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Ubiquitin and ubiquitin-like modifiers modulate NK cell-mediated recognition and killing of damaged cells

  • Received: 28 September 2017 Accepted: 21 November 2017 Published: 23 November 2017
  • Efficient elimination of transformed and virus-infected cells by natural killer (NK) cells mainly depends on the recognition of “induced self” ligands by activating receptors, including NKG2D and DNAM1. The surface expression of these ligands in stressed or diseased cells results from the integration of transcriptional, post-transcriptional and post-translational mechanisms. Among post-translational mechanisms, recent findings indicate that ubiquitin and ubiquitin-like modifications, namely ubiquitination and SUMOylation, contribute to a very rapid negative regulation of NKG2D and DNAM1 ligand surface expression promoting either ligand degradation or ligand intracellular retention. On the other hand, accumulating evidences demonstrate that NKG2D receptor expression is down-regulated by ubiquitin-dependent endocytosis upon ligand stimulation. In this scenario, the overall consequence of the post-translational modifications of activating NK cell receptors and of their ligands on target cells is to impair effector cell-mediated recognition of damaged cells. Our review summarizes recent findings on the role of post-translational modifications in the modulation of target cell susceptibility to NK cell-mediated killing.

    Citation: Rosa Molfetta, Beatrice Zitti, Angela Santoni, Rossella Paolini. Ubiquitin and ubiquitin-like modifiers modulate NK cell-mediated recognition and killing of damaged cells[J]. AIMS Allergy and Immunology, 2017, 1(4): 164-180. doi: 10.3934/Allergy.2017.4.164

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  • Efficient elimination of transformed and virus-infected cells by natural killer (NK) cells mainly depends on the recognition of “induced self” ligands by activating receptors, including NKG2D and DNAM1. The surface expression of these ligands in stressed or diseased cells results from the integration of transcriptional, post-transcriptional and post-translational mechanisms. Among post-translational mechanisms, recent findings indicate that ubiquitin and ubiquitin-like modifications, namely ubiquitination and SUMOylation, contribute to a very rapid negative regulation of NKG2D and DNAM1 ligand surface expression promoting either ligand degradation or ligand intracellular retention. On the other hand, accumulating evidences demonstrate that NKG2D receptor expression is down-regulated by ubiquitin-dependent endocytosis upon ligand stimulation. In this scenario, the overall consequence of the post-translational modifications of activating NK cell receptors and of their ligands on target cells is to impair effector cell-mediated recognition of damaged cells. Our review summarizes recent findings on the role of post-translational modifications in the modulation of target cell susceptibility to NK cell-mediated killing.


    The motion dynamics of incompressible isotropic elastodynamics is characterized as a wave system in Lagrangian coordinates, which inherently satisfies the null condition. Based on this structure, a series of studies have been conducted to establish the global well-posedness of classical solutions to this system; see[1,2]. However, these studies reveal a certain time growth for the highest-order generalized energy. In this paper, we investigate the time growth of the Sobolev norm for classical solutions to three-dimensional inhomogeneous incompressible isotropic elastodynamics with small initial perturbation and establish the uniform bound for the highest-order energy.

    Before presenting the main result of this paper, we briefly review related known results. For three-dimensional elastic waves, John [3] proved that the genuine nonlinearity condition leads to singularity formation even for arbitrarily small spherically symmetric displacement. We also refer readers to [4] regarding large displacement singularity. The existence of almost global solutions was established in [5,6] for three-dimensional quasilinear wave equations with sufficiently small initial data. Significant contributions toward global existence were independently made by Sideris [7,8] and Agemi [9] under the assumption that nonlinearity satisfies the null condition in three dimensions. In terms of three-dimensional incompressible elastodynamics, the only waves presented in the isotropic systems are shear waves, which are linearly degenerate. The global existence of a solution was demonstrated by Sideris and Thomases in [1,2] through two different methods. It is more challenging to obtain the global existence for the two-dimensional incompressible elastodynamics due to the weaker dispersive decay. In [10], the authors proved almost global existence for a two-dimensional incompressible system in Eulerian coordinates. By introducing the concept of strong null condition and observing that the incompressible elastodynamics automatically satisfies such strong null structure in Lagrangian coordinates, Lei [11] successfully proved the global well-posedness for two-dimensional incompressible elastodynamics by the method of Klainerman and Alinhac's ghost weight method [12]. We also see [13] for a different approach using the spacetime resonance method. All the aforementioned works considered the homogeneous fluids. In [14], the authors established the global well-posedness for the three-dimensional inhomogeneous incompressible elastodynamics in Lagrangian coordinates. It is noteworthy that the upper bound of the highest-order generalized energy in those studies depends on time. Utilizing the Klainerman's generalized energy method, an analysis of the inherent structure of the system and the ghost weight method, [15,16] established the uniform bound for the highest-order generalized energy estimates for two-dimensional and three-dimensional incompressible elastodynamics, respectively. Based on the above foundational works, it is natural to verify the uniform bound for the highest-order generalized energy for three-dimensional inhomogeneous incompressible isotropic elastodynamics. To establish the time growth of the Sobolev norm of classical solution, two novel methods are presented in this paper. First, based on the Sobolev embedding inequality and the structure of the system, the refined decay rates were derived for the solution in the domain away from the light cone. Second, we apply the KSS-type estimate to overcome the difficulties posed by insufficient time decay resulting from density perturbation.

    This paper is organized as follows. In Section 2, we introduce the system of three-dimensional inhomogeneous isotropic elastodynamics and define the notations utilized throughout this paper. Besides, the main result along with several useful lemmas are presented in this section. The energy estimates are discussed in Section 3.

    We first formulate the inhomogeneous isotropic elastodynamics and denote some notations that are used frequently in this paper.

    For any given smooth flow map X(t,x), we call it incompressible if

    Ωdx=ΩtdX,Ωt={X(t,x)|xΩ}

    for any smooth bounded connected domain Ω, which yields that

    det(X)=1.

    Denote

    X(t,x)=x+v(t,x).

    Simple calculation shows that the incompressible condition is equivalent to

    v+12[(trv)2tr(v)2]+det(v)=0. (2.1)

    Without loss of generality, we assume that the density of fluid is a small perturbation around the constant state 1, that is, ρ(x)=1+η(x). For the inhomogeneous isotropic material, the motion of the elastic fluid in the Lagrangian coordinate is determined by

    L(X;T,Ω)=T0Ω(12ρ(x)|tX|2W(X)+p(t,x)[det(X)1])dxdt. (2.2)

    Here W(X)C is the strain energy function. p(t,x) is a Lagrangian multiplier that is used to force the flow maps to be incompressible. To simplify the presentation, we only study the typical Hookean case for which the strain energy functional is given by

    W(X)=12|X|2.

