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Research article

Blowup criterion for the Cauchy problem of 2D compressible viscous micropolar fluids with vacuum

  • Received: 19 July 2024 Revised: 29 August 2024 Accepted: 03 September 2024 Published: 06 September 2024
  • MSC : 76N10, 35Q35, 35B40, 35D35

  • In this study, we establish a regular criterion for the 2D compressible micropolar viscous fluids with vacuum that is independent of the velocity of rotation of the microscopic particles. Specifically, we show that if the density verifies ρL(0,T;L)<, then the strong solution will exist globally on R2×(0,T). Consequently, we generalize the results of Zhong (Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), no. 12, 4603–4615) to the compressible case. In particular, we don't need the additional compatibility condition.

    Citation: Dayong Huang, Guoliang Hou. Blowup criterion for the Cauchy problem of 2D compressible viscous micropolar fluids with vacuum[J]. AIMS Mathematics, 2024, 9(9): 25956-25965. doi: 10.3934/math.20241268

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  • In this study, we establish a regular criterion for the 2D compressible micropolar viscous fluids with vacuum that is independent of the velocity of rotation of the microscopic particles. Specifically, we show that if the density verifies ρL(0,T;L)<, then the strong solution will exist globally on R2×(0,T). Consequently, we generalize the results of Zhong (Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), no. 12, 4603–4615) to the compressible case. In particular, we don't need the additional compatibility condition.



    The 2D micropolar equations are a special case of the 3D micropolar equations as follows:

    {ρt+div(ρu)=0,(ρu)t+div(ρuu)+P(ρ)=(μ+κ)Δu+(μ+λκ)divu+2κw,(ρw)t+div(ρuw)+4κw=μΔw+2κ×u, (1.1)

    where ρ=ρ(x1,x2,t), u=(u1(x1,x2,t),u2(x1,x2,t),0), w=w(0,0,w3(x1,x2,t)) and P(ρ) denote the density, velocity, microrotational velocity, and pressure, respectively. The pressure P(ρ)=ργ where γ>1 is the specific heat ratio. The parameter κ>0 is the dynamics micro-rotation viscosity. The viscous constants μ, λ, κ, μ satisfy the physical hypothesis: μ>0, μ>0 and μ+λκ0. Here, and in what follows, =(2,1),×u=1u22u1,w=(2w,1w). In this paper, we consider an initial value problem of system (1.1) with the initial data:

    (ρ,u,w)(x,0)=(ρ0,u0,w0)xR2,(ρ,u,w)(x,t)(0,0,0)as|x|,t0. (1.2)

    The micropolar fluids are non-Newtonian fluids with nonsymmetric stress tensors (called polar fluids), which were first established by Eringen [3] in the 1960s. To put our results, let's introduce some results on the topics of well-posedness for the multidimensional micropolar fluids. For incompressible case, the authors [8,10,11,12] proved a blowup criterion, and the local and global existence of strong solutions for the whole plane and initial boundary value problem. For the compressible case, provied that initial energy is suitably small, the existence of global strong/classical solutions can be found in [4,13]. Recently, Huang et al. [5] investigated compressible micropolar fluids on a time-dependent domain with slip boundary conditions, which contains local strong and global weak solutions. On the other hand, some blowup criteria of the strong solutions for the 3D case are proven in [1,2]. However, the regularity and uniqueness of the weak solution to the micropolar model for large initial data are still open and challenge problems even in the two-dimensional case. Thus, it is important to study the possible mechanism of blow-up to the compressible micropolar model in R2. Motivated by [9,11], we will establish a blowup criterion of the compressible micropolar model (1.1) in terms of the LtLx norm of ρ for the 2D case; this gives the first blowup criterion for strong solutions of the 2D compressible micropolar with a vacuum.

    Denote ˉx(e+|x|2)1/2log1+η0(e+|x|2), with η0>0.

