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Research article

Magnetohydrodynamics approximation of the compressible full magneto- micropolar system

  • Received: 23 May 2022 Revised: 14 June 2022 Accepted: 20 June 2022 Published: 30 June 2022
  • MSC : 35B25, 35Q60, 76W05

  • In this paper, we will use the Banach fixed point theorem to prove the uniform-in-ϵ existence of the compressible full magneto-micropolar system in a bounded smooth domain, where ϵ is the dielectric constant. Consequently, the limit as ϵ0 can be established. This approximation is usually referred to as the magnetohydrodynamics approximation and is equivalent to the neglect of the displacement current.

    Citation: Jishan Fan, Tohru Ozawa. Magnetohydrodynamics approximation of the compressible full magneto- micropolar system[J]. AIMS Mathematics, 2022, 7(9): 16037-16053. doi: 10.3934/math.2022878

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  • In this paper, we will use the Banach fixed point theorem to prove the uniform-in-ϵ existence of the compressible full magneto-micropolar system in a bounded smooth domain, where ϵ is the dielectric constant. Consequently, the limit as ϵ0 can be established. This approximation is usually referred to as the magnetohydrodynamics approximation and is equivalent to the neglect of the displacement current.



    In this paper, we consider the three-dimensional magneto-micropolar fluid equations with fractional dissipation

    {tu+μ(Δ)αuχΔu+uubb+p2χ×v=0,tv+η(Δ)βvκv+4χv+uv2χ×u=0,tb+λ(Δ)γb+ubbu=0,u=0,b=0, (1.1)

    with an initial value

    t=0:u=u0(x),v=v0(x),b=b0(x),xR3. (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 1α=β=γ54. 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 1α=β32, Zhou [30] obtained the Serrin-type criteria uLpTLqx with 2αp+3q2α1 and 32α1<q. Later, Yuan [14] extended the above function space Lq to Bsq,. Recently, the regularity criterion involving u3,bLωTLqx 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 1α=β=γ32 and χ,κ0. Assume that (u0,v0,b0)H1(R3) and u0=b0=0. Furthermore, if

    u3,v,bLϱ(0,T;Lq(R3)),

    with

    2αϱ+3q34(2α1)+3(1ϵ)4q,  3+ϵ2α1<q,  0<ϵ13, (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 uv. 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

    uL3qC1u13L22u13L23u13Lq,  1q<, (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+κt0hv(τ)2L2dτ. (2.2)

    The proof is divided into two cases: 3+ϵ2α1<q< and q=. We first consider the case 3+ϵ2α1<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+κv(t)2L20.

    Integrating the above inequality with respect to t and then obtaining

    u(t)2L2+v(t)2L2+b(t)2L2+2t0(Λαu(τ)2L2+Λαv(τ)2L2+Λαb(τ)2L2+κv(τ)2L2)dτu02L2+v02L2+b02L2.

    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+κhv(t)2L2+χhu(t)2L2+4χhv2L2:=6i=1Ii, (2.3)

    where

    I1=R3(uu)Δhudx,I2=R3(bb)Δhudx,I3=R3(ub)Δhbdx,I4=R3(bu)Δhbdx,I5=R3(uv)Δhvdx,I6=2χR3[(×v)Δhu+(×u)Δhv]dx.

    Thanks to integration by parts and Cauchy's inequality, we arrive at

    I6=4χR3h(×u)hvdxχh(×u)2L2+4χhv2L2=χhu2L2+4χhv2L2. (2.4)

    For I1, we divide it into the following three items: I1i(i=1,2,3) as

    I1=2j,k=1R3ujjukΔhukdx+3j=1R3ujju3Δhu3dx+2k=1R3u33ukΔhukdx:=I11+I12+I13. (2.5)

