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In this paper, we consider the three-dimensional magneto-micropolar fluid equations with fractional dissipation
{∂tu+μ(−Δ)αu−χΔu+u⋅∇u−b⋅∇b+∇p−2χ∇×v=0,∂tv+η(−Δ)βv−κ∇∇⋅v+4χv+u⋅∇v−2χ∇×u=0,∂tb+λ(−Δ)γb+u⋅∇b−b⋅∇u=0,∇⋅u=0,∇⋅b=0, | (1.1) |
with an initial value
t=0:u=u0(x),v=v0(x),b=b0(x),x∈R3. | (1.2) |
Here u=u(x,t), v=v(x,t), b=b(x,t)∈R3, and p=p(x,t)∈R are the velocity, micro-rotational velocity, magnetic fields, and scalar pressure, respectively. μ, χ, and 1λ represent the kinematic viscosity, vortex viscosity, and magnetic Reynolds number, respectively. η and κ are angular viscosities. α, β and γ are the parameters of the fractional dissipations corresponding to the velocity, micro-rotational velocity and magnetic field, respectively. The fractional Laplace operator (−Δ)α is defined through the Fourier transform as
^(−Δ)αf(ξ)=^Λ2αf=|ξ|2αˆf(ξ). |
The incompressible magneto-micropolar fluid equations have made analytic studies a great challenge but offer new opportunities due to their distinctive mathematical features. Regularity criteria for weak solutions are established by Fan and Zhong [1] in pointwise multipliers for 1≤α=β=γ≤54. Local and global well-posedness have been established in [2,3,4], respectively. For α=β=γ=1, we refer to [5,6,7] for the existence of strong solutions and weak solutions, respectively. In the study field of the magneto-micropolar fluid equations, regularity criteria for weak solutions and blow-up criteria for smooth solutions are very important topics. The readers may refer to regularity criteria of weak solutions in Morrey-Campanato space [8], in Lorentz space [9], Besov space [10], Triebel-Lizorkin space [11] and other regularity criteria for weak solutions [12,13,14,15], and [16,17] for blow-up criteria of smooth solutions in different function spaces, respectively. Serrin-type regularity criteria for weak solutions via the velocity fields and the gradient of the velocity field were established in Yuan [13], respectively. We may refer to [18,19,20] for global well-posedness. On the other hand, the global regularity of weak solutions to (1.1) with partial viscosities becomes more complex. In the case of 2D, we may refer to [22,23,24,25], and in the case of 3D, we may refer to [26,27].
If v=0 and χ=0, then (1.1) reduces to MHD equations with fractional dissipation. The MHD equations govern the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas, liquid metals, and salt water. We only recall regularity criteria for our purpose. If α,β>54, some regularity criteria have been established by Wu [28,29], which are given in terms of the velocity u. If 1≤α=β≤32, Zhou [30] obtained the Serrin-type criteria u∈LpTLqx with 2αp+3q≤2α−1 and 32α−1<q≤∞. Later, Yuan [14] extended the above function space Lq to Bsq,∞. Recently, the regularity criterion involving u3,b∈LωTLqx is given in [31]. We also refer to [32,33] for well-posedness and [34] for blow up criterion of smooth solutions.
Motivated by the Serrin-type regularity criterion of weak solutions to Navier-Stokes equations [35,36] and MHD equations [30,31]. The main purpose is to investigate the regularity criterion of weak solutions to the systems (1.1) and (1.2) in this paper and establish the Serrin-type regularity criterion of weak solutions involving partial components. We state our main result as follows:
Theorem 1.1. Let 1≤α=β=γ≤32 and χ,κ≥0. Assume that (u0,v0,b0)∈H1(R3) and ∇⋅u0=∇⋅b0=0. Furthermore, if
u3,v,b∈Lϱ(0,T;Lq(R3)), |
with
2αϱ+3q≤34(2α−1)+3(1−ϵ)4q, 3+ϵ2α−1<q≤∞, 0<ϵ≤13, | (1.3) |
then the solution (u,v,b) to the systems (1.1) and (1.2) remains smooth on [0,T].
Remark 1.2. Since the concrete values of the constants μ, η, and λ play no role in our proof, for this reason, we shall assume them to be all equal to one throughout this paper. For convenience of description, we define horizontal derivatives ∇h:=(∂1,∂2).
Remark 1.3. When v=0 and χ=0, the conclusion in Theorem 1.1 is reduced to the one in [31].