    The local well-posedness result to (1.1)–(1.2) can be similarly obtained by [7] as follows.

    Proposition 2.1. Suppose that the initial data (ρ0,ρ0u0,ρ0w0) satisfy

    ρ00, ˉxaρ0L1H1W1,q, u0L2,w0H1, ρ0w0L2, ρ0u0L2, (2.1)

    with q>2 and a>1. Then there exist T0,N>0, such that the problem (1.1)–(1.2) has a unique strong solution (ρ,u,w) on R2×(0,T0] satisfying

    {ρC([0,T0];L1H1W1,q),ˉxaρL(0,T0;L1H1W1,q),ρu,u,tρutL(0,T0;L2),ρw,tρwtL(0,T0;L2), wL(0,T0;H1)u,wL2(0,T0;H1)Lq+1q(0,T0;W1,q),tu,twL2(0,T0;W1,q),tρut,tρwt,tut,twtL2(R2×(0,T0)), (2.2)

    and

    inf0tT0BNρ(x,t)dx14R2ρ0(x,t)dx. (2.3)

    Our main result of this paper is stated as the following theorem:

    Theorem 2.1. Suppose that the initial data (ρ0,ρ0u0,ρ0w0) verifies

    ρ00, ˉxaρ0L1H1W1,q, u0L2,w0H1, ρ0w0L2, ρ0u0L2,

    with q>2 and a>1. Let (ρ,u,w) be a strong solution to the Cauchy problem (1.1)–(1.2). If 0<T< is the maximal time of existence, then

    limTTρL(0,T;L)=. (2.4)

    Remark 2.1. Chen [2] established a Serrin's blowup criterion for 3D compressible micropolar fluids, i.e.,

    limTT(ρL(0,T;L)+ρuLs(0,T;Lr))=, (2.5)

    where r and s satisfy 2s+3r1, 3<r. It is clear that the blowup criterion in (2.4) for compressible micropolar fluids is weaker than the one in (2.5). Especially, we also notice that (2.4) is no additional growth condition on the micro-rotational velocity and linear velocity. On the other hand, we also extend the incompressible case in Zhong [11] to the compressible case.

    Now, we introduce the viscous flux of the compressible micropolar equations as follows, F:=(2μ+λ)divuP(ρ), V1:=×u, V2:=w, which implies

    ΔF=div(ρ˙u), (μ+κ)ΔV1=(ρ˙u2κV2), μΔV24κV2=(ρ˙w2κV1).

    Similar to [2], we give the estimates of the effective viscous flux below.

    Lemma 2.1. Denote ˙u=ut+uu, ˙w=wt+uw. Let (ρ,u,w) be a strong solution of (1.1)–(1.2). Then for p2 there exists a positive constant C depending only on p, μ, λ, κ and μ such that

    FLp+V1Lp+V2LpC(ρ˙uLp+ρ˙wLp+uLp+wLp+wLp), (2.6)
    FLp(R2)+V1Lp+V2LpC(ρ˙uL2+ρ˙wL2+uL2+wL2+wL2)12p(uL2+wL2+P(ρ)L2)2p, (2.7)
    uLp+wLpC(ρ˙uL2+ρ˙wL2+uL2+wL2+wL2)12p(uL2+wL2+P(ρ)L2)2p+CP(ρ)Lp. (2.8)

    Now we shall prove Theorem 2.1 by contradiction arguments. So, we assume that the opposite holds, i.e.,

    limTTρL(0,T;L)M0<. (2.9)

    To begin with, we have the following basic energy estimate:

    sup0tT(P(ρ)L1+ρu2L2+ρw2L2)+T0(u2L2+w2H1)dtC. (2.10)

    Lemma 2.2. Under the condition (2.9), it holds that for 0tT

    sup0tT(u2L2+w2H1)+T0(ρ˙u2L2+ρ˙w2L2)dtC. (2.11)