    The divergence-free condition and integration by parts entail that

    I11=2i,j,k=1R3ujjuk2iiukdx=2i,j,k=1R3iujjukiukdx+122i,j,k=1R3juj|iuk|2dx=2i,j,k=1R3iujjukiukdx122i,k=1R33u3|iuk|2dx=R31u11u11u1dxR31u11u21u2dxR31u22u11u1dxR31u22u21u2dxR32u11u12u1dxR32u11u22u2dxR32u22u12u1dxR32u22u22u2dx122i,k=1R33u3|iuk|2dx=R31u11u11u1dxR32u22u22u2dx+R33u32u12u1dx+R33u31u21u2dx+R33u32u11u2dx122i,k=1R33u3|iuk|2dx=122j,k=1R33u3kujkujdxR33u31u12u2dx+R33u32u11u2dx=2j,k=1R3u323kujkujdx+R3u3(232u21u1+231u12u2)dxR3u3(232u11u2+231u22u1)dx, (2.6)

    and

    I12=3j=12l=1R3lujju3lu3dx=3j=12l=1R3luju32jlu3dx. (2.7)

    Therefore, we obtain

    |I1|CR3|u3||u||hu|dx. (2.8)

    From Hölder's inequality, Lemma 2.1, the Gagliardo-Nirenberg inequality, and Young's inequality, it follows that

    |I1|CR3|u3||u||hu|dxCu3LquLθ1huLθ2Cu3Lqhu23L2Δu13Lθ13huLθ2Cu3Lqhu2s13L2hΛαu2(1s1)3L2us23L2Λα+1u1s23L2hus3L2hΛαu1s3L2Cu3Lqu2s13L2hΛαu2(1s1)3L2us23L2Λα+1u1s23L2us3L2hΛαu1s3L2Cu3Lqu2s13+s23+s3L2Λα+1u1s23L2hΛαu2(1s1)3+1s3L2C[u3Lqu2s13+s23+s3L2Λα+1u1s23L2]m+16hΛαu(2(1s1)3+1s3)mL2, (2.9)

    where the constants 1<θ1,θ2,m,m< and 0s1,s2,s31 satisfy

    {1θ1+1θ2+1q=1,232=(132)s1+(1+α32)(1s1),23θ1/3=(132)s2+(1+α32)(1s2),23θ2=(132)s3+(1+α32)(1s3),1m+1m=1,(2(1s1)3+1s3)m=2. (2.10)

    Noting that 1α32 and 3+ϵ2α1<q, one solution to (2.10) can be written as

    {θ1=18q5q18ϵ,θ2=18q13q18(1ϵ),s1=11α,s2=19ϵαq,s3=113α3(1ϵ)αq,m=2αqq+3(1ϵ),m=2αq(2α1)q3(1ϵ). (2.11)

    To bound I3, we decompose it into three pieces as

    I3=2j,k=1R3ujjbkΔhbkdx+2j=1R3ujjb3Δhb3dx+3k=1R3u33bkΔhbkdx:=I31+I32+I33. (2.12)

    By using integrating by parts (see[31]), we have

    I31=2j,k,l=1R3[2llujjbkbk+luj2ljbkbk]dx122j,k,l=1R3[2ljujlbkbk+juj2llbkbk]dx. (2.13)

    Similarly, we have

    I32=2j,l=1R3[2llujjb3b3+luj2ljb3b3]dx122j,k,l=1R3[2ljujlb3b3+juj2llb3b3]dx, (2.14)

    and

    I33=3k=12l=1R3[23lu3lbkbk+lu323lbkbk]dx+123k=12j,l=1R3[2ljujlbkbk+juj2llbkbk]dx. (2.15)

    Collecting (2.13)–(2.15), it is easy to derive that

    |I3|CR3|b|(|u|+|b|)(|hu|+|hb|)dx. (2.16)

