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Research article

Global weighted regularity for the 3D axisymmetric non-resistive MHD system

  • Received: 19 April 2024 Revised: 24 June 2024 Accepted: 25 June 2024 Published: 28 June 2024
  • MSC : 35B65, 35Q35, 76D03

  • We consider the regularity criteria of axisymmetric solutions to the non-resistive MHD system with non-zero swirl in R3. By applying a new anisotropic Hardy-Sobolev inequality in mixed Lorentz spaces, we show that strong solutions to this system can be smoothly extended beyond the possible blow-up time T if the horizontal angular component of the velocity belongs to anisotropic Lorentz spaces.

    Citation: Wenjuan Liu, Zhouyu Li. Global weighted regularity for the 3D axisymmetric non-resistive MHD system[J]. AIMS Mathematics, 2024, 9(8): 20905-20918. doi: 10.3934/math.20241017

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  • We consider the regularity criteria of axisymmetric solutions to the non-resistive MHD system with non-zero swirl in R3. By applying a new anisotropic Hardy-Sobolev inequality in mixed Lorentz spaces, we show that strong solutions to this system can be smoothly extended beyond the possible blow-up time T if the horizontal angular component of the velocity belongs to anisotropic Lorentz spaces.


    Generally speaking, the three-dimensional incompressible non-resistive MHD system in Euclidean coordinates reads

    {tu+uu+PΔu=bb,(t,x)R+×R3,tb+ub=bu,divu=divb=0,(u,b)|t=0=(u0,b0), (1.1)

    where the unknowns u=(u1,u2,u3), b=(b1,b2,b3), and P represent the velocity of the fluid, the magnetic field, and the scalar pressure function, respectively. Physically, (1.1) governs the dynamics of the velocity and magnetic fields in electrically conducting fluids, such as plasmas, liquid metals, and salt water. It is frequently applied in astrophysics, geophysics, cosmology, and so forth. One may check the references [7,11,25] for more applications and numerical studies.

    Be aware that system (1.1) reduces to the classical Navier-Stokes equations when b is identically zero. The global regularity of the 3D Navier-Stokes equations with large initial data remains open and it is generally viewed as one of the most challenging open problems in fluid mechanics. As a result, various efforts are made to study the solutions by using axisymmetric methods.

    In this paper, we assume that the solution (u,b) of system (1.1) has the following axisymmetric form:

    {u(t,x)=ur(t,r,z)er+uθ(t,r,z)eθ+uz(t,r,z)ez,b(t,x)=br(t,r,z)er+bθ(t,r,z)eθ+bz(t,r,z)ez.

    Here,

    r=x21+x22,er=(x1r,x2r,0),eθ=(x2r,x1r,0),ez=(0,0,1).

    In the above, uθ is usually called the swirl component. We say u is without swirl if uθ=0.

    In recent years, a great deal of mathematical effort has been dedicated to the study of the 3D axisymmetric Navier-Stokes equations. For the case of uθ=0, Abidi [1] and Ladyzhenskaya [15] independently proved the existence, uniqueness, and regularities of generalized solutions. The first author of this paper in [19] obtained the global well-posedness of the inhomogeneous axisymmetric Navier-Stokes equations. For the case of uθ0, the authors need to impose some smallness conditions on the initial data. For more references, we recommend [4,28] and references therein.

    Many fruitful studies on the well-posedness problem of the MHD system (1.1) have been achieved in recent years, see [5,8,9,13] and references therein. Now we recall some results on the axisymmetric MHD equations. Lei [17] considered a family of special axisymmetric initial data with uθ0=br0=bz0=0 and showed the global well-posedness of system (1.1) without any smallness assumptions. Further improvement was made by Ai and Li [2], who weakened the initial regularity. When the angular velocity is not trivial, Liu [21] obtained the global well-posedness of system (1.1) provided that ruθ0L and bθ0rL32 are small enough. Later on, Zhang and Rao [27] improved this result by removing the smallness of bθ0rL32.

    Researchers are interested in the classical problem of finding regularity criteria of the axisymmetric MHD system. In [23], Li and Liu established the following regularity criteria for the 3D axisymmetric non-resistive MHD system in Lorentz spaces

    uθrsLq(0,T;Lp,(R3))<,3p+2q1+s,31+s<p.