    By calculating the variation of (2.2), we obtain the equation

    ρ2tvΔv=(X)Tp. (2.3)

    Now, we introduce the following derivative vector fields

    t=0,=(1,2,3)and=(0,1,2,3).

    The scaling operator is denoted by

    S=tt+rr.

    Here, the radial derivative is defined by r=xr, r=|x|. The angular momentum operators are denoted by

    Ω=x.

    In the application, we usually use the modified rotational operators and scaling operator; that is, for any vector v and scalar p and ρ, we set

    ˜Sp=Sp,˜Sρ=Sρ,˜Sv=(S1)v,˜Ωip=Ωip,˜Ωiρ=Ωiρ,˜Ωiv=Ωiv+Uiv,i=1,2,3,

    where

    U1=e2e3e3e2,U2=e3e1e1e3,U3=e1e2e2e1.

    Let

    Γ=(Γ1,,Γ8)=(,˜Ω,˜S)

    and for any multi-index α=(α1,α2,,α8)N8, we denote

    Γα=Γα11Γα88.

    We apply the derivatives Γα to the equations (2.1) and (2.3), and then the three-dimensional inhomogeneous isotropic elastodynamics can be written as

    ρ2tΓαvΔΓαv=(X)TΓαpβ+γ=α|β||α|CβαΓγη2tΓβvβ+γ=α|β||α|Cβα(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)=:Nα (2.4)

    with the incompressible condition

    Γαv+β+γ=αi,j=1,2,3,i<jCβα(iΓβvijΓγvjiΓβvjjΓγvi)+β+γ+ι=αCβαCγαβ|1Γβv12Γβv13Γβv11Γβv22Γβv23Γβv21Γβv32Γβv33Γβv3|=0. (2.5)

    Here the binomial coefficient Cβα is given by

    Cβα=α!β!(αβ)!.

    We denote the Klainerman's generalized energy by

    Eκ(t)=|α|κ1R3(|tΓαv|2+|Γαv|2)dx.

    We also define the weighted energy norm

    Xκ(t)=|α|κ2R3tr2|2Γαv|2dx.

    To describe the space of initial data, we introduce the time-independent analogue of Γ as

    Λ=(,rr1,˜Ω)

    and the space of initial data is defined by

    HκΛ={(f,g):|α|κ1(ΛαfL2+ΛαfL2+ΛαgL2)<}.

    As the first step to investigate our problem, we introduce the following lemma, which helps us to solve the additional terms resulting from density perturbation. Let

    LEκ(T)=|α|κ1T0Rnr1+2μr2μ(|Γαv|+|Γαv|r)2dxdt

    with μ(0,12) and μ>μ. Without loss of generality, we choose μ=14 and μ=12 in this paper. In the case of μ=μ, we see LEκ(T) is the KSS norm, and we denote it by KSSκ(T).

    Lemma 1. Let f0=[r/(1+r)]2μ, fk=r/(r+2k) with k1, μ(0,1/2), and v be the solution to the equation 2tvc2Δv+hababv=N in [0,T]×Rn with hab=hba, 0a,bn|hab|min(1,c2)/2 for any integer n3. Then there exists a positive constant C0 that depends only on the dimension n such that

    sup0tTRn|v|2(t)dx+LE1(T)+(ln(2+T))1KSS1(T)C0Rn|v|2(0)dx+C0T0Rn[(|h|+|h|r12μr2μ)|v|(|v|+|v|r)]dxdt+C0|T0RntvNdxdt|+C0supk0|T0Rnfk(rv+n12rv)Ndxdt|,

    where |h|=na,b=0|hab| and |h|=na,b,c=0|chab|.

    This lemma can be found in [17]. See also [18,19] and references therein.

    Based on the previous statement, we are ready to show the main result of this paper.

    Theorem 1. Let W(X)=12|X|2 be an isotropic Hookean strain energy function and (v0,tv0)HκΛ with κ12. Let C0>0 be given constant in Lemma 1. Suppose v0 satisfies the structural constraint condition (2.1) and

    Eκ(0)=|α|κ1(tΛαv02L2+Λαv02L2)ε.

    If

    rΛαηL2δfor|α|κ,

    then there exist two sufficiently small constants, ε0,δ0 and constant C1, that depend only on κ and C0 such that if εε0 and δδ0, the system (2.3) has a unique global classical solution that satisfies

    Eκ(t)C1ε

    uniformly for all t[0,+).

    In this part, we establish several lemmas that are crucial for the energy estimates. Throughout this paper, we denote =(1+||2)12. The notation fg stands for fCg for some generic constant C>0, which may vary from line to line. In the process of deriving the energy estimates, we usually separate the whole integration domain R3 into two parts:

    R={xR3:rt/8},Rc={xR3:r>t/8}.

    We first recall the Sobolev-type inequalities, which were justified in [8].

    Lemma 2. For any vC0(R3)3, r=|x| and ˜r=|y|, we have

    r12v(x)L|α|1˜ΩαvL2,rv(x)L|α|1r˜Ωαv12L2(|y|r)˜Ωαv12L2(|y|r),rtrv(x)L|α|1t˜r˜r˜ΩαvL2(|y|r)+|α|2t˜r˜ΩαvL2(|y|r).

    The following lemma concerns the dispersive decay of solutions in the domain away from the light cone.

    Lemma 3. For any vH2(R3), there holds

    tvL(rt/8)vL2(rt/4)+trvL2(rt/4)+tr2vL2(rt/4).

    Proof. The proof can be found in Lemma 4.3 in [16].

    Let s=8rt. We introduce a radial cutoff function ξ(s)C0 that satisfies

    ξ(s)={1,s1,0,s2.

    It is easy to observe from Lemmas 2 and 3 that

    tvLtvL(R)+r(1ξ(s))vLvL2(rt/4)+trvL2(rt/4)+tr2vL2(rt/4)+ι1,|ι2|1ι1r˜Ωι2[(1ξ(s))v]L2E123(t)+X123(t). (2.6)

    To control the weighted energy norm by the generalized energy norm, the pressure must be estimated via the system (2.4)-(2.5). A similar proof can be found in [14]. For a self-contained presentation, we include its proof below.