    Furthermore, we have

    |I2+I3+I4|CR3|b|(|u|+|b|)(|hu|+|hb|)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|CR3|b|(|u|+|b|)(|hu|+|hb|)dxCbLq|u|+|b|Lθ1|hu|+|hb|Lθ2CbLq(hu23L2Δu13Lθ13+hb23L2Δb13Lθ13)(huLθ2+hbLθ2)CbLq(u2s13L2hΛαu2(1s1)3L2us23L2Λα+1u1s23L2+b2s13L2hΛαb2(1s1)3L2bs23L2Λα+1b1s23L2)(us3L2hΛαu1s3L2+bs3L2hΛαb1s3L2)CbLq(u2s13L2+b2s13L2)(hΛαu2(1s1)3L2+hΛαb2(1s1)3L2)(us23L2+bs23L2)(Λα+1u1s23L2+Λα+1b1s23L2)(us3L2+bs3L2)(hΛαu1s3L2+hΛαb1s3L2)CbLq(uL2+bL2)2s13+s23+s3(Λα+1uL2+Λα+1bL2)1s23(hΛαuL2+hΛαbL2)2(1s1)3+1s3C[bLq(uL2+bL2)2s13+s23+s3(Λα+1uL2+Λα+1bL2)1s23]m+16(hΛαuL2+hΛαbL2)(2(1s1)3+1s3)m, (2.18)

    where the constants 1<θ1,θ2,m,m< and 0s1,s2,s31 satisfy (2.10).

    Similar to I3, we bound I5 as

    |I5|CR3|v|(|u|+|v|)(|hu|+|hv|)dx. (2.19)

    Using the same steps as (2.18), we obtain

    |I5|CR3|v|(|u|+|v|)(|hu|+|hv|)dxC[vLq(uL2+vL2)2s13+s23+s3(Λα+1uL2+Λα+1vL2)1s23]m+16(hΛαuL2+hΛαvL2)(2(1s1)3+1s3)m,

    where the constants 1<θ1,θ2,m,m< and 0s1,s2,s31 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+κhv(t)2L2Cu32αq(2α1)q3(1ϵ)Lqu2((2α1)q3)(2α1)q3(1ϵ)L2Λα+1u6ϵ(2α1)q3(1ϵ)L2+b2αq(2α1)q3(1ϵ)Lq(uL2+bL2)2((2α1)q3)(2α1)q3(1ϵ)(Λα+1uL2+Λα+1bL2)6ϵ(2α1)q3(1ϵ)+v2αq(2α1)q3(1ϵ)Lq(uL2+vL2)2((2α1)q3)(2α1)q3(1ϵ)(Λα+1uL2+Λα+1vL2)6ϵ(2α1)q3(1ϵ)C(u3Lq+bLq+vLq)2αq(2α1)q3(1ϵ)(uL2+bL2+vL2)2((2α1)q3)(2α1)q3(1ϵ)(Λα+1uL2+Λα+1bL2+Λα+1vL2)6ϵ(2α1)q3(1ϵ). (2.20)

    Set

    Θ1=2αq(2α1)q3(1ϵ),Θ2=2((2α1)q3)(2α1)q3(1ϵ),Θ3=6ϵ(2α1)q3(1ϵ). (2.21)

    Integrating (2.20) with respect to t, we obtain

    E(t)CJ0+Ct0(u3Lq+bLq+vLq)Θ1(uL2+bL2+vL2)Θ2(Λα+1uL2+Λα+1bL2+Λα+1vL2)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+κv(t)2L2+χu(t)2L2+4χv(t)2L2:=6i=1Ji, (2.23)

    where

    J1=R3(uu)Δudx,J2=R3(bb)Δudx,J3=R3(ub)Δbdx,J4=R3(bu)Δbdx,J5=R3(uv)Δvdx,J6=2χR3[(×v)Δu+(×u)Δv]dx.

    By integration by parts and Cauchy's inequality, we arrive at

    J6=4χR3(×u)vdxχ(×u)2L2+4χv2L2=χu2L2+4χv2L2. (2.24)

    For J1, we divide it into the following three items: J1i(i=1,2,3)

    J1=R3u33uΔhudx+2j=1R3ujjuΔudx+R3u33u233udx:=J11+J12+J13. (2.25)

    Integrating by parts and using the divergence-free condition yields

    J11=123k=12l=1R33u3luklukdx3k=12l=1R3lu33uklukdx, (2.26)
    J12=123j=13k,l=1R3jujluklukdx2j=13k,l=1R3lujjuklukdx, (2.27)

    and

    J13=123k=1R3(1u1+2u2)3uk3ukdx. (2.28)

    Therefore, we have

    |J1|CR3|hu||u|2dx. (2.29)

