left(1+\| {\text{div}} \mathbf{u}\|_{L^3}^2\right). \end{split} \end{equation} | (2.69) |
Substituting (2.66)–(2.69) into (2.59), one has
\begin{equation} \begin{split} &\Big(\|\rho^{1/3}{\rm div} \mathbf{ u}\|_{L^3}^3\Big)_t+\||{\rm div}\mathbf{ u}|^{1/2}\nabla {\rm div}\mathbf{ u}\|_{L^2}^2\\ &\leqslant C \left(\|(\nabla^2 \mathbf{ u}, \nabla^2 \mathbf{ w}, \nabla^2 \mathbf{ b})\|_{L^2}^2+1\right)\left(\|{\rm div}\mathbf{ u}\|_{L^3}^2+1\right)+C\Big\|\nabla \mathbf{ u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3. \end{split} \end{equation} | (2.70) |
Next, we estimate \|{\rm rot} \mathbf{u}\|_{L^3} . We first multiply (2.58) by |{\rm rot} \mathbf{ u}| {\rm rot} \mathbf{ u} and then deduce from integration by parts that
\begin{equation} \begin{split} &\Big(\frac{1}{3}\|\rho^{1/3}{\rm rot}\mathbf{ u}\|_{L^3}^3\Big)_t+(\eta+\beta)\Big(\||{\rm rot}\mathbf{ u}|^{1/2}\nabla {\rm rot}\mathbf{ u}\|_{L^2}^2+\||{\rm rot}\mathbf{ u}|^{1/2}\nabla |{\rm rot}\mathbf{ u}|\|_{L^2}^2\Big)\\ & = \int \Big[2\beta {\rm rot}({\rm rot} \mathbf{w})-\nabla\rho\times \mathbf{u}_t-\nabla(\rho u^k)\times(\partial_k \mathbf{u})\Big] \cdot (|{\rm rot} \mathbf{u}| {\rm rot} \mathbf{u} ) dx\\ &\quad+\int \Big[(\nabla b^k) \times (\partial_k \mathbf{b})+ \mathbf{b} \cdot \nabla ({\rm rot} \mathbf{b})\Big]\cdot (|{\rm rot} \mathbf{u}| {\rm rot} \mathbf{u} ) dx\triangleq \sum\limits_{i = 1}^5 I_i. \end{split} \end{equation} | (2.71) |
The Hölder's inequality gives
\begin{equation} \begin{split} I_1 & \leqslant C \int |\nabla^2 \mathbf{w}| |\nabla \mathbf{u}| |{\rm rot} \mathbf{u}| dx \leqslant C\|\nabla^2 \mathbf{u}\|_{L^2} \|\nabla^2 \mathbf{w}\|_{L^2} \|{\rm rot} \mathbf{u}\|_{L^3}\\ & \leqslant C \|\nabla^2 \mathbf{w}\|_{L^2}^2\|{\rm rot} \mathbf{u}\|_{L^3}^2+C \|\nabla^2 \mathbf{u}\|_{L^2}^2. \end{split} \end{equation} | (2.72) |
Noting that
(\eta+\beta)\Delta \mathbf{u}+(\eta+\kappa-\beta)\nabla {\text{div}} \mathbf{u} = (2\eta+\kappa)\nabla {\text{div}} \mathbf{u}-(\eta+\beta)\nabla\times({\rm rot} \mathbf{u}), |
thus
\begin{equation} \begin{split} I_2 & = -\int \left( |{\rm rot} \mathbf{u}|{\rm rot} \mathbf{u} \right)\cdot (\nabla \ln \rho)\times \left(2\beta {\rm rot} \mathbf{w}\right) dx\\ &\quad+\int \left( |{\rm rot} \mathbf{u}|{\rm rot} \mathbf{u} \right)\cdot (\nabla \ln \rho)\times [(\eta+\beta)\left(\nabla \times {\rm rot} \mathbf{u} \right)-(2\mu+\lambda)\left(\nabla {\text{div}} \mathbf{u}\right)] dx\\ &\quad+\int \left(|{\rm rot} \mathbf{u}|{\rm rot} \mathbf{u}\right)\cdot (\nabla \ln \rho)\times\left(\nabla P+\rho \mathbf{u}\cdot\nabla \mathbf{u}\right) dx\\ &\quad-\int\left( |{\rm rot} \mathbf{u}|{\rm rot} \mathbf{u} \right) \cdot (\nabla \ln \rho)\times \left( \mathbf{b} \cdot \nabla \mathbf{b}-\frac{1}{2}\nabla | \mathbf{b}|^2\right) dx \triangleq \sum\limits_{i = 1}^6 I_{2, i}. \end{split} \end{equation} | (2.73) |
By virtue of (2.6), (2.23), (2.41), (2.62), we get
\begin{equation} \begin{split} I_{2, 1} &\leqslant C \|\nabla \mathbf{w}\|_{L^3}\|\nabla \rho\|_{L^3} \|\nabla {\rm rot} \mathbf{u}\|_{L^2}^{2} \\ &\leqslant \frac{\eta+\beta}{16}\||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}^2+C\Big(\|\nabla^2 \mathbf{w}\|_{L^2}^2+1\Big) \left(\|{\rm rot} \mathbf{u}\|_{L^3}^2+1\right), \end{split} \end{equation} | (2.74) |