    Later, by using some inequalities in anisotropic Lorentz spaces and the generalized Hardy-Sobolev inequality, this result was extended to anisotropic Lorentz spaces in [16]. More precisely, he proved that if the initial data (u0,b0)H2(R3), br0=bz0=0, and the horizontal swirl component of velocity satisfies

    uθrsLq(0,T;Lp1,x1(R)Lp2,x2(R)Lp3,x3(R))with2q+1p1+1p2+1p3=1+s,12<s0,31+s<pi,21+sq<, (1.2)

    or uθrsL(0,T;Lp1,x1(R)Lp2,x2(R)Lp3,x3(R)) with 1p1+1p2+1p3=1+s and the norm of uθrsL(0,T;Lp1,x1(R)Lp2,x2(R)Lp3,x3(R)) is sufficiently small, then the solution (u,b) can be smoothly extended beyond T. For more regularity criteria on the axisymmetric MHD system, see [12,18,20] and references therein.

    We can rewrite system (1.1) as

    {tur+(urr+uzz)ur(uθ)2r+rP=(Δ1r2)ur+(brr+bzz)br(bθ)2r,tuθ+(urr+uzz)uθ+uruθr=(Δ1r2)uθ+(brr+bzz)bθ+brbθr,tuz+(urr+uzz)uz+zP=Δuz+(brr+bzz)bz,tbr+(urr+uzz)br=(brr+bzz)ur,tbθ+(urr+uzz)bθ+uθbrr=(brr+bzz)uθ+urbθr,tbz+(urr+uzz)bz=(brr+bzz)uz,rur+urr+zuz=0,rbr+brr+zbz=0, (1.3)

    where the operator Δdef=2r2+1rr+2z2.

    In this paper, we consider the following initial data:

    u0=ur0er+uθ0eθ+uz0ez,b0=bθ0eθ.

    Thus, by using the uniqueness of local solutions to system (1.1), we conclude that br=bz=0 for all later times. Then, system (1.3) is equivalent to

    {tur+(urr+uzz)ur(uθ)2r+rP=(Δ1r2)ur(bθ)2r,tuθ+(urr+uzz)uθ+uruθr=(Δ1r2)uθ,tuz+(urr+uzz)uz+zP=Δuz,tbθ+(urr+uzz)bθ=urbθr,rur+urr+zuz=0. (1.4)

    We can also write the vorticity field w in cylindrical coordinates

    w=×u=wr(t,r,z)er+wθ(t,r,z)eθ+wz(t,r,z)ez,

    where

    wr=zuθ,wθ=zurruz,wz=r(ruθ)r.

    According to system (1.4), the quantity (wr,wθ,wz) verifies

    {twr+(urr+uzz)wr=(Δ1r2)wr+(wrr+wzz)ur,twθ+(urr+uzz)wθ=(Δ1r2)wθ+urrwθ+1rz(uθ)21rz(bθ)2,twz+(urr+uzz)wz=Δuz+(wrr+wzz)uz. (1.5)

    We notice that condition (1.2) is concerned with the case 12<s0. Thus, a natural and interesting problem is whether or not the range of indicator s in condition (1.2) can be extended. The goal of this paper is to give a positive answer. Inspired by [16,26], we obtain the regularity criteria of system (1.4) in anisotropic Lorentz spaces with 0s<. Let us state our main result.

    Theorem 1.1. Let (u,b) be an axially symmetric solution to the MHD system (1.1) associated with the initial data (u0,b0)Hm(R3), m3, and br0=bz0=0. If the horizontal swirl component of velocity satisfies

    uθrsLq(0,T;Lp1,x1(R)Lp2,x2(R)Lp3,x3(R))with2q+1p1+1p2+1p3=1+s,0s<,31+s<pi,21+sq<, (1.6)

    or

    uθrsL(0,T;Lp1,x1(R)Lp2,x2(R)Lp3,x3(R))<ϵ with1p1+1p2+1p3=1+s, (1.7)

    where ϵ=ϵ(s,ru0θ)<<1, then (u, b) can be smoothly extended beyond T.