    Lemma 4. For any integer κ6 and multi-index α satisfying |α|κ2, if E[κ/2]+3(t) and rΛαηL2 are small. Then there holds

    tΓαpL2+tρ2tΓαvΔΓαvL2δX12κ(t)+E12κ(t)X12[κ/2]+3(t)+E12[κ/2]+3(t)X12κ(t).

    Proof. It observes from (2.4) that

    Γαp=(X)T(ρ2tΓαvΔΓαv)β+γ=α|β||α|Cβα(X)TΓγη2tΓβvβ+γ=α|β||α|Cβα(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v). (2.7)

    We apply Δ1 to the above equality and take the L2 norm. By the L2 boundness of the Riesz operator, one has

    ΓαpL2Δ1(ρ2tΓαvΔΓαv)L2+(v)T(ρ2tΓαvΔΓαv)L2+β+γ=α|β||α|(X)TΓγη2tΓβvL2+β+γ=α|β||α|(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)L2. (2.8)

    Special attention is paid to the first term on the right-hand side of (2.8). We apply the derivative operator (2tΔ) to the incompressible condition (2.5). Without loss of generality, we assume at least one order derivative operator of (2tΔ) works on the first component of the velocity field in the second line of (2.5). Then we obtain

    (2tΔ)(Γαv)=β+γ=αi,j=1,2,3,i<jCβα(2tΔ)(iΓβvijΓγvjiΓβvjjΓγvi)β+γ+ι=αCβαCγαβ(2tΔ)|1Γβv12Γβv13Γβv11Γβv22Γβv23Γβv21Γβv32Γβv33Γβv3|=β+γ=αm+n=1,i=0,1,2,3Cβα(m+1iΓβv12niΓγv2+m+1iΓβv13niΓγv32miΓβv1n+1iΓγv23miΓβv1n+1iΓγv3m+1iΓβv11niΓγv2+m+1iΓβv23niΓγv3+1miΓβv1n+1iΓγv23miΓβv2n+1iΓγv3m+1iΓβv22niΓγv3m+1iΓβv11niΓγv3+2miΓβv2n+1iΓγv3+1miΓβv1n+1iΓγv3)+β+γ+ι=αm2+m2+m3=1,i=0,1,2,3CβαCγαβ[m1+1iΓβv1(2m2iΓγv23m3iΓιv33m2iΓγv22m3iΓιv33m2iΓγv21m3iΓιv31m2iΓγv23m3iΓιv31m2iΓγv22m3iΓιv32m2iΓγv21m3iΓιv3)].

    Based on the above equality, we handle the first term on the right-hand side of (2.8) as follows

    Δ1(ρ2tΓαvΔΓαv)L2Δ1[(ρ2tΔ)(Γαv)]L2+i=1,2,3Δ1(η2tΓαvi)L2(1+vL)vLρ2tΓαvΔΓαvL2+β+γ=α,|β||α|i,j=1,2,3,ij2ΓβviΓγvjL2+β+γ+ι=α,|β||α|i,j,k=1,2,3,ijk2ΓβviΓγvjΓιvkL2+β+γ=α,|β||α|i,j=1,2,3,ijΔ1(η2ΓβviΓγvj)L2+β+γ+ι=α,|β||α|i,j,k=1,2,3,ijkΔ1(η2ΓβviΓγvjΓιvk)L2+i=1,2,3Δ1(η2tΓαvi)L2+i,j=1,2,3ijΔ1(η2tΓαvivj)L2+i,j,k=1,2,3ijkΔ1(η2tΓαvivjvk)L2. (2.9)

    By (2.8), (2.9), the Sobolev embedding inequality, and the smallness of ηL3, we have

    ΓαpL2(1+vL)vLρ2tΓαvΔΓαvL2+β+γ=α,|β||α|i,j=1,2,3,ij2ΓβviΓγvjL2+β+γ+ι=α,|β||α|1,j,k=1,2,3,ijk2ΓβviΓγvjΓιvkL2+β+γ=α|β||α|Γγη2tΓβvL2+β+γ=α|β||α|(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)L2+i=1,2,3η2tΓαviL65+i,j=1,2,3ijη2tΓαvivjL65+i,j,k=1,2,3ijη2tΓαvivjvkL65.

    By (2.4), the above inequality and the smallness of E[κ/2]+3(t), one obtains

    ΓαpL2+ρ2tΓαvΔΓαvL2β+γ=α,|β||α|i,j=1,2,3,ij2ΓβviΓγvjL2+β+γ+ι=α,|β||α|i,j,k=1,2,3,ijk2ΓβviΓγvjΓιvkL2+β+γ=α|β||α|Γγη2tΓβvL2+β+γ=α|β||α|(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)L2+i=1,2,3η2tΓαviL65+i,j=1,2,3ijη2tΓαvivjL65+i,j,k=1,2,3ijkη2tΓαvivjvkL65.

    We deduce from Lemma 2 that

    tΓαpL2+tρ2tΓαvΔΓαvL2β+γ=α|γ|<|β|<|α|tr2ΓβvL2rΓγvL+β+γ=α|β||γ|rtr2ΓβvLΓγvL2+β+γ+ι=α|γ|,|ι|<|β|<|α|tr2ΓβvL2rΓγvLΓιvL+β+γ+ι=α|β|,|ι||γ|rtr2ΓβvLΓγvL2ΓιvL+β+γ+ι=α|β|,|γ||ι|rtr2ΓβvLΓγvLΓιvL2+β+γ=α|γ|<|β|<|α|rΓγηLtr2tΓβvL2+β+γ=α|β||γ|ΓγηL2rtr2tΓβvL+rηL3(1+vL)2tr2tΓαvL2+β+γ=α|γ|<|β|<|α|rΓγvLtr(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)L2+β+γ=α|β||γ|ΓγvL2rtr(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)LδX12κ(t)+E12κ(t)X12[κ/2]+3(t)+E12[κ/2]+3(t)X12κ(t).

    It completes the proof.

    As an application of the above result, we establish the estimate of weighted energy.

    Lemma 5. Let vHκΓ(R3) be the solution to the system (2.3) with the constraint condition (2.1). For any integer κ6, if Eκ(t) and rΛαηL2 are small, then we have

    Xκ(t)Eκ(t).

    Proof. For any multi-index α satisfying |α|κ2, we apply Γα to Lemma 3.3 in [11] to get

    Xκ(t)Eκ(t)+t2tΓαvΔΓαv2L2.