    From Hölder's inequality and Lemma 2.1, it follows that

    |J1|ChuL2u2L4ChuL2u232αL2Λαu32αL6ChuL2u232αL2hΛαu1αL2Λα+1u12αL2. (2.30)

    By using integrating by parts and the divergence-free condition, we have

    J3=3j,k,l=1R3l(ujjbk)lbkdx=3j,k,l=1R3(lujjbklbk+uj2ljbklbk)dx=3j,k,l=1R3bkl(lujjbk)dx=3j,k,l=1R3(bk2llujjbk+bkluj2jlbk)dx. (2.31)

    Then we arrive at

    |J3|CR3|b|(|u|+|b|)(|Δu|+|Δb|)dx. (2.32)

    Furthermore, we have

    |J2+J3+J4|CR3|b|(|u|+|b|)(Δu|+|Δb|)dx. (2.33)

    It follows from the same procedure (2.18) that

    |J2+J3+J4|CR3|b|(|u|+|b|)(|Δu|+|Δb|)dxCbLq|u|+|b|Lθ1|Δu|+|Δb|Lθ2CbLq(Δu23L2Δu13Lθ13+Δb23L2Δb13Lθ13)(ΔuLθ2+ΔbLθ2)CbLq(u2s13L2Λα+1u2(1s1)3L2us23L2Λα+1u1s23L2+b2s13L2Λα+1b2(1s1)3L2bs23L2Λα+1b1s23L2)×(us3L2Λα+1u1s3L2+bs3L2Λα+1b1s3L2)CbLq(uL2+bL2)2s13+s23+s3(Λα+1uL2+Λα+1bL2)2(1s1)3+1s23+1s3Cb2αq(2α1)q3Lq(u2L2+b2L2)+18(Λα+1u2L2+Λα+1b2L2), (2.34)

    where the constants 1<θ1,θ2,m,m< and 0s1,s2,s31 satisfy (2.10).

    Similar to J3, we bound J5 as

    |J5|CR3|v|(|u|+|v|)(|Δu|+|Δv|)dx. (2.35)

    The same procedure leads to (2.34) yields

    |J5|CR3|v|(|u|+|v|)(|Δu|+|Δv|)dxCv2αq(2α1)q3Lq(u2L2+v2L2)+18(Λα+1u2L2+Λα+1v2L2).

    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)+34Λα+1b(t)2L2+κv(t)2L2C(b2αq(2α1)q3Lq+v2αq(2α1)q3Lq)(u2L2+b2L2+v2L2)+ChuL2u232αL2hΛαu1αL2Λα+1u12αL2. (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)+34t0(Λα+1u(τ)2L2+Λα+1v(τ)2L2+Λα+1b(τ)2L2)dτ+t0κv(τ)2L2dτC+Ct0(b2αq(2α1)q3Lq+v2αq(2α1)q3Lq)(u2L2+b2L2+v2L2)dτ+Ct0huL2u232αL2hΛαu1αL2Λα+1u12αL2dτC+Ct0(b2αq(2α1)q3Lq+v2αq(2α1)q3Lq)(u2L2+b2L2+v2L2)dτ+Csup0τthuL2t0u232αL2hΛαu1αL2Λα+1u12αL2dτ. (2.37)

    From Young's inequality, it follows that

    Csup0τthuL2t0u232αL2hΛαu1αL2Λα+1u12αL2dτCsup0τthuL2[t0u2L2dτ]134α[t0hΛαu2L2dτ]12α[t0Λα+1u2L2dτ]14αCsup0τthuL2[t0u2α1+αL2Λα+1u21+αL2dτ]134α[t0hΛαu2L2dτ]12α[t0Λα+1u2L2dτ]14αCsup0τthuL2[t0hΛαu2L2dτ]12α[t0Λα+1u2L2dτ]14α+4α34α(1+α)Csup0τthuL2[(t0hΛαu2L2dτ)12+1][(t0Λα+1u2L2dτ)14+1]CE(t)[t0Λα+1u2L2dτ]14+Csup0τthuL2[t0Λα+1u2L2dτ]14+CE(t)+Csup0τthuL2CE(t)[t0Λα+1u2L2dτ]14+C(sup0τthu2L2+1)[t0Λα+1u2L2dτ]14+CE(t)+Csup0τthu2L2+CCE(t)[t0Λα+1u2L2dτ]14+C[t0Λα+1u2L2dτ]14+CE(t)+C. (2.38)