and
\begin{equation} \begin{split} I_{2, 2} &\leqslant C \|{\rm rot} \mathbf{u}\|_{L^9}^{1/2}\Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}} \||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}\|\nabla \rho\|_{L^r} \\ &\leqslant \frac{\eta+\beta}{16}\||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}^2+C\Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3. \end{split} \end{equation} | (2.75) |
For I_{2, 3} , we utilize the fact
\int(\nabla \phi \times\nabla \psi)\cdot v dx = -\int \psi(\nabla \phi )\cdot(\nabla\times v) dx, |
and take \phi = \ln\rho , \psi = {\rm div} \mathbf{ u} , and v = |{\rm rot} \mathbf{ u}|{\rm rot} \mathbf{ u} , then
\begin{equation} \begin{split} I_{2, 3} &\leqslant C \|\nabla \rho\|_{L^r}\Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}\|{\rm rot} \mathbf{u}\|_{L^9}^{1/2} \||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}\\ &\leqslant \frac{\eta+\beta}{16}\||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}^2+C\Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3. \end{split} \end{equation} | (2.76) |
The inequality (2.41), together with (2.4), leads to
\begin{equation} I_{2, 4}\leqslant C \int|{\rm rot} \mathbf{u}|^2 |\nabla \rho|^2 dx\leqslant C \|\nabla^2 \mathbf{u}\|_{L^2}^2 \|\nabla \rho\|_{L^3}^2 \leqslant C \|\nabla^2 \mathbf{u}\|_{L^2}^2, \end{equation} | (2.77) |
\begin{equation} \begin{split} I_{2, 5} &\leqslant C \int |\nabla \mathbf{u}||\nabla \rho| |{\rm rot} \mathbf{u}|^2|\rho| | \mathbf{u}|dx\leqslant C\|\nabla^2 \mathbf{u} \|_{L^2} \|{\rm rot} \mathbf{u}\|_{L^6}^2dx\\ &\leqslant \frac{\eta+\kappa}{32}\||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}^2+\|\nabla^2 \mathbf{u}\|_{L^2}^2 \left(\|{\rm rot} \mathbf{u}\|_{L^3}^2+1\right), \end{split} \end{equation} | (2.78) |
and
\begin{equation} \begin{split} I_{2, 6} &\leqslant C \int |\nabla \mathbf{b}| |{\rm curl } \mathbf{u}|^2 |\nabla \rho| | \mathbf{b}| dx\leqslant C \| \mathbf{b}\|_{L^6}\|\nabla \rho\|_{L^3} \|\nabla \mathbf{b}\|_{L^6} \|{\rm curl} \mathbf{u}\|_{L^6}^{2} \\ &\leqslant \frac{\eta}{32}\||{\rm curl} \mathbf{u}|^{1/2} \nabla {\rm curl} \mathbf{u}\|_{L^2}^2+\|\nabla^2 \mathbf{b}\|_{L^2}^2 \left(\|{\rm curl} \mathbf{u}\|_{L^3}^2+1\right). \end{split} \end{equation} | (2.79) |
Taking (2.74)–(2.79) into (2.73), one has
\begin{equation} \begin{split} I_{2} &\leqslant \frac{\eta+\beta}{4}\||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}^2+C \Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3\\ &\quad+C\left(1+\|(\nabla^2 \mathbf{u}, \nabla^2 \mathbf{w}, \nabla^2 \mathbf{B})\|_{L^2}^2\right) \left(\|{\rm rot} \mathbf{u}\|_{L^3}^2+1\right). \end{split} \end{equation} | (2.80) |
Similarly,
\begin{equation} \begin{split} I_{3} &\leqslant C \int |{\rm rot} \mathbf{u}|^{2}\left( | \mathbf{u}| |\nabla \rho| |\nabla \mathbf{u}|+ |\nabla \mathbf{u}|^2|\rho| \right) dx\leqslant C\|\nabla^2 \mathbf{u} \|_{L^2} \|{\rm rot} \mathbf{u}\|_{L^6}^2dx\\ &\leqslant \frac{\eta+\beta}{8}\||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}^2+\|\nabla^2 \mathbf{u}\|_{L^2}^2 \left(\|{\rm rot} \mathbf{u}\|_{L^3}^2+1\right), \end{split} \end{equation} | (2.81) |
and