    Proof. Multiplying (1.1)2 and (1.1)3 by ˙u and ˙w respectively, then integrating the resulting equations by parts over R2, we obtain after adding them together that

    (ρ|˙u|2+ρ|˙w|2)dx=˙uP(ρ)dx+μ˙uΔudx+(μ+λ)˙udivudxκ˙urot(rotu)dx+μ˙wΔwdx4κw˙wdx+2κw˙udx+2κrotu˙wdx=:8i=1Ii. (2.12)

    Similar to the estimates of [9], Ii(i=1,,6) can be given as

    6i=1Iiddt(P(ρ)divuμ2|u|2μ2|w|2μ+λ2(divu)2κ2|rotu|22κ|w|2)dx+C(w3L3+u3L3+w3L3),

    and that

    I7+I8=2κ(wrotut+wtrotu)dx+2κ(w(uu)+rotu(uw))dx=2κddtwrotudx,

    where we have used the fact that

    2κ(w(uu)+rotu(uw))dx=2κ(u11u12w+u22u12wu11u21wu22u21w)dx+2κ(u11u21w+u21u22wu12u11wu22u12w)dx=2κ(u11u12wu22u21w+u21u22wu12u11w)dx=2κ(1u12u1w+u1221u1w)dx+2κ(1u22u2w+u2212u2w)dx2κ(1u22u2w+u2221u2w)dx+2κ(1u12u1w+u1212u1w)dx=0.

    Now, putting Ii into (2.12), by taking advantage of (2.8), (2.9), Hölder's and Young inequalities, we are led to

    B(t)+(ρ|˙u|2+ρ|˙w|2)dxC(u2L2+w2L2)+C(u3L3+w3L3)+Cw4L212(ρ˙u2L2+ρ˙w2L2)+C(u4L2+w4H1+1), (2.13)

    where

    B(t):=μ2u2L2+μ+λ2divu2L2+μ2w2L2+κ2rotu2w2L2divuP(ρ)dxμ4u2L2+μ+λ2divu2L2+μ2w2L2+κ2rotu2w2L2C.

    Then, using (2.10) and Gronwall's inequality leads to (2.11), the conclusion follows.

    Lemma 2.3. Under the condition (2.9), it holds that

    sup0tTt(ρ˙u2L2+ρ˙w2L2)+T0(t˙u2L2+t˙w2H1)dtC. (2.14)

    Proof. Operating ˙uj[t+div(u)] and ˙w[t+div(u)] to (1.1)2 and (1.1)3 respectively, summing up and integrating by parts over R2, one has

    12ddt(ρ|˙u|2+ρ|˙w|2)dx=(μ+κ)˙uj[Δutj+div(uΔuj)]dx+μ˙w[Δwt+div(uΔw)]dx+(μ+λκ)˙uj[tjdivu+div(ujdivu)]dx˙uj[jPt(ρ)+div(jP(ρ)u)]dx+2κ˙uj[wt+i(uiw)]dx+2κ˙w[×ut+i(ui×u)]dx4κ˙w[wt+div(uw)]dx=:7i=1Ji. (2.15)

    By virtue of estimates in [9], one can similarly give Ji(i=1,,4) as follows

    4i=1Ji(μ|˙u|2(λ+μ)(div ˙u)2κ|×˙u|2μ2|˙w|2)+C(u4L4+w4L4+w4L4)

    and

    J5=2κ˙w(×˙u)dx2κ(uw(×˙u)+u˙uw)dx2κ˙w(×˙u)dx+κ4˙u2L2+Cu4L4+Cw4L4,J6=2κ˙w(×˙u)dx2κ(uu(˙w)+u˙w(×u))dx=2κ˙w(×˙u)dx,J7=4κ|˙w|2dx+4κ(u1˙w1w+u2˙w2w+u1w1˙w+u2w2˙w)dx=4κ|˙w|2dx+4κ(u1˙w1w+u2˙w2wu22w˙wu11w˙w)dx4κ(w˙w2u2+w˙w1u1)dx4κ|˙w|2dx+δ˙w2L2+Cu4L4+Cw4L4,