    Remark 1.1. In [26], the authors established several new anisotropic Hardy-Sobolev inequalities in mixed Lebesgue spaces and mixed Lorentz spaces. They also derived regularity criteria of the 3D axisymmetric Navier-Stokes system. We extend the related regularity criteria to the MHD system. In addition, compared to the results in [23], thanks to the new anisotropic Hardy-Sobolev inequality, we generalize the result to the anisotropic Lorentz space.

    Remark 1.2. We extend the results in [16] to the case of 0s<.

    The remaining of this paper is organized as follows: In Section 2, we provide the definition of anisotropic Lorentz spaces and gather some elementary inequalities. The proof of Theorem 1.1 is given in Section 3.

    Notations. We shall always denote R3dx=2π0Rrdrdz and the letter C as a generic constant which may vary from line to line. The Fourier transform ˆf of a Schwartz function f on Rn is defined as ˆf(ξ)=(2π)n2Rneixξf(x) dx. Furthermore, for s0, we define Λsf by ^Λsf(ξ)=(ni=1|ξi|2)s2ˆf(ξ), where the notation Λ stands for the square root of the negative Laplacian (Δ)12. Similarly, we denote ^Λsxif(ξ)=|ξi|sˆf(ξ) and ^Λsx1,x2,xkf(ξ)=(ki=1|ξi|2)s2ˆf(ξ).

    First, we recall the definition of Lorentz spaces, see [24] for details. Given 1p<,1q, a measurable function f then belongs to the Lorentz spaces Lp,q(R3) if fLp,q(R3)<, where

    fLp,q(R3):={(0tq1|xR3:|f(x)>t|qpdt)1q,if q<,supt>0t|{xR3:|f(x)|>t}|1p,if q=.

    The anisotropic Lorentz space Lp,q(R3) was first introduced in [3,10,14], and its norm is determined by

    fLp,q(R3):=fLp1,q11Lp2,q22Lp3,q33(R3)=fLp1,q1x1(R)Lp2,q2x2(R)Lp3,q3x3(R).

    For the convenience of the reader, we present some technical lemmas which will be useful later.

    Lemma 2.1 (Hölder's inequality [10,14]). Let fLr1,s1(R3) and gLr2,s2(R3). Then, there exists a constant C>0 such that

    fgLr,s(R3)CfLr1,s1(R3)gLr2,s2(R3),

    where 1r=1r1+1r2, 1s=1s1+1s2, 0<r1,r2,s1,s2.

    Lemma 2.2 (Sobolev inequality [10,14]). Assume that 1l. It then holds that

    fLp,l(R3)CΛsfLr,l(R3),

    with 1<ri<pi< and

    3i=1(1ri1pi)=s.

    The subsequent lemmas are crucial in substantiating our findings.

    Lemma 2.3 ([26]). Suppose that Rn=Rk1×Rk2Rki×Rnij=1kj and nij=1kj. Let r1=x21+x2k1, r2=x2k1+1+x2k1+k2, , ri=x2i1j=1kj+1+x2ij=1kj, 0<p,q, 1<(pj)l, 1(qj)l and 0<αj<kj(pj)l, 1ji, 1lkj. Then, for all fC0(Rn), we have that it holds that

    f(x1,x2,xn)ij=1|rj|αjLp1,q1(Rk1)Lpi,qi(Rki)Lp,q(Rnij=1kj)CΛij=1αjx1,xij=1kjfLp1,q1(Rk1)Lpi,qi(Rki)Lp,q(Rnij=1kj).

    Lemma 2.4 (Gagliardo-Nirenberg inequality [6]). Let 0σ<s< and 1q,r. Then we have

    ΛσuLp(R3)CuθLq(R3)Λsu1θLr(R3),

    where 3pσ=θ3q+(1θ)(3rs) and 0θ1σs(θ0 if sσ3r).

    Lemma 2.5 ([22]). Assume that u is a smooth axisymmetric vector field and w=×u. Then it holds that

    urr=Δ1z(wθr)2rrΔ2z(wθr).

    In addition, for 1<p<, it is valid that

    urrLp(Rn)CwθrLp(Rn),

    and

    2urrLp(Rn)CzwθrLp(Rn).

    This section is devoted to the proof of Theorem 1.1. To begin, motivated by Li and Liu [23], the following lemma can be obtained.