\begin{equation} I_4 \leqslant C \int |{\rm rot } \mathbf{u}|^2 |\nabla \mathbf{b}|^2 dx \leqslant C \|\nabla^2 \mathbf{b}\|_{L^2}^2 \|{\rm rot} \mathbf{u}\|_{L^3}^2. \end{equation} | (2.82) |
We notice that {\text{div}} \mathbf{b} = 0 , thus
\begin{equation} \begin{split} I_5 &\leqslant C \int |{\rm rot} \mathbf{u}| | \mathbf{b}| |\nabla {\rm rot } \mathbf{u}| |\nabla \mathbf{b}|dx\\ &\leqslant C \| \mathbf{b}\|_{L^3} \|\nabla^2 \mathbf{b}\|_{L^2} \|{\rm rot} \mathbf{u}\|_{L^3}^{1/2} \||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}\\ &\leqslant \frac{\eta}{8}\||{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u}\|_{L^2}^2+\|\nabla^2 \mathbf{b}\|_{L^2}^2 \left(\|{\rm rot} \mathbf{u}\|_{L^3}^2+1\right). \end{split} \end{equation} | (2.83) |
Putting (2.72), (2.73) and (2.80)–(2.83) into (2.71), we obtain
\begin{equation} \begin{split} &\Big(\|\rho^{1/3}{\rm rot} \mathbf{ u}\|_{L^3}^3\Big)_t+\||{\rm rot}\mathbf{ u}|^{1/2}\nabla {\rm rot}\mathbf{ u}\|_{L^2}^2\\ &\leqslant C \left(\|(\nabla^2 \mathbf{ u}, \nabla^2 \mathbf{ w}, \nabla^2 \mathbf{ b})\|_{L^2}^2+1\right)\left(\|{\rm rot}\mathbf{ u}\|_{L^3}^2+1\right)+C\Big\|\nabla \mathbf{ u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3. \end{split} \end{equation} | (2.84) |
We close the estimations. The inequality (2.70), together with (2.84), gives
\begin{equation} \begin{split} &\left( \|( \rho^{1/3} {\text{div}} \mathbf{u}, \rho^{1/3} {\rm rot} \mathbf{u} ) \|_{L^3}^3\right)_t+\|(| {\text{div}} \mathbf{u}|^{1/2}\nabla {\text{div}} \mathbf{u}, |{\rm rot} \mathbf{u}|^{1/2} \nabla {\rm rot} \mathbf{u})\|_{L^2}^2\\ & \leqslant \left( 1+\|(\nabla^2 \mathbf{w}, \nabla^2 \mathbf{u}, \nabla^2 \mathbf{b})\|_{L^2}^2\right)\left(1+\|( {\text{div}} \mathbf{u}, {\rm rot} \mathbf{u})\|_{L^3}^2\right)+C\Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3. \end{split} \end{equation} | (2.85) |
Due to
1 < \frac{18-2r}{r}\leqslant 2, \quad {\rm and} \quad 1\leqslant \frac{5r-18}{r} < 2, \quad {\rm for}\quad \frac{9}{2}\leqslant r < 6, |
thus
\begin{equation} \Big\|\nabla \mathbf{u} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3\leqslant C \Big\|\nabla \mathbf{u} \Big\|_{L^3}^{\tfrac{5r-18}{r}}\Big\|\nabla \mathbf{u} \Big\|_{L^6}^{\tfrac{18-2r}{r}}\leqslant C \left(1+\|\nabla \mathbf{u}\|_{L^3}^2\right)\left(1+\|\nabla \mathbf{u}\|_{H^1}^2\right). \end{equation} | (2.86) |
Thanks to \|\nabla \mathbf{u}\|_{L^r}\leqslant C \left(\|{\rm div} \mathbf{u} \|_{L^r}+\|{\rm rot} \mathbf{u}\|_{L^r}\right)\; (\forall\; r > 1) , we obtain from (2.6), (2.51), (2.85), and (2.86) that
\begin{equation*} \begin{split} &\sup\limits_{0\leqslant t \leqslant T}\|\nabla \mathbf{u}\|_{L^3}^3 \leqslant C\sup\limits_{0\leqslant t \leqslant T}\left( \| {\text{div}} \mathbf{u} \|_{L^3}^3+\| {\rm rot} \mathbf{u} \|_{L^3}^3 \right)\\ & \leqslant C(T)+C \int_0^T\|(\nabla^2 \mathbf{w}, \nabla^2 \mathbf{u}, \nabla^2 \mathbf{b})\|_{L^2}^2\left(1+\|\nabla \mathbf{u}\|_{L^3}^2\right)dt\\ &\quad +C \int_0^T \left(1+\|\nabla \mathbf{u}\|_{H^1}^2\right) \left(1+\|\nabla \mathbf{u}\|_{L^3}^2\right) dt\\ & \leqslant C(T)+C \int_0^T\|(\nabla^2 \mathbf{w}, \nabla^2 \mathbf{u}, \nabla^2 \mathbf{b})\|_{L^2}^2\|\nabla \mathbf{u}\|_{L^3}^2dt, \end{split} \end{equation*} |
which, allied with (2.51), gives (2.56).