    Then, we have

    12(u(t)2L2+v(t)2L2+b(t)2L2)+34t0(Λα+1u(τ)2L2+Λα+1v(τ)2L2+Λα+1b(τ)2L2)dτ+t0κv(τ)2L2dτC+Ct0(b2αq(2α1)q3Lq+v2αq(2α1)q3Lq)(u2L2+b2L2+v2L2)dτ+CE(t)[t0Λα+1u2L2dτ]14+C[t0Λα+1u2L2dτ]14+CE(t)+C. (2.39)

    By using Hölder's inequality, Young's inequality, and (2.22), we deduce that

    CE(t)C+Ct0(u3Lq+bLq+vLq)Θ1(uL2+bL2+vL2)Θ2(Λα+1uL2+Λα+1bL2+Λα+1vL2)Θ3dτC+C[t0(u3Lq+bLq+vLq)2αq(2α1)q3(uL2+bL2+vL2)2dτ]Θ2[t0(Λα+1uL2+Λα+1bL2+Λα+1vL2)2dτ]12Θ3C+Ct0(u3Lq+bLq+vLq)2αq(2α1)q3(uL2+bL2+vL2)2dτ+116t0(Λα+1uL2+Λα+1bL2+Λα+1vL2)2dτ. (2.40)

    Similarly, it follows from (2.22) and Hölder's inequality and Young's inequality that

    CE(t)[t0Λα+1u2L2dτ]14C[t0Λα+1u2L2dτ]14+C[t0Λα+1u2L2dτ]14t0(u3Lq+bLq+vLq)Θ1(uL2+bL2+vL2)Θ22(Λα+1uL2+Λα+1bL2+Λα+1vL2)Θ3dτC[t0Λα+1u2L2dτ]14+C[t0Λα+1u2L2dτ]14[t0(u3Lq+bLq+vLq)2αq(2α1)q3(uL2+bL2+vL2)2dτ]Θ22[t0(Λα+1uL2+Λα+1bL2+Λα+1vL2)2dτ]Θ32C[t0Λα+1u2L2dτ]14+C[t0(Λα+1uL2+Λα+1bL2+Λα+1vL2)2dτ]2Θ3+14[t0(u3Lq+bLq+vLq)2αq(2α1)q3(uL2+bL2+vL2)2dτ]Θ2C[t0Λα+1u2L2dτ]14+C[t0(Λα+1uL2+Λα+1bL2+Λα+1vL2)2dτ]2Θ3+14[t0(u3Lq+bLq+vLq)Θ4(uL2+bL2+vL2)2dτ]3(2α1)q+3(1ϵ)124[(2α1)q3(1ϵ)]C+Ct0(u3Lq+bLq+vLq)Θ4(uL2+bL2+vL2)2dτ+116t0(Λα+1uL2+Λα+1bL2+Λα+1vL2)2dτ, (2.41)

    where Θ4=8αq3(2α1)q+3(1ϵ)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)+34t0(Λα+1u(τ)2L2+Λα+1v(τ)2L2+Λα+1b(τ)2L2)dτ+t0κv(τ)2L2dτC+Ct0(b2αq(2α1)q3Lq+v2αq(2α1)q3Lq)(u2L2+b2L2+v2L2)dτ+Ct0(u3Lq+bLq+vLq)Θ4(uL2+bL2+vL2)2dτ+Ct0(u3Lq+bLq+vLq)2αq(2α1)q3(uL2+bL2+vL2)2dτ+18[t0(Λα+1uL2+Λα+1bL2+Λα+1vL2)2dτ]C+Ct0(u3Θ4Lq+bΘ4Lq+vΘ4Lq)(uL2+bL2+vL2)2dτ+14t0(Λα+1u2L2+Λα+1b2L2+Λα+1v2L2)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+t0κv(τ)2L2dτC+Ct0(u3Θ4Lq+bΘ4Lq+vΘ4Lq)(uL2+bL2+vL2)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+t0κv(τ)2L2dτCexp[Ct0(u3Θ4Lq+bΘ4Lq+vΘ4Lq)dτ]<. (2.44)