    Lemma 3.1 (Continuation criterion [23]). Assume that (u0,b0)Hm(R3) with m3 and br0=bz0=0. Let (u,b) be an axially symmetric local solution of system (1.1). If

    supt[0,T)wθr(,t)L2(R3)<+,

    then the solution (u,b) can be smoothly extended beyond T.

    Now, we introduce the following new variables:

    Γ:=ruθ,Π:=bθr.

    By taking advantage of system (1.4), we obtain

    {tΓ+(urr+uzz)Γ=(Δ2rr)Γ,tΠ+(urr+uzz)Π=0. (3.1)

    The following proposition states fundamental estimates of system (1.1) which do not need the axisymmetric assumption.

    Proposition 3.1. Let (u,b) be a smooth solution of system (1.1) with (u0,b0)Hm(m3). Then, we have for any tR+,

    u(t)2L2+b(t)2L2+t0u(τ)2L2dτu02L2+b02L2, (3.2)
    Π(t)LpΠ0Lp,p[2,+], (3.3)

    and

    Γ(t)LpΓ0Lp,p[2,+]. (3.4)

    Proof. Taking the inner product of (1.1)1 and (1.1)2 with u and b, respectively, integrating by parts, and summing the results together, we get

    12ddt(u2L2+b2L2)+u2L20.

    Applying Gronwall's inequality leads to the desired result (3.2). The estimates for Γ and Π are classical for the heat equation when p<, and follow from the maximum principle when p=. We omit the details here, see [22].

    Now we are in a position to derive the estimates of (Ω,J). We deduce from system (1.5) that the pair (Ω,J)def=(wθr,wrr) satisfies

    {tΩ+(urr+uzz)Ω=(Δ+2rr)ΩzΠ22uθrJ,tJ+(urr+uzz)J=(Δ+2rr)J+(wrr+wzz)urr. (3.5)

    Proposition 3.2. Under the assumptions of Theorem 1.1, the following estimate of (Ω,J) holds:

    sup0tT(Ω(t)2L2+J(t)2L2)+T0(Ω(t)2L2+J(t)2L2) dt<.

    Proof. Multiplying (3.5)1 and (3.5)2 by Ω and J, respectively, integrating over R3, and using the divergence-free condition, we observe

    12ddt(Ω(t)2L2+J(t)2L2)+Ω(t)2L2+J(t)2L2=(wrr+wzz)urrJ dxzΠ2Ω dx2uθrJΩ dx=2π+0zuθrurrJ rdrdz+2π+0r(ruθ)rzurrJ rdrdzzΠ2Ω dx2uθrJΩ dx=uθrurrzJ dxuθzurrrJ dxzΠ2Ω dx2uθrJΩ dx|uθ||urr||J| dxzΠ2Ω dx2uθrJΩ dx:=I1+I2+I3. (3.6)

    For I2, integration by parts, Young's inequality, and (3.3) yield

    |I2|=|Π2zΩ dx|CΠLΠL2zΩL2CΠ02LΠ02L2+14Ω2L2.

    For any 31+s<pi, by using Lemmas 2.1 and 2.2, and (3.4) we achieve

    |I1|=|ruθ|ss+1|uθrs|11+s|urr||J| dxuθrs11+sLp1,x1Lp2,x2Lp3,x3(R3)ruθss+1Lx1Lx2Lx3(R3)urrL2p1(1+s)p1(1+s)2,2x1L2p2(1+s)p2(1+s)2,2x2L2p3(1+s)p3(1+s)2,2x3(R3)JL218J2L2+Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Γ02ss+1Lx1Lx2Lx3(R3)urr2L2p1(1+s)p1(1+s)2,2x1L2p2(1+s)p2(1+s)2,2x2L2p3(1+s)p3(1+s)2,2x3(R3)18J2L2+Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Λ3i=11pi(1+s)urr2L2.

    (i) under the assumption that (1.6) holds.

    We get from Lemmas 2.4 and 2.5, and Young's inequality that

    |I1|18J2L2+Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)urr23i=12pi(1+s)L2Λurr3i=12pi(1+s)L218J2L2+Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Ω23i=12pi(1+s)L2Ω3i=12pi(1+s)L218(J2L2+Ω2L2)+Cuθrs21+s3i=11piLp1,x1Lp2,x2Lp3,x3(R3)Ω2L2.