Lemma 2.8. Under the circumstance of Theorem 1.1, then
\begin{equation} \sup\limits_{t\in [0, T]}\|\nabla \mathbf{ w}\|_{L^3}^3+\int_0^T\|(|{\rm div}\mathbf{ w}|^{1/2}\nabla {\rm div}\mathbf{ w}, |{\rm rot}\mathbf{ w}|^{1/2}\nabla {\rm rot}\mathbf{ w})\|_{L^2}^2 dt\leqslant C(T). \end{equation} | (2.87) |
Proof. Operating div and rot to (2.1) _3 , one has
\begin{equation} \begin{split} &\rho ({\rm div} \mathbf{w})_t+\rho \mathbf{u}\cdot\nabla({\rm div} \mathbf{w})-(2\eta^{\prime}+\kappa^{\prime})\Delta ({\rm div} \mathbf{w}) +(\nabla \rho)\cdot \mathbf{w}_t +\partial_k(\rho u^m)\partial_m w^k+4\beta {\rm div} \mathbf{w} = 0, \end{split} \end{equation} | (2.88) |
and
\begin{equation} \begin{split} \rho({\rm rot} \mathbf{w})_t+\rho \mathbf{u}\cdot\nabla({\rm rot} \mathbf{w})-\eta^{\prime}\Delta({\rm rot} \mathbf{w})+(\nabla\rho)\times \mathbf{w}_t+\nabla(\rho u^m)\times(\partial_m \mathbf{w}) +4\beta {\rm rot} \mathbf{w}-2\beta {\rm rot}({\rm rot} \mathbf{u}) = 0. \end{split} \end{equation} | (2.89) |
We multiply (2.88) by | {\text{div}} \mathbf{w}| {\text{div}} \mathbf{w} and integrate by parts over \mathbb{R}^3 , and one has
\begin{equation} \begin{split} &\Big(\frac{1}{3}\| \rho^{1/3}{\rm div}\mathbf{ w}\|_{L^3}^3 \Big)_t+4\beta \|{\rm div}\mathbf{ w}\|_{L^3}^3+(2\eta^{\prime}+\kappa^{\prime})\|(|{\rm div}\mathbf{ w}|^{1/2}\nabla {\rm div}\mathbf{ w}, |{\rm div}\mathbf{ w}|^{1/2}\nabla |{\rm div}\mathbf{ w}|)\|_{L^2}^2\\ & = -\int \left( | {\text{div}} \mathbf{w}| {\text{div}} \mathbf{w}\right)\left( \mathbf{w}_t\cdot\nabla \rho \right) dx-\int \left(| {\text{div}} \mathbf{w}| {\text{div}} \mathbf{w}\right) \partial_k (\rho u^m)\partial_m w^k dx\triangleq N_1+N_2. \end{split} \end{equation} | (2.90) |
The inequalitis (2.1) _2 and (2.60) give
\begin{equation} \begin{split} N_1 &\leqslant \frac{3(2\eta^{\prime}+\kappa^{\prime})}{8}\|| {\text{div}} \mathbf{w}|^{1/2}\nabla {\text{div}} \mathbf{w}\|_{L^2}^2+C \Big\|\nabla \mathbf{w} \Big\|_{L^{\tfrac{9r}{4r-9}}}^3\\ &\quad+C \|(\nabla^2 \mathbf{w}, \nabla^2 \mathbf{u})\|_{L^2}^2\left(\| {\text{div}} \mathbf{u}\|_{L^3}^2+1\right), \quad 3 < r < 6. \end{split} \end{equation} | (2.91) |
Similar to the estimation of (2.67), we get
\begin{equation} N_2\leqslant \frac{2\eta^{\prime}+\kappa^{\prime}}{8}\|| {\text{div}} \mathbf{w}|^{1/2}\nabla {\text{div}} \mathbf{w}\|_{L^2}^2+C\|(\nabla^2 \mathbf{u}, \nabla^2 \mathbf{w})\|_{L^2}^2\left(\| {\text{div}} \mathbf{w}\|_{L^3}^2+1\right). \end{equation} | (2.92) |
Substituting (2.91) and (2.92) into (2.90), we have
\begin{equation} \begin{split} &\Big(\|\rho^{1/3}{\rm div} \mathbf{w}\|_{L^3}^3\Big)_t+\|| {\text{div}} \mathbf{w}|^{1/2}\nabla {\text{div}} \mathbf{w}\|_{L^2}^2+\|{\rm div} \mathbf{w}\|_{L^3}^3\\ & \leqslant C\left(\|\nabla^2 \mathbf{w}\|_{L^2}^2+\|\nabla^2 \mathbf{u}\|_{L^2}^2 \right)\left(1+\| {\text{div}} \mathbf{w}\|_{L^3}^2\right)+C\Big\|\nabla \mathbf{w} \Big\|_{L^{\tfrac{9p}{4p-9}}}^3. \end{split} \end{equation} | (2.93) |
Multiplying (2.89) by |{\rm rot} \mathbf{w}| {\rm rot} \mathbf{w} and integrating by parts over \mathbb{R}^3 , we deduce
\begin{equation*} \begin{split} &\Big(\frac{1}{3}\|\rho^{1/3}{\rm rot} \mathbf{w}\|_{L^3}^3\Big)_t+\eta^{\prime}\|(|{\rm rot} \mathbf{w}|^{1/2}\nabla {\rm rot} \mathbf{w}, |{\rm rot} \mathbf{w}|^{1/2}\nabla |{\rm rot} \mathbf{w}|)\|_{L^2}^2+4\beta \|{\rm rot} \mathbf{w}\|_{L^3}^3\\ & = \int \Big[2\beta {\rm rot}({\rm rot} \mathbf{u})-\nabla\rho\times \mathbf{w}_t -\nabla(\rho u^k)\times(\partial_k \mathbf{w})\Big]\cdot (|{\rm rot} \mathbf{w}| {\rm rot} \mathbf{w} ) dx. \end{split} \end{equation*} |
Hence
\begin{equation} \begin{split} &\Big(\|\rho^{1/3}{\rm rot} \mathbf{w}\|_{L^3}^3\Big)_t+\||{\rm rot} \mathbf{w}|^{1/2} \nabla {\rm rot} \mathbf{w}\|_{L^2}^2+ \|{\rm rot} \mathbf{w}\|_{L^3}^3\\ &\leqslant C \left(\|\nabla^2 \mathbf{u}\|_{L^2}^2+\|\nabla^2 \mathbf{w}\|_{L^2}^2\right) \left(1+\|{\rm rot} \mathbf{w}\|_{L^3}^2\right)+C \Big\|\nabla \mathbf{w} \Big\|_{L^{\tfrac{9p}{4p-9}}}^3. \end{split} \end{equation} | (2.94) |
Similar to the estimation of (2.85), we can get (2.87) from (2.93) and (2.94).