Remark 1.4. Compared with [31], the main difficulty in this paper comes from the nonlinear term u⋅∇v. In order to overcome the difficulty caused by the nonlinear term, owing to the energy functional (see (2.2)), we first use integrating by parts and ∇⋅u=0 to transform it into a control of the horizontal derivative, and then use Hölder's inequality, multiplicative Sobolev inequality, the Gagliardo-Nirenberg inequality, and Young's inequality to control the nonlinear term.
In this section, our main purpose is to complete the proof of Theorem 1.1. To this end, we introduce the following lemma:
Lemma 2.1. ([37]) The multiplicative Sobolev inequality
‖∇u‖L3q≤C‖∂1∇u‖13L2‖∂2∇u‖13L2‖∂3∇u‖13Lq, 1≤q<∞, | (2.1) |
holds.
In what follows, we prove Theorem 1.1.
Proof. Let
E(t):=‖∇hu(t)‖2L2+‖∇hv(t)‖2L2+‖∇hb(t)‖2L2+∫t0(‖∇hΛαu(τ)‖2L2+‖∇hΛαv(τ)‖2L2+‖∇hΛαb(τ)‖2L2)dτ+κ∫t0‖∇h∇⋅v(τ)‖2L2dτ. | (2.2) |
The proof is divided into two cases: 3+ϵ2α−1<q<∞ and q=∞. We first consider the case 3+ϵ2α−1<q<∞.
Taking the inner product of the first three equations of (1.1) with (u,v,b), and adding them up, using integrating by parts, the divergence-free condition, and Cauchy inequality, we obtain
12ddt(‖u(t)‖2L2+‖v(t)‖2L2+‖b(t)‖2L2)+‖Λαu(t)‖2L2+‖Λαv(t)‖2L2+‖Λαb(t)‖2L2+κ‖∇⋅v(t)‖2L2≤0. |
Integrating the above inequality with respect to t and then obtaining
‖u(t)‖2L2+‖v(t)‖2L2+‖b(t)‖2L2+2∫t0(‖Λαu(τ)‖2L2+‖Λαv(τ)‖2L2+‖Λαb(τ)‖2L2+κ‖∇⋅v(τ)‖2L2)dτ≤‖u0‖2L2+‖v0‖2L2+‖b0‖2L2. |
By multiplying the first three equations of (1.1) by Δhu, Δhv, and Δhb, respectively, and adding them up, using integrating by parts and the divergence-free condition, we have
12ddt(‖∇hu(t)‖2L2+‖∇hv(t)‖2L2+‖∇hb(t)‖2L2)+‖∇hΛαu(t)‖2L2+‖∇hΛαv(t)‖2L2+‖∇hΛαb(t)‖2L2+κ‖∇h∇⋅v(t)‖2L2+χ‖∇h∇u(t)‖2L2+4χ‖∇hv‖2L2:=6∑i=1Ii, | (2.3) |
where
I1=∫R3(u⋅∇u)⋅Δhudx,I2=−∫R3(b⋅∇b)⋅Δhudx,I3=∫R3(u⋅∇b)⋅Δhbdx,I4=−∫R3(b⋅∇u)⋅Δhbdx,I5=∫R3(u⋅∇v)⋅Δhvdx,I6=−2χ∫R3[(∇×v)⋅Δhu+(∇×u)⋅Δhv]dx. |
Thanks to integration by parts and Cauchy's inequality, we arrive at