    The term I3 can be bounded by

    |I3||uθ|r|Ω|2 dx+|uθ|r|J|2 dx:=I31+I32. (3.7)

    We shall estimate I31 and I32 in the following two cases:

    Case 1. 0s<1

    Lemma 2.1 yields that

    |I31|=||uθ|rs|Ω|2r1s dx|CuθrsLp1,x1Lp2,x2Lp3,x3(R3)Ω2r1sLp1p11,1x1Lp2p21,1x2Lp3p31,1x3(R3)CuθrsLp1,x1Lp2,x2Lp3,x3(R3)Ωr1s22L2p1p11,2x1L2p2p21,2x2L2p3p31,2x3(R3).

    Due to 0<1s2<p11p1,p21p2, we invoke Lemma 2.3 with k1=2, k2=1, i=1 to get

    Ωr1s2L2p1p11,2x1L2p2p21,2x2L2p3p31,2x3(R3)CΛ1s2x1,x2ΩL2p1p11,2x1L2p2p21,2x2L2p3p31,2x3(R3).

    Noting that 03i=112pi+1s2<1 when 31+s<pi, Lemmas 2.2 and 2.4 allow us to conclude that

    Λ1s2x1,x2ΩL2p1p11,2x1L2p2p21,2x2L2p3p31,2x3(R3)CΛ3i=112pi+1s2ΩL2CΩ1+s3i=11pi2L2Ω1s+3i=11pi2L2,

    which implies

    |I31|CuθrsLp1,x1Lp2,x2Lp3,x3(R3)Ω1+s3i=11piL2Ω1s+3i=11piL2. (3.8)

    Similarly, we have

    |I32|CuθrsLp1,x1Lp2,x2Lp3,x3(R3)J1+s3i=11piL2J1s+3i=11piL2. (3.9)

    From (3.8), (3.9), and Young's inequality, we obtain

    |I3|CuθrsLp1,x1Lp2,x2Lp3,x3(R3)Ω1+s3i=11piL2Ω1s+3i=11piL2+uθrsLp1,x1Lp2,x2Lp3,x3(R3)J1+s3i=11piL2J1s+3i=11piL2Cuθrs21+s3i=11piLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2)+18(Ω2L2+J2L2).

    Substituting the above estimates into (3.6), we know that, for 0s<1,

    ddt(Ω(t)2L2+J(t)2L2)+Ω(t)2L2+J(t)2L2Cuθrs21+s3i=11piLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2). (3.10)

    Case 2. s1

    By applying Lemmas 2.1, 2.2, 2.4, and (3.4) we get

    |I31|=|uθrs|21+s|ruθ|s1s+1|Ω|2 dxCuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)ruθs1s+1Lx1Lx2Lx3(R3)Ω2Lp1(1+s)p1(1+s)2,1x1Lp2(1+s)p2(1+s)2,1x2Lp3(1+s)p3(1+s)2,1x3(R3)Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Γ0s1s+1Lx1Lx2Lx3(R3)Ω2L2p1(1+s)p1(1+s)2,2x1L2p2(1+s)p2(1+s)2,2x2L2p3(1+s)p3(1+s)2,2x3(R3)Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Ω23i=12pi(1+s)L2Ω3i=12pi(1+s)L2.

    Along the same line as the proof of I31, we infer that

    |I32|Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)J23i=12pi(1+s)L2J3i=12pi(1+s)L2.

    Collecting all estimates above, we conclude from Young's inequality that

    |I3|Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Ω23i=12pi(1+s)L2Ω3i=12pi(1+s)L2+uθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)J23i=12pi(1+s)L2J3i=12pi(1+s)L2Cuθrs21+s3i=11piLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2)+18(Ω2L2+J2L2).

    Putting all the estimates above into (3.6) yields for s1,

    ddt(Ω(t)2L2+J(t)2L2)+Ω(t)2L2+J(t)2L2Cuθrs21+s3i=11piLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2). (3.11)

    We obtain from (3.10) and (3.11) that, for 0s<,

    ddt(Ω(t)2L2+J(t)2L2)+Ω(t)2L2+J(t)2L2Cuθrs21+s3i=11piLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2), (3.12)

    which along with Gronwall's inequality leads to the desired result.

    (ii) under the assumption that (1.7) holds.