In this section, we mainly focus on proving in Theorem 1.1 holds, based on the global a priori estimates that have been obtained in Section 2. Actually, the global existence can be established by modifying the method in [6]. In the next step, by modifying the ideas of [19], we will prove the uniqueness of solutions holds.
Proof of uniqueness. Let (\rho_1, \mathbf{u}_1, \mathbf{w}_1, \mathbf{b}_1) and (\rho_2, \mathbf{u}_2, \mathbf{w}_2, \mathbf{b}_2) be two solutions to the system (2.1), (1.2) and (1.3) on \mathbb{R}^3\times[0, T] that satisfy (1.13) and have the same initial data. Define
\varphi \triangleq \rho_1-\rho_2, \quad \mathbf{v} \triangleq \mathbf{u}_1- \mathbf{u}_2, \quad \varpi \triangleq \mathbf{w}_1- \mathbf{w}_2, \quad \phi\triangleq \mathbf{b}_1- \mathbf{b}_2, |
then it follows from (2.1) _1 that
\begin{equation} \varphi_t+ \mathbf{u}_2\cdot\nabla \varphi+\varphi {\text{div}} \mathbf{u}_2+\rho_1 {\text{div}} \mathbf{v}+ \mathbf{v}\cdot\nabla\rho_1 = 0. \end{equation} | (3.1) |
Multiplying (3.1) by \varphi , one can deduce from integration by parts that
\begin{equation} \begin{split} \frac{d}{dt}\|\varphi\|_{L^2}^2 &\leqslant C\| {\text{div}} \mathbf{u}_2\|_{L^\infty}\|\varphi\|_{L^2}^2+C\left(\|\nabla \mathbf{v}\|_{L^2}+\| \mathbf{v}\|_{L^6}\|\nabla \rho_1\|_{L^3}\right)\|\varphi\|_{L^2}\\ &\leqslant C\| {\text{div}} \mathbf{u}_2\|_{L^\infty}\|\varphi\|_{L^2}^2+C \|\nabla \mathbf{v}\|_{L^2}\|\varphi\|_{L^2}. \end{split} \end{equation} | (3.2) |
Due to (1.13), we know that \| {\text{div}} \mathbf{u}_2\|_{L^\infty}\in L^1(0, T) . Thus, the inequality (3.2) yields
\begin{equation} \|\varphi(t)\|_{L^2} \leqslant C\int_0^t\|\nabla \mathbf{v}\|_{L^2} ds \leqslant C t^{1/2}\left(\int_0^t\|\nabla \mathbf{v}\|_{L^2}^2 ds \right)^{1/2}, \quad \forall\ t\in[0, T]. \end{equation} | (3.3) |
Since it holds that \dot{ \mathbf{u}}_2 = \mathbf{u}_{2t}+ \mathbf{u}_2 \cdot \nabla \mathbf{u}_2 , we get
\begin{equation} \begin{split} &\rho_1 \mathbf{v}_t+\rho_1 \mathbf{u}_1\cdot\nabla \mathbf{v}-(\eta+\beta)\Delta \mathbf{v}-(\eta+\kappa-\beta)\nabla {\text{div}} \mathbf{v}\\ & = -\varphi\dot{ \mathbf{u}}_2-\rho_1 \mathbf{v}\cdot\nabla \mathbf{u}_2-\nabla\left(P(\rho_1)-P(\rho_2)\right)+2\beta {\rm rot}\varpi+ \mathbf{b}_1 \cdot \nabla \phi-\frac{1}{2}\nabla(| \mathbf{b}_1|^2-| \mathbf{b}_2|^2). \end{split} \end{equation} | (3.4) |
Multiplying (3.4) by \mathbf{v} and integrating by parts, we get