I6=4χ∫R3∇h(∇×u)⋅∇hvdx≤χ‖∇h(∇×u)‖2L2+4χ‖∇hv‖2L2=χ‖∇h∇u‖2L2+4χ‖∇hv‖2L2. | (2.4) |
For I1, we divide it into the following three items: I1i(i=1,2,3) as
I1=2∑j,k=1∫R3uj∂jukΔhukdx+3∑j=1∫R3uj∂ju3Δhu3dx+2∑k=1∫R3u3∂3ukΔhukdx:=I11+I12+I13. | (2.5) |
The divergence-free condition and integration by parts entail that
I11=2∑i,j,k=1∫R3uj∂juk∂2iiukdx=−2∑i,j,k=1∫R3∂iuj∂juk∂iukdx+122∑i,j,k=1∫R3∂juj|∂iuk|2dx=−2∑i,j,k=1∫R3∂iuj∂juk∂iukdx−122∑i,k=1∫R3∂3u3|∂iuk|2dx=−∫R3∂1u1∂1u1∂1u1dx−∫R3∂1u1∂1u2∂1u2dx−∫R3∂1u2∂2u1∂1u1dx−∫R3∂1u2∂2u2∂1u2dx−∫R3∂2u1∂1u1∂2u1dx−∫R3∂2u1∂1u2∂2u2dx−∫R3∂2u2∂2u1∂2u1dx−∫R3∂2u2∂2u2∂2u2dx−122∑i,k=1∫R3∂3u3|∂iuk|2dx=−∫R3∂1u1∂1u1∂1u1dx−∫R3∂2u2∂2u2∂2u2dx+∫R3∂3u3∂2u1∂2u1dx+∫R3∂3u3∂1u2∂1u2dx+∫R3∂3u3∂2u1∂1u2dx−122∑i,k=1∫R3∂3u3|∂iuk|2dx=122∑j,k=1∫R3∂3u3∂kuj∂kujdx−∫R3∂3u3∂1u1∂2u2dx+∫R3∂3u3∂2u1∂1u2dx=−2∑j,k=1∫R3u3∂23kuj∂kujdx+∫R3u3(∂232u2∂1u1+∂231u1∂2u2)dx−∫R3u3(∂232u1∂1u2+∂231u2∂2u1)dx, | (2.6) |
and
I12=−3∑j=12∑l=1∫R3∂luj∂ju3∂lu3dx=3∑j=12∑l=1∫R3∂luju3∂2jlu3dx. | (2.7) |
Therefore, we obtain
|I1|≤C∫R3|u3||∇u||∇h∇u|dx. | (2.8) |
From Hölder's inequality, Lemma 2.1, the Gagliardo-Nirenberg inequality, and Young's inequality, it follows that
|I1|≤C∫R3|u3||∇u||∇h∇u|dx≤C‖u3‖Lq‖∇u‖Lθ1‖∇h∇u‖Lθ2≤C‖u3‖Lq‖∇h∇u‖23L2‖Δu‖13Lθ13‖∇h∇u‖Lθ2≤C‖u3‖Lq‖∇hu‖2s13L2‖∇hΛαu‖2(1−s1)3L2‖∇u‖s23L2‖Λα+1u‖1−s23L2‖∇hu‖s3L2‖∇hΛαu‖1−s3L2≤C‖u3‖Lq‖∇u‖2s13L2‖∇hΛαu‖2(1−s1)3L2‖∇u‖s23L2‖Λα+1u‖1−s23L2‖∇u‖s3L2‖∇hΛαu‖1−s3L2≤C‖u3‖Lq‖∇u‖2s13+s23+s3L2‖Λα+1u‖1−s23L2‖∇hΛαu‖2(1−s1)3+1−s3L2≤C[‖u3‖Lq‖∇u‖2s13+s23+s3L2‖Λα+1u‖1−s23L2]m′+16‖∇hΛαu‖(2(1−s1)3+1−s3)mL2, | (2.9) |
where the constants 1<θ1,θ2,m,m′<∞ and 0≤s1,s2,s3≤1 satisfy
{1θ1+1θ2+1q=1,2−32=(1−32)s1+(1+α−32)(1−s1),2−3θ1/3=(1−32)s2+(1+α−32)(1−s2),2−3θ2=(1−32)s3+(1+α−32)(1−s3),1m+1m′=1,(2(1−s1)3+1−s3)m=2. | (2.10) |
Noting that 1≤α≤32 and 3+ϵ2α−1<q≤∞, one solution to (2.10) can be written as
{θ1=18q5q−18ϵ,θ2=18q13q−18(1−ϵ),s1=1−1α,s2=1−9ϵαq,s3=1−13α−3(1−ϵ)αq,m=2αqq+3(1−ϵ),m′=2αq(2α−1)q−3(1−ϵ). | (2.11) |