    By virtue of Lemma 2.5 and Young's inequality, we obtain

    |I1|18J2L2+Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Λurr2L2Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Ω2L2+18J2L2.

    For I3, similar to that in (3.7), Young's inequality yields that

    |I3||uθ|r|Ω|2 dx+|uθ|r|J|2 dx:=I31+I32.

    We will estimate I31 and I32 in the following two cases:

    Case 1. 0s<1

    Using Lemma 2.1, one finds

    |I31|CuθrsLp1,x1Lp2,x2Lp3,x3(R3)Ω2r1sLp1p11,1x1Lp2p21,1x2Lp3p31,1x3(R3)CuθrsLp1,x1Lp2,x2Lp3,x3(R3)Ωr1s22L2p1p11,2x1L2p2p21,2x2L2p3p31,2x3(R3).

    Since 0<1s2<p11p1,p21p2, we can apply Lemma 2.3 with k1=2, k2=1, i=1 to get

    Ωr1s2L2p1p11,2x1L2p2p21,2x2L2p3p31,2x3(R3)CΛ1s2x1,x2ΩL2p1p11,2x1L2p2p21,2x2L2p3p31,2x3(R3).

    From Lemma 2.2, we infer that

    Λ1s2x1,x2ΩL2p1p11,2x1L2p2p21,2x2L2p3p31,2x3(R3)CΩL2,

    which implies

    |I31|CuθrsLp1,x1Lp2,x2Lp3,x3(R3)Ω2L2. (3.13)

    Analogously to the treatments of (3.13), we get

    |I32|CuθrsLp1,x1Lp2,x2Lp3,x3(R3)J2L2. (3.14)

    (3.13) and (3.14) lead us to conclude that

    |I3|CuθrsLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2).

    Substituting the above estimates into (3.6), we get that, for 0s<1,

    ddt(Ω(t)2L2+J(t)2L2)+Ω(t)2L2+J(t)2L2Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Ω2L2+CuθrsLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2). (3.15)

    Case 2. s1

    Applying Lemmas 2.1 and 2.2, and (3.4), we infer that

    |I31|Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)ruθs1s+1Lx1Lx2Lx3(R3)Ω2L2p1(1+s)p1(1+s)2,2x1L2p2(1+s)p2(1+s)2,2x2L2p3(1+s)p3(1+s)2,2x3(R3)Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)Ω2L2.

    Similarly, we obtain

    |I32|Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)J2L2.

    Thus, we can see that

    |I3|Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2).

    Plugging the above estimates into (3.6), we know that, for s1,

    ddt(Ω(t)2L2+J(t)2L2)+Ω(t)2L2+J(t)2L2Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2). (3.16)

    We infer from (3.15) and (3.16) that, for 0s<,

    ddt(Ω(t)2L2+J(t)2L2)+Ω(t)2L2+J(t)2L2Cuθrs21+sLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2)+CuθrsLp1,x1Lp2,x2Lp3,x3(R3)(Ω2L2+J2L2). (3.17)

    We choose

    ϵ=(4C)max{1,1+s2}, (3.18)

    where C is a sufficiently large constant and C=C(s,ruθ0). Together with Gronwall's inequality, we obtain

    sup0tT(Ω(t)2L2+J(t)2L2)+T0(Ω(t)2L2+J(t)2L2) dtC.

    Thus, we complete the proof of Proposition 3.2.

    Now we are in a position to complete the proof of Theorem 1.1.

    Proof of Theorem 1.1. With the help of Lemma 3.1 and Proposition 3.2, we naturally infer that the solution (u,b) can be smoothly extended beyond T.

    Wenjuan Liu and Zhouyu Li: Conceptualization, Methodology, Validation, Writing-original draft, Writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank the anonymous referees for their suggestions which make the paper more readable. The work is partially supported by the National Natural Science Foundation of China (No. 11801443), Scientic Research Program Funded by Shaanxi Provincial Education Department (No. 22JK0475), Young Talent Fund of Association for Science and Technology in Shaanxi, China (No. 20230525) and Shaanxi Fundamental Science Research Project for Mathematics and Physics (Nos. 23JSQ046 and 22JSQ031).

    The authors have no relevant financial or non-financial interests to disclose. The authors have no competing interests to declare that are relevant to the content of this article.



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