\begin{equation*} \begin{split} &\frac{1}{2}\frac{d}{dt}\|\rho_1^{1/2} \mathbf{v}\|_{L^2}^2+(\eta+\beta)\|\nabla \mathbf{v}\|_{L^2}^2+(\eta+\kappa-\beta)\| {\text{div}} \mathbf{v}\|_{L^2}^2\\ &\quad \leqslant C\|\varphi\|_{L^2}\|\dot{ \mathbf{u}}_2\|_{L^3}\| \mathbf{v}\|_{L^6}+C\| \mathbf{v}\|_{L^2}\|\nabla \mathbf{u}_2\|_{L^3}\| \mathbf{v}\|_{L^6}\\ &\qquad+C\|\varphi\|_{L^2}\|\nabla \mathbf{v}\|_{L^2}+C \|\varpi\|_{L^2}\|\nabla \mathbf{v}\|_{L^2}\\ &\qquad +C \left( \| \mathbf{b}_1\|_{L^\infty}+\| \mathbf{b}_2\|_{L^\infty} \right)\|\phi\|_{L^2}\|\nabla \mathbf{v}\|_{L^2}\\ & \leqslant \frac{\eta+\beta}{2}\|\nabla \mathbf{v}\|_{L^2}^2+C\left(1+\|\dot{ \mathbf{u}}_2\|_{L^3}^2 \right)\|\varphi\|_{L^2}^2\\ &\quad+C\left( 1+\| \mathbf{b}_1\|_{L^\infty}^2+\| \mathbf{b}_2\|_{L^\infty}^2+\|\nabla \mathbf{u}_2\|_{L^3}^2\right) \left(\| \mathbf{v}\|_{L^2}^2 +\| \varpi\|_{L^2}^2+\|\phi\|_{L^2}^2\right). \end{split} \end{equation*} |
Thus, the inequality (1.13), together with (2.50) and (2.56), leads to
\begin{equation} \begin{split} \frac{d}{dt}\|\rho_1^{1/2} \mathbf{v}\|_{L^2}^2+\|\nabla \mathbf{v}\|_{L^2}^2 \leqslant C\left(1+\|\dot{ \mathbf{u}}_2\|_{L^3}^2 \right)\|\varphi\|_{L^2}^2+C\left(1+ \|\rho_1^{1/2} \mathbf{v}\|_{L^2}^2+\|\rho_1^{1/2}\varpi\|_{L^2}^2+\|\phi\|_{L^2}^2 \right). \end{split} \end{equation} | (3.5) |
On the other hand, the inequality (2.1) _3 gives
\begin{equation} \begin{split} \rho_1 \varpi_t+\rho_1 \mathbf{u}_1\cdot\nabla \varpi-\eta^{\prime}\Delta \varpi-(\eta^{\prime}+\kappa^{\prime})\nabla {\text{div}} \varpi = -\varphi\dot{ \mathbf{w}}_2-\rho_1 \varpi\cdot\nabla \mathbf{w}_2-4\beta \varpi+2\beta {\rm rot} \mathbf{v}. \end{split} \end{equation} | (3.6) |
Multiply (3.6) by \varpi and integrate by parts, and we get
\begin{equation*} \begin{split} &\frac{1}{2}\frac{d}{dt}\|\rho_1^{1/2} \varpi\|_{L^2}^2+\eta^{\prime}\|\nabla \varpi\|_{L^2}^2+(\eta^{\prime}+\kappa^{\prime})\| {\text{div}} \varpi\|_{L^2}^2\\ & \leqslant C\|\varphi\|_{L^2}\|\dot{ \mathbf{w}}_2\|_{L^3}\|\varpi\|_{L^6}+C\| \mathbf{v}\|_{L^2}\|\nabla \mathbf{w}_2\|_{L^3}\| \varpi\|_{L^6} +C\| \mathbf{v}\|_{L^2}\|\nabla \varpi\|_{L^2}\\ & \leqslant \frac{\eta^{\prime}}{2}\|\nabla \varpi\|_{L^2}^2+C\|\dot{ \mathbf{w}}_2\|_{L^3}^2\|\varphi\|_{L^2}^2+C\left( 1+\|\nabla \mathbf{w}_2\|_{L^3}^2\right) \| \mathbf{v}\|_{L^2}^2, \end{split} \end{equation*} |
thus
\begin{equation} \frac{d}{dt}\|\rho_1^{1/2} \varpi\|_{L^2}^2+\|\nabla \varpi\|_{L^2}^2\leqslant C\|\dot{ \mathbf{w}}_2\|_{L^3}^2\|\varphi\|_{L^2}^2+C\left( 1+\|\nabla \mathbf{w}_2\|_{L^3}^2\right) \| \mathbf{v}\|_{L^2}^2. \end{equation} | (3.7) |
Note that
\begin{equation} \phi_t-\alpha \Delta \phi = - \mathbf{u}_1 \cdot \nabla \phi- \mathbf{v} \cdot \nabla \mathbf{b}_2+\phi\cdot \nabla \mathbf{u}_1+ \mathbf{b}_2 \cdot \nabla \mathbf{v}-\phi{\rm div} \mathbf{u}_1- \mathbf{b}_2 {\rm div} \mathbf{v}. \end{equation} | (3.8) |
Multiply (3.8) by \phi , and one can get form integration by parts that
\begin{equation} \begin{split} \frac{1}{2}\frac{d}{dt}\|\phi\|_{L^2}^2+\|\nabla \phi\|_{L^2}^2 & = \int ( - \mathbf{u}_1 \cdot \nabla \phi- \mathbf{v} \cdot \nabla \mathbf{b}_2+\phi \cdot \nabla \mathbf{u}_1+ \mathbf{b}_2 \cdot \nabla \mathbf{v}\\ &-\phi{\rm div} \mathbf{u}_1- \mathbf{b}_2 {\rm div} \mathbf{v} )\cdot \phi \; dx \triangleq\sum\limits_{i = 1}^6 J_i. \end{split} \end{equation} | (3.9) |