    It follows from the above inequality and Lemma 4 that

    Xκ(t)Eκ(t)+δ2Xκ(t)+E[κ/2]+3(t)Xκ(t)+Eκ(t)X[κ/2]+3(t).

    By the smallness of δ and Eκ(t), we arrive at the lemma.

    In preparation for the energy estimates, more detailed analysis of pressure is needed. In what follows, we always assume that Eκ(t) and rΛαηL2 are small.

    Lemma 6. For any integer κ8 and multi-index α satisfying |α|κ1, we have

    ΓαpL2E12κ(t) (2.10)

    and

    tΓαpL2(Rc)+tρ2tΓαvΔΓαvL2(Rc)E12κ(t). (2.11)

    Proof. Following the calculations in Lemma 4, we arrive at (2.9). Special attention is paid to the last three terms on the right-hand side of (2.9). From (2.4), one has

    2tΓαv=ρ1ΔΓαvρ1(X)TΓαpβ+γ=α|β||α|Cβαρ1Γγη2tΓβvβ+γ=α|β||α|Cβαρ1(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v).

    We use the above equality and the Sobolev embedding inequality to solve the last three terms on the right-hand side of (2.9) by

    Δ1[ρ1η(1+v)2ΔΓαv]L2+Δ1[ρ1η(1+v)2(X)TΓαp]L2+β+γ=α|β||α|Δ1[ρ1η(1+v)2(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι22tΓι1v)]L2+β+γ=α|β||α|Δ1[ρ1η(1+v)2Γγη2tΓβv]L2η(1+v)2ΓαvL2+i+j+k=1i(ρ1)jηk(1+v)2ΓαvL65+η(1+v)2(X)TΓαpL65+β+γ=α|β||α|η(1+v)2(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)L65+β+γ=α|β||α|η(1+v)2Γγη2tΓβvL65.

    Utilizing (2.4), (2.8), the above estimate, and the Sobolev embedding inequality, we arrive at

    ΓαpL2+ρ2tΓαvΔΓαvL2β+γ=α|γ|<|β|<|α|2ΓβvL2ΓγvL+β+γ=α|β||γ|2ΓβvLΓγvL2+β+γ+ι=α|γ|,|ι|<|β|<|α|2ΓβvL2ΓγvLΓιvL+β+γ+ι=α|β|,|ι||γ|2ΓβvLΓγvL2ΓιvL+β+γ+ι=α|β|,|γ||ι|2ΓβvLΓγvLΓιvL2+β+γ=α|γ|<|β|<|α|ΓγηL2tΓβvL2+β+γ=α|β||γ|ΓγηL22tΓβvL+1|ι|3ιηL2ΓαvL2+β+γ=α|γ|<|β|<|α|ΓγvL2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1vL2+β+γ=α|β||γ|ΓγvL22tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1vLδE12κ(t)+E12κ(t)E12[κ/2]+4(t). (2.12)

    The smallness of δ and Eκ(t) leads to (2.10).

    To verify (2.11), we use (2.12), Lemma 2, and the smallness of Eκ(t) to get

    tΓαpL2(Rc)+tρ2tΓαvΔΓαvL2(Rc)β+γ=α|γ|<|β|<|α|2ΓβvL2rΓγvL+β+γ=α|β||γ|r2ΓβvLΓγvL2+β+γ+ι=α|β|,|ι||γ|2ΓβvLΓγvL2rΓιvL+β+γ+ι=α|β|,|γ||ι|2ΓβvLrΓγvLΓιvL2+β+γ+ι=α|γ|,|ι|<|β|<|α|2ΓβvL2rΓγvLΓιvL+β+γ=α|γ|<|β|<|α|rΓγηL2tΓβvL2+β+γ=α|β||γ|ΓγηL2r2tΓβvL+1|ι|3rιηL2ΓαvL2+β+γ=α|γ|<|β|<|α|rΓγvL2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1vL2+β+γ=α|β||γ|ΓγvL2r(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)LE12κ(t)E12[κ/2]+4(t)+δE12κ(t).

    The smallness of δ and Eκ(t) implies (2.11).

    In the subsequent part, we present the improved decay properties for the third-order spatial derivatives of unknown variables in the domain away from the light cone.

    Lemma 7. For any integer κ10 and multi-index α satisfying |α|[κ/2], it holds that

    t23ΓαvL2(R)E12[κ/2]+5(t).

    Proof. We apply the derivative operator to the equation (2.4) to get

    ΔΓαv=η2tΓαv+ρ2tΓαv+(X)TΓαp+(X)TΓαp+β+γ=α|β||α|CβαΓγη2tΓβv+β+γ=α|β||α|CβαΓγη2tΓβv+β+γ=α|β||α|Cβα[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)].

    By multiplying the above equality by t2ξ(s) and taking the L2 inner product, one has

    R3|t2ξ(s)ΔΓαv|2dx7R3|t2ξ(s)η2tΓαv|2dx+7R3|t2ξ(s)ρ2tΓαv|2dx+7R3|t2ξ(s)(X)TΓαp|2dx+7R3|t2ξ(s)(X)T2Γαp|2dx+β+γ=α|β||α|7R3|Cβαt2ξ(s)Γγη2tΓβv|2dx+β+γ=α|β||α|7R3|Cβαt2ξ(s)Γγη2tΓβv|2dx+β+γ=α|β||α|7R3|Cβαt2ξ(s)[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]|2dx. (2.13)

    Since for any fH2(R3), the following Sobolev embedding inequality holds true

    fLf12L22f12L2. (2.14)

    By Lemma 2, the first term on the right-hand side of (2.13) is estimated by

    7R3|t2ξ(s)η2tΓαv|2dxR3|trξ(s)ηrtΓαv|2dx+R3|tξ(s)ηt˜SΓαv|2dx(ξ(s)rtrrtΓαv2L+tξ(s)t˜SΓαv2L)η2L2(rtrrtΓαv2L+(tξ(s)t˜SΓαv)L22(tξ(s)t˜SΓαv)L2)η2L2[rtrrtΓαv2L+(ξ(s)t˜SΓαvL2+ξ(s)trt˜SΓαvL2)(t1ξ(s)t˜SΓαvL2+ξ(s)t˜SΓαvL2+ξ(s)trt2˜SΓαvL2)]η2L2δ2E[κ/2]+4(t). (2.15)

    By a similar argument, we handle the second term on the right-hand side of (2.13) as follows

    7R3|t2ξ(s)ρ2tΓαv|2dx7R3|trξ(s)ρrtΓαv|2dx+CR3(|tξ(s)ρt˜SΓαv|2+|tξ(s)ρtΓαv|2)dx7R3|r2ξ(s)ρ2rΓαv|2dx+CR3(|tξ(s)ρt˜SΓαv|2+|tξ(s)ρtΓαv|2+|ξ(s)ρrr˜SΓαv|2+|ξ(s)ρrrΓαv|2)dx7R3|r2ξ(s)ρ3Γαv|2dx+C(trΓαv2L2(R)+tr˜SΓαv2L2(R))716R3|t2ξ(s)3Γαv|2dx+CE[κ/2]+3(t). (2.16)

    In view of (2.7), the smallness of δ and Eκ(t), one has

    ΓαpL2(R)+tΓαpL2(R)(1+t)(X)T(2tΓαvΔΓαv)L2(R)+β+γ=α(1+t)(X)TΓγη2tΓβvL2(R)+β+γ=α(1+t)(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)L2(R)E12[κ/2]+2(t).