To bound I3, we decompose it into three pieces as
I3=2∑j,k=1∫R3uj∂jbkΔhbkdx+2∑j=1∫R3uj∂jb3Δhb3dx+3∑k=1∫R3u3∂3bkΔhbkdx:=I31+I32+I33. | (2.12) |
By using integrating by parts (see[31]), we have
I31=2∑j,k,l=1∫R3[∂2lluj∂jbkbk+∂luj∂2ljbkbk]dx−122∑j,k,l=1∫R3[∂2ljuj∂lbkbk+∂juj∂2llbkbk]dx. | (2.13) |
Similarly, we have
I32=2∑j,l=1∫R3[∂2lluj∂jb3b3+∂luj∂2ljb3b3]dx−122∑j,k,l=1∫R3[∂2ljuj∂lb3b3+∂juj∂2llb3b3]dx, | (2.14) |
and
I33=3∑k=12∑l=1∫R3[∂23lu3∂lbkbk+∂lu3∂23lbkbk]dx+123∑k=12∑j,l=1∫R3[∂2ljuj∂lbkbk+∂juj∂2llbkbk]dx. | (2.15) |
Collecting (2.13)–(2.15), it is easy to derive that
|I3|≤C∫R3|b|(|∇u|+|∇b|)(|∇h∇u|+|∇h∇b|)dx. | (2.16) |
Furthermore, we have
|I2+I3+I4|≤C∫R3|b|(|∇u|+|∇b|)(|∇h∇u|+|∇h∇b|)dx. | (2.17) |
Similar to (2.13), it follows from Hölder's inequality, Lemma 2.1, Gagliardo-Nirenberg inequality, and Young's inequality that
|I2+I3+I4|≤C∫R3|b|(|∇u|+|∇b|)(|∇h∇u|+|∇h∇b|)dx≤C‖b‖Lq‖|∇u|+|∇b|‖Lθ1‖|∇h∇u|+|∇h∇b|‖Lθ2≤C‖b‖Lq(‖∇h∇u‖23L2‖Δu‖13Lθ13+‖∇h∇b‖23L2‖Δb‖13Lθ13)⋅(‖∇h∇u‖Lθ2+‖∇h∇b‖Lθ2)≤C‖b‖Lq(‖∇u‖2s13L2‖∇hΛαu‖2(1−s1)3L2‖∇u‖s23L2‖Λα+1u‖1−s23L2+‖∇b‖2s13L2‖∇hΛαb‖2(1−s1)3L2‖∇b‖s23L2‖Λα+1b‖1−s23L2)⋅(‖∇u‖s3L2‖∇hΛαu‖1−s3L2+‖∇b‖s3L2‖∇hΛαb‖1−s3L2)≤C‖b‖Lq(‖∇u‖2s13L2+‖∇b‖2s13L2)(‖∇hΛαu‖2(1−s1)3L2+‖∇hΛαb‖2(1−s1)3L2)⋅(‖∇u‖s23L2+‖∇b‖s23L2)(‖Λα+1u‖1−s23L2+‖Λα+1b‖1−s23L2)⋅(‖∇u‖s3L2+‖∇b‖s3L2)(‖∇hΛαu‖1−s3L2+‖∇hΛαb‖1−s3L2)≤C‖b‖Lq(‖∇u‖L2+‖∇b‖L2)2s13+s23+s3(‖Λα+1u‖L2+‖Λα+1b‖L2)1−s23⋅(‖∇hΛαu‖L2+‖∇hΛαb‖L2)2(1−s1)3+1−s3≤C[‖b‖Lq(‖∇u‖L2+‖∇b‖L2)2s13+s23+s3(‖Λα+1u‖L2+‖Λα+1b‖L2)1−s23]m′+16(‖∇hΛαu‖L2+‖∇hΛαb‖L2)(2(1−s1)3+1−s3)m, | (2.18) |
where the constants 1<θ1,θ2,m,m′<∞ and 0≤s1,s2,s3≤1 satisfy (2.10).
Similar to I3, we bound I5 as
|I5|≤C∫R3|v|(|∇u|+|∇v|)(|∇h∇u|+|∇h∇v|)dx. | (2.19) |
Using the same steps as (2.18), we obtain
|I5|≤C∫R3|v|(|∇u|+|∇v|)(|∇h∇u|+|∇h∇v|)dx≤C[‖v‖Lq(‖∇u‖L2+‖∇v‖L2)2s13+s23+s3(‖Λα+1u‖L2+‖Λα+1v‖L2)1−s23]m′+16(‖∇hΛαu‖L2+‖∇hΛαv‖L2)(2(1−s1)3+1−s3)m, |
where the constants 1<θ1,θ2,m,m′<∞ and 0≤s1,s2,s3≤1 satisfy (2.10).