    Finally, we consider the case q=. By repeating the above procedure, we derive that

    E(t)CJ0+Ct0(u3L+bL+vL)2α2α1(uL2+bL2+vL2)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+t0κv(τ)2L2dτCexp[Ct0(u38α3(2α1)L+b8α3(2α1)L+v8α3(2α1)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 1α=β=γ32 is studied. However, the regularity of the weak solution of the magnetic micropolar fluid equation when 1α,β,γ32 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.



    [1] G. Łukaszewicz, Micropolar fluids: theory and applications, Boston: Birkhäuser, 1999.
    [2] S. Kawashima, Smooth global solutions for two-dimensional equations of electro-magneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 207. https://doi.org/10.1007/BF03167869 doi: 10.1007/BF03167869
    [3] S. Kawashima, Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math., 10 (1986), 131–149. https://doi.org/10.21099/tkbjm/1496160397 doi: 10.21099/tkbjm/1496160397
    [4] S. Kawashima, Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid II, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 181–184. https://doi.org/10.3792/pjaa.62.181 doi: 10.3792/pjaa.62.181
    [5] S. Jiang, F. C. Li, Rigorous derivation of the compressible magnetohydrodynamic equations from the electromagnetic fluid system, Nonlinearity, 25 (2012), 1735–1752. https://doi.org/10.1088/0951-7715/25/6/1735 doi: 10.1088/0951-7715/25/6/1735
    [6] S. Jiang, F. C. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, 2013, arXiv: 1309.3668.
    [7] A. Milani, On a singular perturbation problem for the linear Maxwell equations, Rend. Sem. Mat. Univ. Politec. Torino, 38 (1980), 99–110.
    [8] A. Milani, Local in time existence for the complete Maxwell equations with monotone characteristic in a bounded domain, Annali di Matematica pura ed applicata, 131 (1982), 233–254. https://doi.org/10.1007/BF01765154 doi: 10.1007/BF01765154
    [9] A. Milani, The quasi-stationary Maxwell equations as singular limit of the complete equations: the quasi-linear case, J. Math. Anal. Appl., 102 (1984), 251–274. https://doi.org/10.1016/0022-247X(84)90218-X doi: 10.1016/0022-247X(84)90218-X
    [10] M. Stedry, O. Vejvoda, Small time-periodic solutions of equations of magnetohydrodynamics as a singularity perturbed problem, Aplikace matematiky, 28 (1983), 344–356. https://doi.org/10.21136/am.1983.104046 doi: 10.21136/am.1983.104046
    [11] M. Stedry, O. Vejvoda, Equations of magnetohydrodynamics of compressible fluid: periodic solutions, Aplikace matematiky, 30 (1985), 77–91. https://doi.org/10.21136/am.1985.104130 doi: 10.21136/am.1985.104130
    [12] M. Stedry, O. Vejvoda, Equations of magnetohydrodynamics: periodic solutions, Časopis pro pěstováni matematiky, 111 (1986), 177–184. https://doi.org/10.21136/cpm.1986.118275
    [13] D. Lauerová, The Rothe method and time periodic solutions to the Navier-Stokes equations and equations of magnetohydrodynamics, Aplikace matematiky, 35 (1990), 89–98. https://doi.org/10.21136/am.1990.104392 doi: 10.21136/am.1990.104392
    [14] F. Li, Y. Mu, Low Mach number limit of the full compressible Navier-Stokes-Maxwell system, J. Math. Anal. Appl., 412 (2014), 334–344. https://doi.org/10.1016/j.jmaa.2013.10.064 doi: 10.1016/j.jmaa.2013.10.064
    [15] R. Wei, B. Guo, Y. Li, Global existence and optimal convergence rates of solutions for 3D compressible magneto-micropolar fluid equations, J. Differ. Equations, 263 (2017), 2457–2480. https://doi.org/10.1016/j.jde.2017.04.002 doi: 10.1016/j.jde.2017.04.002