    We substitute the above inequality into the third term on the right-hand side of (2.13) to get

    7R3|t2ξ(s)(X)TΓαp|2dxt22v2L(R)t2v2L(R)Γαp2L2(R)+t22v2L(R)t2Γαp2L2(R)E[κ/2]+2(t).

    We come back to the fourth term on the right-hand side of (2.13). We apply the divergence operator to the equality (2.7) and multiply t2 on both sides of the resulting equality. By taking the L2 inner product, one has

    t2ξ(s)ΔΓαpL2t2ξ(s)[(v)T(ρ2tΓαvΔΓαv)]L2+β+γ=α|β||α|t2ξ(s)[(v)TΓγη2tΓβv]L2+t2ξ(s)(ρ2tΓαvΔΓαv)L2+β+γ=α|β||α|t2ξ(s)(Γγη2tΓβv)L2+β+γ=α|β||α|t2ξ(s)[(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]L2. (2.17)

    For the first two terms on the right-hand side of (2.17), we deduce from Lemma 3 that

    t2ξ(s)[(v)T(ρ2tΓαvΔΓαv)]L2+β+γ=α|β||α|t2ξ(s)[(v)TΓγη2tΓβv]L2i+j=1ti(v)TL(R)trj(ρ2tΓαvΔΓαv)L2(R)+β+γ=α,|β||γ| i+j+k=1ti(v)TL(R)jΓγηL2(R)t2tkΓβvL(R)+β+γ=α,|γ|<|β|<|α| i+j+k=1ti(v)TL(R)jΓγηL(R)tr2tkΓβvL2(R)E124(t)E12[κ/2]+3(t)+δE124(t)E12[κ/2]+5(t).

    For the third and fourth terms on the right-hand side of (2.17), we have

    t2ξ(s)(ρ2tΓαvΔΓαv)L2+β+γ=α|β||α|t2ξ(s)(Γγη2tΓβv)L2t2ξ(s)η2tΓαvL2+t2ξ(s)[(ρ2tΔ)(Γαv)]L2+β+γ=α,|β||α|i+j=1t2ξ(s)iΓγη2t(jΓβv)L2. (2.18)

    By the definition of ˜S and the Sobolev embedding inequality, we solve the first term on the right-hand side of (2.18) by

    t2ξ(s)η2tΓαvL2ηL3tξ(s)t˜SΓαvL6+trξ(s)ηrtΓαvL2ηL3(t˜SΓαvL2(R)+trt˜SΓαvL2(R))+rηLtrrtΓαvL2(R)δE12[κ/2]+3(t).

    In terms of (2.5) and the definition of ˜S, the last two terms on the right-hand side of (2.18) are estimated by

    t2ξ(s)[(ρ2tΔ)(Γαv)]L2+β+γ=α,|β||α|i+j=1t2ξ(s)iΓγη2t(jΓβv)L2β+γ=αi+j=1triΓβvL2(R)tjΓγvL(R)+β+γ+ι=αi+j+k=1triΓβvL2(R)tjΓγvL(R)kΓιvL(R)+β+γ+ι=αj+k=1ΓγηL(R)trjΓβvL2(R)tkΓιvL(R)+β+γ+ι=αi+j+k=1ΓγηL(R)triΓβvL2(R)tjΓγvL(R)kΓιvL(R)+β+γ=α|β||α|ξ(s)ΓγηL3tξ(s)t˜SΓβvL6+β+γ=α|β||α|rΓγηL(R)trtrΓβvL2(R)(1+δ)E12[κ/2]+3(t)(E12[κ/2]+4(t)+E[κ/2]+4(t))+δE12[κ/2]+3(t).

    Along the same line, the last term on the right-hand side of (2.17) is dealt with by

    β+γ=α|β||α|t2ξ(s)[(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]L2β+γ=α|β||α|tr2ΓγvL2(R)t2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1vL(R)+β+γ=α|β||α|tΓγvL(R)tr(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)L2(R)E[κ/2]+4(t).

    Combining all the estimates, Lemma 4, and the fact

    t2ξ(s)2ΓαpL2t2ξ(s)ΔΓαpL2+tξ(s)ΓαpL2,

    the fourth term on the right-hand side of (2.13) is estimated by

    7R3|t2ξ(s)(X)T2Γαp|2dxt2v2L(R)t2v2L(R)2Γαp2L2(R)+t2v2L(R)t22Γαp2L2(R)+t22Γαp2L2(R)E[κ/2]+5(t).

    We employ the similar method as (2.15) and (2.16) to estimate the fifth and sixth terms on the right-hand side of (2.13) by

    β+γ=α|β||α|7R3|Cβαt2ξ(s)Γγη2tΓβv|2dx+β+γ=α|β||α|7R3|Cβαt2ξ(s)Γγη2tΓβv|2dxβ+γ=α|β||α|R3(|tξ(s)Γγηt˜SΓβv|2dx+|tξ(s)rΓγηrtΓβv|2+|tξ(s)Γγηt˜SΓβv|2+|tξ(s)ΓγηtΓβv|2+|tξ(s)ΓγηrrtΓβv|2)dxβ+γ=α|β||α|(Γγη2L3tξ(s)t˜SΓβv2L6+rξ(s)Γγη2LtrΓβv2L2(R)+Γγη2L(R)tr˜SΓβv2L2(R)+Γγη2L(R)trtΓβv2L2(R)+rξ(s)Γγη2Ltr2Γβv2L2(R))δ2E[κ/2]+3(t).