Combining (2.3), (2.4), (2.9), (2.18), and (2.20), we arrive at
ddt(‖∇hu(t)‖2L2+‖∇hv(t)‖2L2+‖∇hb(t)‖2L2)+‖∇hΛαu(t)‖2L2+‖∇hΛαv(t)‖2L2+‖∇hΛαb(t)‖2L2+κ‖∇h∇⋅v(t)‖2L2≤C‖u3‖2αq(2α−1)q−3(1−ϵ)Lq‖∇u‖2((2α−1)q−3)(2α−1)q−3(1−ϵ)L2‖Λα+1u‖6ϵ(2α−1)q−3(1−ϵ)L2+‖b‖2αq(2α−1)q−3(1−ϵ)Lq(‖∇u‖L2+‖∇b‖L2)2((2α−1)q−3)(2α−1)q−3(1−ϵ)(‖Λα+1u‖L2+‖Λα+1b‖L2)6ϵ(2α−1)q−3(1−ϵ)+‖v‖2αq(2α−1)q−3(1−ϵ)Lq(‖∇u‖L2+‖∇v‖L2)2((2α−1)q−3)(2α−1)q−3(1−ϵ)(‖Λα+1u‖L2+‖Λα+1v‖L2)6ϵ(2α−1)q−3(1−ϵ)≤C(‖u3‖Lq+‖b‖Lq+‖v‖Lq)2αq(2α−1)q−3(1−ϵ)(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2((2α−1)q−3)(2α−1)q−3(1−ϵ)(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)6ϵ(2α−1)q−3(1−ϵ). | (2.20) |
Set
Θ1=2αq(2α−1)q−3(1−ϵ),Θ2=2((2α−1)q−3)(2α−1)q−3(1−ϵ),Θ3=6ϵ(2α−1)q−3(1−ϵ). | (2.21) |
Integrating (2.20) with respect to t, we obtain
E(t)≤CJ0+C∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)Θ1(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)Θ2(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)Θ3dτ, | (2.22) |
where J0=‖∇u(0)‖2L2+‖∇v(0)‖2L2+‖∇b(0)‖2L2.
By taking the inner product of the first three equations of (1.1) with (−Δu,−Δv,−Δb) and integrating by parts, the divergence-free condition, we have
12ddt(‖∇u(t)‖2L2+‖∇v(t)‖2L2+‖∇b(t)‖2L2)+‖Λα+1u(t)‖2L2+‖Λα+1v(t)‖2L2+‖Λα+1b(t)‖2L2+κ‖∇∇⋅v(t)‖2L2+χ‖∇∇u(t)‖2L2+4χ‖∇v(t)‖2L2:=6∑i=1Ji, | (2.23) |
where
J1=∫R3(u⋅∇u)⋅Δudx,J2=−∫R3(b⋅∇b)⋅Δudx,J3=∫R3(u⋅∇b)⋅Δbdx,J4=−∫R3(b⋅∇u)⋅Δbdx,J5=∫R3(u⋅∇v)⋅Δvdx,J6=−2χ∫R3[(∇×v)⋅Δu+(∇×u)⋅Δv]dx. |
By integration by parts and Cauchy's inequality, we arrive at
J6=4χ∫R3∇(∇×u)⋅∇vdx≤χ‖∇(∇×u)‖2L2+4χ‖∇v‖2L2=χ‖∇∇u‖2L2+4χ‖∇v‖2L2. | (2.24) |
For J1, we divide it into the following three items: J1i(i=1,2,3)
J1=∫R3u3∂3u⋅Δhudx+2∑j=1∫R3uj∂ju⋅Δudx+∫R3u3∂3u⋅∂233udx:=J11+J12+J13. | (2.25) |
Integrating by parts and using the divergence-free condition yields
J11=123∑k=12∑l=1∫R3∂3u3∂luk∂lukdx−3∑k=12∑l=1∫R3∂lu3∂3uk∂lukdx, | (2.26) |
J12=123∑j=13∑k,l=1∫R3∂juj∂luk∂lukdx−2∑j=13∑k,l=1∫R3∂luj∂juk∂lukdx, | (2.27) |
and
J13=123∑k=1∫R3(∂1u1+∂2u2)∂3uk∂3ukdx. | (2.28) |
Therefore, we have
|J1|≤C∫R3|∇hu||∇u|2dx. | (2.29) |
From Hölder's inequality and Lemma 2.1, it follows that
|J1|≤C‖∇hu‖L2‖∇u‖2L4≤C‖∇hu‖L2‖∇u‖2−32αL2‖Λαu‖32αL6≤C‖∇hu‖L2‖∇u‖2−32αL2‖∇hΛαu‖1αL2‖Λα+1u‖12αL2. | (2.30) |
By using integrating by parts and the divergence-free condition, we have