    [16] Z. Wu, W. Wang, The pointwise estimates of diffusion wave of the compressible micropolar fluids, J. Differ. Equations, 265 (2018), 2544–2576. https://doi.org/10.1016/j.jde.2018.04.039 doi: 10.1016/j.jde.2018.04.039
    [17] P. Zhang, Blow-up criterion for 3D compressible viscous magneto-micropolar fluids with initial vacuum, Bound. Value Probl., 2013 (2013), 160. https://doi.org/10.1186/1687-2770-2013-160 doi: 10.1186/1687-2770-2013-160
    [18] C. Jia, Z. Tan, J. Zhou, Well-posedness of compressible magneto-micropolar fluid equations, 2019, arXiv: 1906.06848v2.
    [19] Z. Song, The global well-posedness for the 3-D compressible micropolar system in the critical Besov space, Z. Angew. Math. Phys., 72 (2021), 160. https://doi.org/10.1007/s00033-021-01591-x doi: 10.1007/s00033-021-01591-x
    [20] T. Tang, J. Sun, Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum, Discrete Contin. Dyn. Syst. B, 26 (2021), 6017–6026. https://doi.org/10.3934/dcdsb.2020377 doi: 10.3934/dcdsb.2020377
    [21] J. Fan, Z. Zhang, Y. Zhou, Local well-posedness for the incompressible full magneto-micropolar system with vacuum, Z. Angew. Math. Phys., 71 (2020), 42. https://doi.org/10.1007/s00033-020-1267-z doi: 10.1007/s00033-020-1267-z
    [22] M. A. Fahmy, A new boundary element algorithm for a general solution of nonlinear space-time fractional dual phase-lag bio-heat transfer problems during electromagnetic radiation, Case Stud. Therm. Eng., 25 (2021), 100918. https://doi.org/10.1016/j.csite.2021.100918 doi: 10.1016/j.csite.2021.100918
    [23] M. A. Fahmy, A new boundary element formulation for modeling and simulation of three-temperature distributions in carbon nanotube fiber reinforced composites with inclusions, Math. Method. Appl. Sci., in press. https://doi.org/10.1002/mma.7312
    [24] M. A. Fahmy, A new BEM modeling algorithm for size-dependent thermopiezoelectric problems in smart nanostructures, Comput. Mater. Con., 69 (2021), 931–944. https://doi.org/10.32604/cmc.2021.018191 doi: 10.32604/cmc.2021.018191
    [25] M. A. Fahmy, Boundary element modeling of 3T nonlinear transient magneto-thermoviscoelastic wave propagation problems in anisotropic circular cylindrical shells, Compos. Struct., 277 (2021), 114655. https://doi.org/10.1016/j.compstruct.2021.114655 doi: 10.1016/j.compstruct.2021.114655
    [26] M. A. Fahmy, M. M. Almehmadi, F. M. Al Subhi, A. Sohail, Fractional boundary element solution of three-temperature thermoelectric problems, Sci. Rep., 12 (2022), 6760. https://doi.org/10.1038/s41598-022-10639-5 doi: 10.1038/s41598-022-10639-5
    [27] M. A. Fahmy, 3D Boundary element model for ultrasonic wave propagation fractional order boundary value problems of functionally graded anisotropic fiber-reinforced plates, Fractal Fract., 6 (2022), 247. https://doi.org/10.3390/fractalfract6050247 doi: 10.3390/fractalfract6050247
    [28] M. A. Fahmy, Boundary element and sensitivity analysis of anisotropic thermoelastic metal and alloy discs with holes, Materials, 15 (2022), 1828. https://doi.org/10.3390/ma15051828 doi: 10.3390/ma15051828
    [29] M. A. Fahmy, M. M. Almehmadi, Boundary element analysis of rotating functionally graded anisotropic fiber-reinforced magneto-thermoelastic composites, Open Eng., 12 (2022), 313–322. https://doi.org/10.1515/eng-2022-0036 doi: 10.1515/eng-2022-0036
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