    For the last term on the right-hand side of (2.13), we have

    β+γ=α|β||α|7R3|Cβαt2ξ(s)[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]|2dxβ+γ=α,|β||α|i+j1t2i(Γγv)T2L(R)trj(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)2L2(R)E[κ/2]+4(t).

    Collecting the above estimates and the fact

    R3|t2ξ(s)3Γαv|2dxR3|t2ξ(s)ΔΓαv|2dx+R3|tξ(s)2Γαv|2dx,

    we complete the proof.

    By Lemma 7, we obtain the following estimate.

    Lemma 8. For any integer κ12 and multi-index α satisfying |α|[κ/2], there holds

    t32ΓαvL(R)E12[κ/2]+5(t), (2.19)
    t22ΓαvL(R)E12[κ/2]+6(t). (2.20)

    Proof. By the definition of ˜S and Lemma 7, we have

    t2t2ΓαvL2(R)trt2ΓαvL2(R)+tr2˜SΓαvL2(R)+tr2ΓαvL2(R)+t23ΓαvL2(R)E12[κ/2]+5(t). (2.21)

    The inequalities (2.14), (2.21), and Lemma 7 yield that

    t32ΓαvL(R)t32(ξ(s)Γαv)12L22(ξ(s)Γαv)12L2(ξ(s)Γαv12L2+ξ(s)trΓαv12L2) (ξ(s)Γαv12L2+ξ(s)trΓαv12L2+tξ(s)2Γαv12L2)E12[κ/2]+5(t).

    To consider (2.20), the definition of ˜S, combined with inequalities (2.14) and (2.21) and Lemmas 3 and 7, implies that

    t22tΓαvL(R)+t2ΓαvL(R)t2tΓαvL(R)+ttΓαvL(R)+tt˜SΓαvL(R)+trrtΓαvL(R)+t2(ξ(s)Γαv)12L22(ξ(s)Γαv)12L2E12[κ/2]+4(t)+i,|ι|1trir˜Ωι(ξ(s)rtΓαv)L2+(ξ(s)trΓαv12L2+t2ξ(s)2Γαv12L2)(ξ(s)Γαv12L2+tξ(s)2Γαv12L2+t2ξ(s)3Γαv12L2)E12[κ/2]+6(t),

    which implies the desired.

    Before concluding this section, we formulate the following two lemmas, which are utilized in the process of deriving energy estimates.

    Lemma 9. For any integer κ12 and multi-index α satisfying |α|[κ/2], we have

    t2ρ2tΓαvΔΓαvLE12[κ/2]+5(t).

    Proof. We separate two cases to consider this lemma. For the case xR, we use the Sobolev embedding inequality (2.14) to get

    ξ(s)(ρ2tΓαvΔΓαv)L[ξ(s)(ρ2tΓαvΔΓαv)]12L22[ξ(s)(ρ2tΓαvΔΓαv)]12L2.

    For the integer i satisfying 1i2, one has

    t2i[ξ(s)(ρ2tΓαvΔΓαv)]L2j+k=i1jit2jξ(s)k(ρ2tΓαvΔΓαv)L2+1i2t2ξ(s)i(ρ2tΓαvΔΓαv)L2. (2.22)

    We estimate the first term on the right-hand side of (2.22) by

    j+k=i1jit2jξ(s)k(ρ2tΓαvΔΓαv)L2tξ(s)(ρ2tΓαvΔΓαv)L2+ρ2tΓαvΔΓαvL2E12[κ/2]+3(t).

    By (2.4), we solve the second term on the right-hand side of (2.22) as follows

    1i2t2ξ(s)i(ρ2tΓαvΔΓαv)L21i2j+k=i,1jt2ξ(s)j(X)TkΓαpL2+1i2t2ξ(s)(X)TiΓαpL2+β+γ=α,|β||α| j+k=i,1i2t2ξ(s)jΓγη2tkΓβvL2+β+γ=α,|β||α|1i2t2ξ(s)i[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]L2. (2.23)

    For the first term on the right-hand side of (2.23), by Lemmas 3 and 4, we have

    1i2j+k=i,1jt2ξ(s)j(X)TkΓαpL21i2j+k=i,1jtj(X)TL(R)tkΓαpL2(R)X125(t)(E[κ/2]+5(t)+δE12[κ/2]+4(t)).

    Adopting the same method as was used in (2.17), we estimate the second term on the right-hand side of (2.23) by

    1i2t2ξ(s)(X)TiΓαpL21i2tvL(R)(1+vL(R))tiΓαpL2(R)+1i2t2iΓαpL2(R)δE12[κ/2]+5(t)+E[κ/2]+5(t).

    For the third term on the right-hand side of (2.23), by the definition of ˜S and (2.6), we obtain

    β+γ=α,|β||α| j+k=i,1i2t2ξ(s)jΓγη2tkΓβvL2β+γ=α,|β||α| j+k=i,1i2t(ξ(s)jΓγη2tkΓβvL2+ξ(s)jΓγηrrtkΓβvL2)+β+γ=α,|β||α|1i2tξ(s)iΓγηt˜SΓβvL2+β+γ=α,|β||α| j+k=i,1ki2t(ξ(s)jΓγηt˜Sk1ΓβvL2+ξ(s)jΓγηtkΓβvL2)β+γ=α,|β||α| j+k=i,1i2rξ(s)jΓγηLtrtkΓβvL2(R)+β+γ=α,|β||α|1i2iΓγηL2(R)tt˜SΓβvL(R)+β+γ=α,|β||α| j+k=i,1ki2jΓγηL(R)(trt˜Sk1ΓβvL2(R)+trtkΓβvL2(R))δE12[κ/2]+4(t).

    The last term on the right-hand side of (2.23) is estimated by

    β+γ=α,|β||α|1i2t2ξ(s)i[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]L2β+γ=α,|β||α|1i2,j+k+l=il(X)TL(R)tj(Γγv)TL(R)trk(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)L2(R)E[κ/2]+5(t).

    We verify the case xRc. By Lemmas 2 and 6, we have

    t2ρ2tΓαvΔΓαvL(Rc)tr(1ξ(s))(ρ2tΓαvΔΓαv)Li,|ι|1tir˜Ωι[(1ξ(s))(ρ2tΓαvΔΓαv)]L2i+j+k1|ι1|+|ι2|+|ι3|1j+|ι2|ir˜Ωι1(1ξ(s))L(Rc)rjr˜Ωι2ρL(Rc)kr˜Ωι32tΓβvL2(Rc)+i+j1|ι1|+|ι2|1ir˜Ωι1(1ξ(s))L(Rc)t(ρ2tjr˜Ωι2ΓβvΔjr˜Ωι2Γβv)L2(Rc)E12[κ/2]+4(t).