J3=−3∑j,k,l=1∫R3∂l(uj∂jbk)∂lbkdx=−3∑j,k,l=1∫R3(∂luj∂jbk∂lbk+uj∂2ljbk∂lbk)dx=3∑j,k,l=1∫R3bk∂l(∂luj∂jbk)dx=3∑j,k,l=1∫R3(bk∂2lluj∂jbk+bk∂luj∂2jlbk)dx. | (2.31) |
Then we arrive at
|J3|≤C∫R3|b|(|∇u|+|∇b|)(|Δu|+|Δb|)dx. | (2.32) |
Furthermore, we have
|J2+J3+J4|≤C∫R3|b|(|∇u|+|∇b|)(Δu|+|Δb|)dx. | (2.33) |
It follows from the same procedure (2.18) that
|J2+J3+J4|≤C∫R3|b|(|∇u|+|∇b|)(|Δu|+|Δb|)dx≤C‖b‖Lq‖|∇u|+|∇b|‖Lθ1‖|Δu|+|Δb|‖Lθ2≤C‖b‖Lq(‖Δu‖23L2‖Δu‖13Lθ13+‖Δb‖23L2‖Δb‖13Lθ13)(‖Δu‖Lθ2+‖Δb‖Lθ2)≤C‖b‖Lq(‖∇u‖2s13L2‖Λα+1u‖2(1−s1)3L2‖∇u‖s23L2‖Λα+1u‖1−s23L2+‖∇b‖2s13L2‖Λα+1b‖2(1−s1)3L2‖∇b‖s23L2‖Λα+1b‖1−s23L2)×(‖∇u‖s3L2‖Λα+1u‖1−s3L2+‖∇b‖s3L2‖Λα+1b‖1−s3L2)≤C‖b‖Lq(‖∇u‖L2+‖∇b‖L2)2s13+s23+s3(‖Λα+1u‖L2+‖Λα+1b‖L2)2(1−s1)3+1−s23+1−s3≤C‖b‖2αq(2α−1)q−3Lq(‖∇u‖2L2+‖∇b‖2L2)+18(‖Λα+1u‖2L2+‖Λα+1b‖2L2), | (2.34) |
where the constants 1<θ1,θ2,m,m′<∞ and 0≤s1,s2,s3≤1 satisfy (2.10).
Similar to J3, we bound J5 as
|J5|≤C∫R3|v|(|∇u|+|∇v|)(|Δu|+|Δv|)dx. | (2.35) |
The same procedure leads to (2.34) yields
|J5|≤C∫R3|v|(|∇u|+|∇v|)(|Δu|+|Δv|)dx≤C‖v‖2αq(2α−1)q−3Lq(‖∇u‖2L2+‖∇v‖2L2)+18(‖Λα+1u‖2L2+‖Λα+1v‖2L2). |
Combining (2.23), (2.24), (2.30), (2.34), and (2.36), we have
12ddt(‖∇u(t)‖2L2+‖∇v(t)‖2L2+‖∇b(t)‖2L2)+34(‖Λα+1u(t)‖2L2+‖Λα+1v(t)‖2L2)+34‖Λα+1b(t)‖2L2+κ‖∇∇⋅v(t)‖2L2≤C(‖b‖2αq(2α−1)q−3Lq+‖v‖2αq(2α−1)q−3Lq)(‖∇u‖2L2+‖∇b‖2L2+‖∇v‖2L2)+C‖∇hu‖L2‖∇u‖2−32αL2‖∇hΛαu‖1αL2‖Λα+1u‖12αL2. | (2.36) |
Integrating (2.36) over the interval (0,t) and using Hölder's inequality, it was deduced that
12(‖∇u(t)‖2L2+‖∇v(t)‖2L2+‖∇b(t)‖2L2)+34∫t0(‖Λα+1u(τ)‖2L2+‖Λα+1v(τ)‖2L2+‖Λα+1b(τ)‖2L2)dτ+∫t0κ‖∇∇⋅v(τ)‖2L2dτ≤C+C∫t0(‖b‖2αq(2α−1)q−3Lq+‖v‖2αq(2α−1)q−3Lq)(‖∇u‖2L2+‖∇b‖2L2+‖∇v‖2L2)dτ+C∫t0‖∇hu‖L2‖∇u‖2−32αL2‖∇hΛαu‖1αL2‖Λα+1u‖12αL2dτ≤C+C∫t0(‖b‖2αq(2α−1)q−3Lq+‖v‖2αq(2α−1)q−3Lq)(‖∇u‖2L2+‖∇b‖2L2+‖∇v‖2L2)dτ+Csup0≤τ≤t‖∇hu‖L2∫t0‖∇u‖2−32αL2‖∇hΛαu‖1αL2‖Λα+1u‖12αL2dτ. | (2.37) |
From Young's inequality, it follows that
Csup0≤τ≤t‖∇hu‖L2∫t0‖∇u‖2−32αL2‖∇hΛαu‖1αL2‖Λα+1u‖12αL2dτ≤Csup0≤τ≤t‖∇hu‖L2[∫t0‖∇u‖2L2dτ]1−34α[∫t0‖∇hΛαu‖2L2dτ]12α[∫t0‖Λα+1u‖2L2dτ]14α≤Csup0≤τ≤t‖∇hu‖L2[∫t0‖u‖2α1+αL2‖Λα+1u‖21+αL2dτ]1−34α[∫t0‖∇hΛαu‖2L2dτ]12α[∫t0‖Λα+1u‖2L2dτ]14α≤Csup0≤τ≤t‖∇hu‖L2[∫t0‖∇hΛαu‖2L2dτ]12α[∫t0‖Λα+1u‖2L2dτ]14α+4α−34α(1+α)≤Csup0≤τ≤t‖∇hu‖L2[(∫t0‖∇hΛαu‖2L2dτ)12+1][(∫t0‖Λα+1u‖2L2dτ)14+1]≤CE(t)[∫t0‖Λα+1u‖2L2dτ]14+Csup0≤τ≤t‖∇hu‖L2[∫t0‖Λα+1u‖2L2dτ]14+CE(t)+Csup0≤τ≤t‖∇hu‖L2≤CE(t)[∫t0‖Λα+1u‖2L2dτ]14+C(sup0≤τ≤t‖∇hu‖2L2+1)[∫t0‖Λα+1u‖2L2dτ]14+CE(t)+Csup0≤τ≤t‖∇hu‖2L2+C≤CE(t)[∫t0‖Λα+1u‖2L2dτ]14+C[∫t0‖Λα+1u‖2L2dτ]14+CE(t)+C. | (2.38) |