    where we have used the following facts that

    2κ(uw(×˙u)+u˙uw)dx=2κ(u11˙u21w+u21˙u22wu12˙u11wu22˙u12w)dx+2κ(u11˙u12w+u22˙u12wu11˙u21wu22˙u21w)dx=2κ(u21˙u22wu22˙u21w)dx+2κ(u11˙u12wu12˙u11w)dx=2κ(1u22˙u2w+u2221˙u2w2u21˙u2wu2212˙u2w)dx+2κ(1u12˙u1w+u1221˙u1w2u11˙u1wu1212˙u1w)dx=2κ(1u22˙u2w2u21˙u2w)dx+2κ(1u12˙u1w2u11˙u1w)dx2κ|u||˙u||w|dx,2κ(uu˙w+u˙w(×u))dx=2κ(u11u12˙w+u22u12˙wu11u21˙wu22u21˙w)dx+2κ(u11u21˙w+u21u22˙wu12u11˙wu22u12˙w)dx=2κ(u11u12˙wu12u11˙w)dx+2κ(u21u22˙wu22u21˙w)dx=2κ(1u12u1˙w+u1221u1˙w1u12u1˙wu1212u1˙w)dx+2κ(1u22u2˙w+u2221u2˙w1u22u2˙wu2212u2˙w)dx=0.

    Then, inserting Ji(i=1,,7) into (2.15), and choosing δ small enough, leads to

    ddt(ρ|˙u|2+ρ|˙w|2)dx+μ2˙u2L2+μ2˙w2L2+2κ˙w2L2C(ρ˙u2L2+ρ˙w2L2)+Cu2L2+Cw2L2+Cw2L2w2L2+C, (2.16)

    where we have used (2.8), and (2.9). Then, multiplying (2.28) by t, and integrating resultant over (0,T), we deduce (2.14) from (2.16), (2.10), (2.11) and Gronwall's inequality.

    Lemma 2.4. Under the condition (2.9), it holds that

    sup0tT(ρH1W1,q+tuH1+twH1)+T0(2uLq+2wLq)dtC. (2.17)

    Proof. First, |ρ|q(q>2) satisfies

    ddtρLqC(1+uL)ρLq+C2uLqC(1+uL)ρLq+Cρ˙uLq+CwLq, (2.18)

    where we used the fact that

    2uLqC(ρ˙uLq+PLq+wLq),2wLqC(ρ˙wLq+wLq+uLq), (2.19)

    which follows from the standard Lp-estimate for the following elliptic systems:

    (μ+κ)Δu+(μ+λκ)divu=ρ˙u+P(ρ)2κw, μΔw4κw=ρ˙w2κ×u.

    Next, it follows from the Gargliardo–Nirenberg inequality and Lemma 2.1 that

    divuL+V1LCF1L+CPL+CV1LC(q)+C(q)Fq2(q1)Lq+C(q)V1q2(q1)LqC(q)+C(q)(ρ˙uLq+wLq)q2(q1). (2.20)

    By using the Beale-Kato-Majda type inequality (cf. [6]), and (2.19), we are led to

    uLC(divuL+V1L)log(e+2uLq)+CuL2+CC(1+ρ˙uLq+wLq)log(e+ρLp). (2.21)

    Next, by taking advantage of Hölder's and Gagliardo–Nirenberg inequalities, (2.11) and (2.12) that

    T0ρ˙uLqdtT0(ρ˙uq+1qLq+tρ˙u2Lq)dtCT0(ρ˙u2L2+t˙u2L2+tq3+q22q2q3q22q+1)dtC, (2.22)

    and

    T0(u2Lq+w2Lq)dtCT0w4qL22w24qL2dt+CT0u4qL22u24qL2dtCT0(ρ˙w2L2+w2H1+u2L2+ρ˙u2L2)dtC, (2.23)

    which along with Gronwall's inequality, (2.23) and (2.19) that

    sup0tTρLqC,T0(2uq+1qLq+2wq+1qLq+t2u2Lq+t2w2Lq)dtC. (2.24)