    Collecting all of the estimates together, we justify the lemma.

    Lemma 10. For any integer κ12 and multi-index α satisfying |α|[κ/2], one has

    β+γ=αt2Γγη2tΓβvLδE12[κ/2]+4(t).

    Proof. We deduce from the definition of ˜S that

    β+γ=αΓγη2tΓβv=β+γ=α11+t(Γγη2tΓβv+Γγηt˜SΓβvΓγηrrtΓβv).

    By Lemma 2 and (2.6), one has

    β+γ=αt2Γγη2tΓβvLβ+γ=αΓγηLrtr2tΓβvL+β+γ=αΓγηLtt˜SΓβvL+β+γ=αrΓγηLrtrrtΓβvLδE12[κ/2]+4(t).

    This completes the proof.

    This section is devoted to the energy estimates. For any integer κ12 and multi-index αN8 satisfying |α|κ1, we apply Lemma 1 to the system (2.4)-(2.5) to get

    sup0tTR3|Γαv|2(t)dx+LEκ(T)C0R3|Γαv|2(0)dx+C0T0R3[(|η|+|η|r12r12)|Γαv|(|Γαv|+|Γαv|r)]dxdt+C0|T0R3tΓαvNαdxdt|+C0supk0|T0R3fk(rΓαv+Γαvr)Nαdxdt|. (3.1)

    For the second term on the right-hand side of (3.1), by Lemma 2, one has

    C0T0R3[(|η|+|η|r12r12)|Γαv|(|Γαv|+|Γαv|r)]dxdtT0(rrηL+rηL)r14r12|Γαv|L2r14r12(|Γαv|+|Γαv|r)L2dti,j,k,|ι|1T0rijr˜ΩιkηL2r14r12|Γαv|L2r14r12(|Γαv|+|Γαv|r)L2dtδLEκ(T).

    By utilizing (2.4), we formulate the third term on the right-hand side of (3.1) as follows

    T0R3tΓαvNαdxdt=T0R3tΓαv(X)TΓαpdxdtβ+γ=α|β||α|CβαT0R3tΓαvΓγη2tΓβvdxdtβ+γ=α|β||α|CβαT0R3tΓαv[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]dxdt. (3.2)

    We handle term by term on the right-hand side of (3.2). For the first term on the right-hand side of (3.2), since (X)T is composed of elements of the form lvjmvk, jvj, ivj and the constant 1, where j,k,l,m=1,2,3, it follows that

    T0R3tΓαv(X)TΓαpdxdti,j,k,l,m,n=1,2,3ijkT0R3|tΓαvilvjmvknΓαp|dxdt+i,j=1,2,3ijT0R3|tΓαvijvjiΓαp|dxdt+i,j=1,2,3ijT0R3|tΓαviivjjΓαp|dxdt+|T0R3tΓαvΓαpdxdt|. (3.3)

    For the first term on the right-hand side of (3.3), by (2.6) and Lemma 6, we have

    i,j,k,l,m,n=1,2,3ijkT0R3|tΓαvilvjmvknΓαp|dxdtT0t2tΓαvL2t2v2LΓαpL2dtT0t2Eκ(t)E[κ/2]+2(t)dt.

    To handle the second and third terms on the right-hand side of (3.3), we apply (2.11) and (2.19) to show

    i,j=1,2,3ijT0R3|tΓαvijvjiΓαp|dxdt+1,j=1,2,3ijT0R3|tΓαviivjjΓαp|dxdtT0t32tΓαvL2(R)t32vL(R)ΓαpL2(R)dt+T0t2tΓαvL2(Rc)r(1ξ(s))vL(Rc)tΓαpL2(Rc)dtT0t32E32κ(t)dt.

    Along the same line, the last term on the right-hand side of (3.3) is estimated by

    |T0R3tΓαvΓαpdxdt|=|T0R3t(Γαv)Γαpdxdt|=|T0R3[β+γ+ι=αtΓβv1(2Γγv23Γιv33Γγv22Γιv33Γγv21Γιv31Γγv23Γιv31Γγv22Γιv32Γγv21Γιv3)β+γ=α(tΓβv12Γγv2+tΓβv13Γγv32Γβv1tΓγv23Γβv1tΓγv3tΓβv11Γγv2+tΓβv23Γγv3+1Γβv1tΓγv23Γβv2tΓγv3tΓβv22Γγv3tΓβv11Γγv3+2Γβv2tΓγv3+1Γβv1tΓγv3)]Γαpdxdt|β+γ+ι=α|γ|,|ι||β|T0ΓβvL2ΓγvL(1+ΓιvL)ΓαpL2dtβ+γ+ι=α|γ|,|ι||β|T0t32ΓβvL2(1+ΓιvL)(t32ΓγvL(R)ΓαpL2(R)+r(1ξ(s))ΓγvLtΓαpL2(Rc))dtT0t32Eκ(t)E12[κ/2]+3(t)dt.

    For the second term on the right-hand side of (3.2), by the definition of ˜S and (2.6), one has

    β+γ=α|β||α|CβαT0R3tΓαvΓγη2tΓβvdxdtβ+γ=α|γ|<|β|<|α|T0t1r14r12|tΓαv|L2rΓγηLt˜SΓβvL2dt+β+γ=α|γ|<|β|<|α|T0t2tΓαvL2r2ΓγηL(tr2tΓβvL2+trtrΓβvL2)dt+β+γ=α|β||γ|T0t2tΓαvL2rΓγηL2(t2tΓβvL+tt˜SΓβvL+ttrΓβvL)dtδLEκ(T)+δT0t2Eκ(t)dt.

    It is left to estimate the last term on the right-hand side of (3.2). Two cases are considered. By Lemmas 2, 4, and (2.6), we solve the case |γ|<|β|<|α| by

    β+γ=α|γ|<|β|<|α|CβαT0R3tΓαv[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]dxdtβ+γ=α|γ|<|β|<|α|T0t2tΓαvL2tΓγvLtρ2tΓβvΔΓβvL2dt+β+γ=α,ι1+ι2=β|γ|<|β|<|α|,|ι1||ι2|T0t2tΓαvL2tΓγvLΓι2ηL2rtr2tΓι1vLdt+β+γ=α,ι1+ι2=β|γ|<|β|<|α|,|ι2|<|ι1|T0t2tΓαvL2tΓγvLrΓι2ηLtr2tΓι1vL2dtT0t2E32κ(t)dt.