Then, we have
12(‖∇u(t)‖2L2+‖∇v(t)‖2L2+‖∇b(t)‖2L2)+34∫t0(‖Λα+1u(τ)‖2L2+‖Λα+1v(τ)‖2L2+‖Λα+1b(τ)‖2L2)dτ+∫t0κ‖∇∇⋅v(τ)‖2L2dτ≤C+C∫t0(‖b‖2αq(2α−1)q−3Lq+‖v‖2αq(2α−1)q−3Lq)(‖∇u‖2L2+‖∇b‖2L2+‖∇v‖2L2)dτ+CE(t)[∫t0‖Λα+1u‖2L2dτ]14+C[∫t0‖Λα+1u‖2L2dτ]14+CE(t)+C. | (2.39) |
By using Hölder's inequality, Young's inequality, and (2.22), we deduce that
CE(t)≤C+C∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)Θ1(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)Θ2(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)Θ3dτ≤C+C[∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)2αq(2α−1)q−3(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ]Θ2[∫t0(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)2dτ]12Θ3≤C+C∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)2αq(2α−1)q−3(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ+116∫t0(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)2dτ. | (2.40) |
Similarly, it follows from (2.22) and Hölder's inequality and Young's inequality that
CE(t)[∫t0‖Λα+1u‖2L2dτ]14≤C[∫t0‖Λα+1u‖2L2dτ]14+C[∫t0‖Λα+1u‖2L2dτ]14∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)Θ1(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)Θ22(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)Θ3dτ≤C[∫t0‖Λα+1u‖2L2dτ]14+C[∫t0‖Λα+1u‖2L2dτ]14[∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)2αq(2α−1)q−3(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ]Θ22[∫t0(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)2dτ]Θ32≤C[∫t0‖Λα+1u‖2L2dτ]14+C[∫t0(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)2dτ]2Θ3+14⋅[∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)2αq(2α−1)q−3(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ]Θ2≤C[∫t0‖Λα+1u‖2L2dτ]14+C[∫t0(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)2dτ]2Θ3+14⋅[∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)Θ4(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ]3(2α−1)q+3(1−ϵ)−124[(2α−1)q−3(1−ϵ)]≤C+C∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)Θ4(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ+116∫t0(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)2dτ, | (2.41) |
where Θ4=8αq3(2α−1)q+3(1−ϵ)−12.