    Finally, taking q=2 in (2.18), one gets from (2.11), (2.21), (2.24), and Gronwall's inequality that

    sup0tTρL2C. (2.25)

    The standard L2-estimate for the elliptic systems, (2.10), (2.12), (2.14) lead to

    sup0tT(t2u2L2+t2w2L2)Csup0tTt(ρ˙u2L2+ρ˙w2L2+P2L2+u2L2+w2H1)C.

    From which, by virtue of (2.24) and (2.25), the conclusion follows.

    With the aid of (2.17), the following x-weighted estimate of ρ(x,t) can be found in [9,Lemma 3.7].

    Lemma 2.5. Under the condition (2.9), it holds that for a>1, q>2 and 0TT,

    sup0tTˉxaρL1H1W1,qC. (2.26)

    Lemma 2.6. Under the condition (2.9), it holds that for 0TT,

    sup0tTt(ρut2L2+ρwt2L2)+T0(tut2L2+twt2L2)dtC. (2.27)

    Proof. It follows from (2.19), Hölder's and Young's inequalites that

    12ddt(ρ|ut|2+ρ|wt|2)dx+[(μ+κ)|ut|2+μ|wt|2]dx+(μ+λκ)(divut)2dx+4κ|wt|2dx4κ|wt|2dx+κ|ut|2dx+C(P|divu|+|P||u||)|divut|dx+CuL2ρut2L4+CwL2ρutL3ρwtL6+CρuL6ρwtL3wtL2+CwL2ρuL8uL4ρwtL8+Cρ14u2L12ρwtL32wL2+Cρu2L8wL4wtL2+CρuL6ρutL3utL2+CρuL6ρutL3u2L4+Cρ14u2L12ρutL32uL2+Cρu2L8uL4utL24κwt2L2+κut2l2+δ(ut2L2+wt2L2)+C(δ)(2u2L2+2w2L2)+C(ρut2L2+ρwt2L2)+Cˉxau2L2qq2ρ2(γ1)Lˉxaρ2Lq+C4κwt2L2+κut2L2+δ(ut,wt)2L2+C(ρut,ρwt,ρ˙u,ρ˙w)2L2+C, (2.28)

    where we have used the following fact that for any σ(0,1], and any s>2, see [9, (3.72)],

    ρσvLsσCρ3σ4sL4s2σρσ4sˉxσa4sL4sσvˉxσa4sL4sσC(ρvL2+vL2).

    Then, multiplying (2.28) by t and choosing δ small enough, we obtain (2.27) after using Gronwall's inequality, (2.11), and (2.14). The conclusion follows.

    Proof of Theorem 2.1. Similar to [12], by taking advantage of the estimates in Lemmas 2.2–2.6, we can define (ρ,u,w)(x,T)limtT(ρ,u,w)(x,t) as a new initial data, which starts from T. In view of the local well-posedness presented in Proposition 2.1, we can extend the local strong solution beyond T, which contradicts the definition of T. Thus, we complete the proof of Theorem 2.1.

    In this work, we show that if the density verifies ρL(0,T;L)<, then the strong solution of the 2D compressible micropolar viscous fluids with vacuum will exist globally on R2×(0,T). Consequently, we generalize the results of Zhong (Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), no. 12, 4603–4615) to the compressible case. In particular, we don't need the additional compatibility condition.

    Dayong Huang: Responsible for the review and editing of the manuscript, as well as the supervision of the research project; Guoliang Hou: Responsible for visualization and preparation of the original draft of the manuscript. All authors have read and approved the final version of the manuscript for publication.

    The authors declare that they have no competing interests.



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