We deduce from (2.56) that
\begin{equation} J_1 = -\frac{1}{2}\int \mathbf{u}_1 \cdot \nabla (|\phi|^2)dx = \frac{1}{2}\int {\rm div} \mathbf{u}_1 |\phi|^2dx \leqslant \frac{\alpha}{8}\|\nabla \phi\|_{L^2}^2+\| \phi\|_{L^2}^2, \end{equation} | (3.10) |
\begin{equation} \begin{split} J_2 &\leqslant C \| \mathbf{v}\|_{L^2} \|\nabla \mathbf{b}_2\|_{L^3} \|\phi\|_{L^6}\leqslant \frac{\alpha}{8}\|\nabla \phi\|_{L^2}^2+C\left(1+\|\nabla^2 \mathbf{b}_2\|_{L^2}^2\right)\|\sqrt{\rho_1} \mathbf{v}\|_{L^2}^2, \end{split} \end{equation} | (3.11) |
and
\begin{equation} J_3+J_4+J_5 \leqslant \frac{\alpha}{4}\|\nabla \phi\|_{L^2}^2+C\left(1+\|\nabla^2 \mathbf{b}_2\|_{L^2}^2\right)\|\sqrt{\rho_1} \mathbf{v}\|_{L^2}^2+\| \phi\|_{L^2}^2. \end{equation} | (3.12) |
We have from (1.13) that
\begin{equation} \begin{split} J_6 &\leqslant C \| \mathbf{b}_2\|_{L^\infty} \|\nabla \mathbf{v}\|_{L^2} \|\phi\|_{L^2}\leqslant C_1 \|\nabla \mathbf{v}\|_{L^2}^2+C \| \mathbf{b}_2\|_{L^6}\|\nabla \mathbf{b}_2\|_{L^6} \|\phi\|_{L^2}^2\\ &\leqslant C_1 \|\nabla \mathbf{v}\|_{L^2}^2+C \left(1+\|\nabla^2 \mathbf{b}_2\|_{L^2}^2 \right) \|\phi\|_{L^2}^2. \end{split} \end{equation} | (3.13) |
Taking (3.10)–(3.13) into (3.9), one has
\begin{equation*} \begin{split} &\frac{d}{dt}\|\phi\|_{L^2}^2+\|\nabla \phi\|_{L^2}^2\leqslant C_1 \|\nabla \mathbf{v}\|_{L^2}^2+C \left(1+\|\nabla^2 \mathbf{b}_2\|_{L^2}^2 \right)\left( \|\sqrt{\rho_1} \mathbf{v}\|_{L^2}^2+\|\phi\|_{L^2}^2\right), \end{split} \end{equation*} |
which, combined with (3.3), (3.5), and (3.7), gives
\begin{equation} \begin{split} &\Big(\|\rho_1^{1/2} \mathbf{v}\|_{L^2}^2+\|\rho_1^{1/2} \varpi\|_{L^2}^2+\|\phi\|_{L^2}^2\Big)_t+\|\nabla \mathbf{v}\|_{L^2}^2+\|\nabla \varpi\|_{L^2}^2+\|\nabla \phi\|_{L^2}^2\\ &\leqslant C \left( \|\rho_1^{1/2} \mathbf{v}\|_{L^2}^2+\|\rho_1^{1/2}\varpi\|_{L^2}^2+\|\phi\|_{L^2}^2\right)+ Ct \left(1+\|\dot{ \mathbf{u}}_2\|_{L^3}^2+\|\dot{ \mathbf{w}}_2\|_{L^3}^2\right) \left(\int_0^t\|\nabla \mathbf{v}\|_{L^2}^2 ds\right). \end{split} \end{equation} | (3.14) |
Let
\Phi(t)\triangleq\left(\|\rho_1^{1/2} \mathbf{v}\|_{L^2}^2+\|\rho_1^{1/2}\varpi\|_{L^2}^2+\|\phi\|_{L^2}^2\right)+\int_0^t \left(\|\nabla \mathbf{v}\|_{L^2}^2+\|\nabla \varpi\|_{L^2}^2+\|\nabla \phi\|_{L^2}^2\right) ds, |
then, the inequality (3.14) gives
\begin{equation} \ell'(t)\leqslant \psi(t) \ell (t)\quad{\rm with}\quad \ell(0) = 0, \end{equation} | (3.15) |
where
\begin{equation*} \psi(t) \triangleq Ct \left(1+\|(\dot{\mathbf{ u}}_2, \dot{\mathbf{ w}}_2)\|_{L^3}^2\right)+C. \end{equation*} |
We know from (1.13) that \psi(t)\in L^1(0, T) ; thus, the inequality (3.15), combined with Gronwall's inequality, gives