We substitute (2.40) and (2.41) into (2.39) and then use Young's inequality to obtain
12(‖∇u(t)‖2L2+‖∇v(t)‖2L2+‖∇b(t)‖2L2)+34∫t0(‖Λα+1u(τ)‖2L2+‖Λα+1v(τ)‖2L2+‖Λα+1b(τ)‖2L2)dτ+∫t0κ‖∇∇⋅v(τ)‖2L2dτ≤C+C∫t0(‖b‖2αq(2α−1)q−3Lq+‖v‖2αq(2α−1)q−3Lq)(‖∇u‖2L2+‖∇b‖2L2+‖∇v‖2L2)dτ+C∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)Θ4(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ+C∫t0(‖u3‖Lq+‖b‖Lq+‖v‖Lq)2αq(2α−1)q−3(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ+18[∫t0(‖Λα+1u‖L2+‖Λα+1b‖L2+‖Λα+1v‖L2)2dτ]≤C+C∫t0(‖u3‖Θ4Lq+‖b‖Θ4Lq+‖v‖Θ4Lq)(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ+14∫t0(‖Λα+1u‖2L2+‖Λα+1b‖2L2+‖Λα+1v‖2L2)dτ. | (2.42) |
Then we have
‖∇u(t)‖2L2+‖∇v(t)‖2L2+‖∇b(t)‖2L2+∫t0(‖Λα+1u(τ)‖2L2+‖Λα+1v(τ)‖2L2+‖Λα+1b(τ)‖2L2)dτ+∫t0κ‖∇∇⋅v(τ)‖2L2dτ≤C+C∫t0(‖u3‖Θ4Lq+‖b‖Θ4Lq+‖v‖Θ4Lq)(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ. | (2.43) |
Thanks to Gronwall's inequality and condition (1.3), we obtain
‖∇u(t)‖2L2+‖∇v(t)‖2L2+‖∇b(t)‖2L2+∫t0(‖Λα+1u(τ)‖2L2+‖Λα+1v(τ)‖2L2+‖Λα+1b(τ)‖2L2)dτ+∫t0κ‖∇∇⋅v(τ)‖2L2dτ≤Cexp[C∫t0(‖u3‖Θ4Lq+‖b‖Θ4Lq+‖v‖Θ4Lq)dτ]<∞. | (2.44) |
Finally, we consider the case q=∞. By repeating the above procedure, we derive that
E(t)≤CJ0+C∫t0(‖u3‖L∞+‖b‖L∞+‖v‖L∞)2α2α−1(‖∇u‖L2+‖∇b‖L2+‖∇v‖L2)2dτ. |
Thanks to Gronwall's inequality and condition (1.3), we obtain
‖∇u(t)‖2L2+‖∇v(t)‖2L2+‖∇b(t)‖2L2+∫t0(‖Λα+1u(τ)‖2L2+‖Λα+1v(τ)‖2L2+‖Λα+1b(τ)‖2L2)dτ+∫t0κ‖∇∇⋅v(τ)‖2L2dτ≤Cexp[C∫t0(‖u3‖8α3(2α−1)L∞+‖b‖8α3(2α−1)L∞+‖v‖8α3(2α−1)L∞)dτ]<∞. | (2.45) |
By the above steps, we establish a higher-order a priori estimate of the solutions, and then we obtain that the higher-order norm of the solutions is bounded, thus proving the smoothness of the solutions. This completes the proof of Theorem 1.1.
In this paper, the regularity criterion of the weak solution of the three-dimensional magnetic micropolar fluid equation when 1≤α=β=γ≤32 is studied. However, the regularity of the weak solution of the magnetic micropolar fluid equation when 1≤α,β,γ≤32 on R3 is still an open problem, and it is hoped that the method in this paper can provide inspiration for the solution of this problem.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This research was supported by [the Basic Research Project of Key Scientific Research Project Plan of Universities in Henan Province (Grant No. 20ZX002)].
The authors declare there is no conflict of interest.
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