\begin{equation*} \left( \| \mathbf{v}\|_{L^2}^2+\|\varpi\|_{L^2}^2+\|\phi\|_{L^2}^2\right)+\int_0^T\left(\|\nabla \mathbf{v}\|_{L^2}^2+\|\nabla \varpi\|_{L^2}^2+\|\nabla \phi\|_{L^2}^2\right)dt = 0, \quad \forall\; t \in [0, T], \end{equation*} |
so
\mathbf{v}(x, t) = 0, \quad \varpi(x, t) = 0, \quad \phi(x, t) = 0, \quad {\rm a.e.\; on}\; \mathbb{R}^3\times [0, T], |
which, combined with (3.3), gives
\varphi(x, t) = 0, \quad {\rm a.e.\; on}\; \mathbb{R}^3\times [0, T]. |
Thus, Theorem 1.1 is proved.
From the discussion in the previous sections, we have concluded that the three-dimensional compressible magneto-micropolar fluid system (1.1) possesses a global and unique solution in \mathbb{R}^3 , as follows:
For s\in [9/2, 6) , assume that (\rho_0, \mathbf{ u}_0, \mathbf{ w}_0, \mathbf{ b}_0) satisfies
\begin{equation*} \inf \rho_0(x) > 0, \quad \rho_0-1\in H^1\mathop{\cap}W^{1, s}, \quad (\mathbf{ u}_0, \mathbf{w}_0) \in H^1\mathop{\cap}W^{1, 3}, \quad \mathbf{ b}_0 \in H^1. \end{equation*} |
There exists a constant \varepsilon > 0 , depending on \eta, \kappa, \beta, \eta^{\prime}, \kappa^{\prime}, \gamma, \alpha, A, \inf \rho_0 , \sup \rho_0 , \|\nabla \mathbf{ u}_0\|_{L^2} , \|\nabla \mathbf{ w}_0\|_{L^2} , and \|\nabla \mathbf{ b}_0\|_{L^2} , such that if
\begin{equation*} \mathcal{S}_0\triangleq \|(\rho_0-1, \mathbf{ u}_0, \mathbf{ w}_0, \mathbf{ b}_0)\|_{L^2}^2\leqslant \varepsilon, \end{equation*} |
the systems (1.1)–(1.3) possess a global uniqueness solution (\rho, \mathbf{ u}, \mathbf{ w}, \mathbf{ b}) in \mathbb{R}^3\times (0, \infty) satisfying
\begin{equation*} \begin{cases} \rho-1\in C([0, T]; H^1 \mathop{\cap} W^{1, s}), \quad \inf \rho(x, t) > 0, \\ (\mathbf{ u}, \; \mathbf{ w}, \mathbf{ b})\in C([0, T]; L^2 \mathop{\cap} L^a) \quad 2\leqslant a < 6, \\ (\mathbf{ u}, \mathbf{ w})\in L^\infty(0, T; H^1 \mathop{\cap} W^{1, 3}) \mathop{\cap} L^2(0, T; H^2)\mathop{\cap} L^\ell(0, T; W^{1, \infty}), \\ \mathbf{ b}\in L^\infty([0, T]; H^1) \mathop{\cap} L^2(0, T; H^2), \\ (t^{1/2}\mathbf{ \dot{u}}, t^{1/2}\mathbf{ \dot{w}})\in L^\infty(0, T; L^2), \quad (t^{1/2}\nabla \mathbf{ \dot{u}}, t^{1/2}\nabla \mathbf{ \dot{w}})\in L^2(0, T; L^2), \\ (t^{1/2}\mathbf{ b}_t, \; t^{1/2}\nabla^2\mathbf{ b})\in L^\infty([0, T]; L^2), \quad (t^{1/2}\nabla \mathbf{ b}_t, \; t^{1/2}\nabla^3\mathbf{ b})\in L^2([0, T]; L^2), \end{cases} \end{equation*} |
with 1 < \ell < (4s)/(5s-6) .
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
We sincerely thank the Associate Editor and the anonymous referees for their carefully reading and helpful suggestions that led to the improvement of the paper.
M.Y. Zhang was partially supported by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2021QA049) and the Science and Technology Project of Weifang (2022GX006).
The author declares no conflict of interest.
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