    Utilizing Lemmas 9 and 10, we solve the case |β||γ| by

    β+γ=α|β||γ|CβαT0R3tΓαv[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]dxdtβ+γ=α|β||γ|T0t2tΓαvL2ΓγvL2t2ρ2tΓβvΔΓβvLdt+β+γ=α,ι1+ι2=β|β||γ|T0t2tΓαvL2ΓγvL2t2Γι2η2tΓι1vLdtT0t2E32κ(t)dt.

    By summing up the above estimates, we deduce that

    |T0R3tΓαvNαdxdt|T0t32E32κ(t)dt+δT0t32Eκ(t)dt+δLEκ(T).

    We continue to handle the last term on the right-hand side of (3.1). The identity (2.4) yields that

    T0R3fk(rΓαv+Γαvr)Nαdxdt=β+γ=α|β||α|CβαT0R3fk(rΓαv+Γαvr)Γγη2tΓβvdxdtβ+γ=α|β||α|CβαT0R3fk(rΓαv+Γαvr)[(X)T(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]dxdtT0R3fk(rΓαv+Γαvr)(X)TΓαpdxdt. (3.4)

    We use Lemmas 9 and 10 to handle the first two terms on the right-hand side of (3.4) by

    β+γ=α|γ|<|β|<|α|T0t1r14r12(|rΓαv|+|Γαv|r)L2r2ΓγηLtr2tΓβvL2dt+β+γ=α|β||γ|T0t1r14r12(|rΓαv|+|Γαv|r)L2rΓγηL2rtr2tΓβvLdt+β+γ=α|γ|<|β|<|α|T0t2|rΓαv|+|Γαv|rL2tΓγvL(t2tΓβvΔΓβvL2+ι1+ι2=βrtrΓι2η2tΓι1vL2)dt+β+γ=α|β||γ|T0t2|rΓαv|+|Γαv|rL2ΓγvL2(t2 ρ2tΓβvΔΓβvL+ι1+ι2=βt2Γι2η2tΓι1vL)dtδLEκ(T)+δT0t2Eκ(t)dt+T0t2E32κ(t)dt.

    We rewrite the last term on the right-hand side of (3.4) as follows

    T0R3fk(rΓαv+Γαvr)(X)TΓαpdxdti,j=1,2,3ijT0R3(|fk(rΓαvi+Γαvir)jvjiΓαp|+|fk(rΓαvi+Γαvir)ivjjΓαp|)dxdt+j,l,m,n,s=1,2,3ijlT0R3|fk(rΓαvi+Γαvir)mvjnvlsΓαp|dxdt+|T0R3fk(rΓαv+Γαvr)Γαpdxdt|. (3.5)

    For the first two terms on the right-hand side of (3.5), the Lemma 6 and (2.19) imply that

    T0t2|rΓαv|+|Γαv|rL2(1+vL)r(1ξ(s))vLtΓαpL2(Rc)dt+T0t32|rΓαv|+|Γαv|rL2(1+vL)t32vL(R)ΓαpL2(R)dtT0t32E32κ(t)dt.

    We use (2.7) to formulate the last term on the right-hand side of (3.5) by

    |T0R3fk(rΓαv+Γαvr)Γαpdxdt|T0R3|fk(rΓαv+Γαvr)Δ1[(v)T(ρ2tΓαvΔΓαv)]|dxdt+β+γ=α|β||α|T0R3|fk(rΓαv+Γαvr)Δ1[(X)TΓγη2tΓβv]|dxdt+β+γ=α|β||α|T0R3|fk(rΓαv+Γαvr)Δ1[(Γγv)T(2tΓβvΔΓβv+ι1+ι2=βCι1βΓι2η2tΓι1v)]|dxdt+|T0R3fk(rΓαv+Γαvr)Δ1(ρ2tΓαvΔΓαv)dxdt|. (3.6)

    By (2.19) and Lemma 6, we estimate the first term on the right-hand side of (3.6) by

    T0R3|fk(rΓαv+Γαvr)Δ1[(v)T(ρ2tΓαvΔΓαv)]|dxdtT0t32|rΓαv|+|Γαv|rL2t32vL(R)ρ2tΓαvΔΓαvL2(R)dt+T0t2|rΓαv|+|Γαv|rL2r(1ξ(s))vL tρ2tΓαvΔΓαvL2(Rc)dtT0t32E32κ(t)dt. (3.7)

    The second and third terms on the right-hand side of (3.6) can be solved using the same method employed by the first two terms of (3.4).

    For the last term on the right-hand side of (3.6), we observe that

    |T0R3fk(rΓαv+Γαvr)Δ1(ρ2tΓαvΔΓαv)dxdt||T0R3fk(rΓαv+Γαvr)Δ1(ρ2tΓαvΔΓαv)dxdt|+T0R3|fk(rΓαv+Γαvr)Δ1(η2tΓαv)|dxdt. (3.8)

    In view of (2.5), we write the first term on the right-hand side of (3.8) by

    i,j=1,2,3ijT0R3|fk(rΓαv+Γαvr)Δ1[(ρ2tΓαviΔΓαvi)vj]|dxdt+i,j,l=1,2,3ijlT0R3|fk(rΓαv+Γαvr)Δ1[(ρ2tΓαviΔΓαvi)vjvl]|dxdt+β+γ+ι=α,|β||α|i,j,l=1,2,3,ijlT0R3|fk(rΓαv+Γαvr)Δ1(2ΓβviΓγvjΓιvl)|dxdt+β+γ+ι=α,|β||α|i,j,l=1,2,3,ijmT0R3|fk(rΓαv+Γαvr)Δ1(η2ΓβviΓγvjΓιvl)|dxdt+β+γ=α,|β||α|i,j=1,2,3,ij|T0R3fk(rΓαv+Γαvr)Δ1(2ΓβviΓγvj)dxdt|+β+γ=α,|β||α|i,j=1,2,3,ijT0R3|fk(rΓαv+Γαvr)Δ1(η2ΓβviΓγvj)|dxdt. (3.9)

    The first two terms on the right-hand side of (3.9) are dealt with by the same method as (3.7). By (2.6), the third and fourth terms on the right-hand side of (3.9) are estimated by

    T0t2|rΓαv|+|Γαv|rL2(1+ηL3)(β+γ+ι=α|γ|,|ι|<|β|<|α|2ΓβvL2tΓγvLtΓιvL+β+γ+ι=α|β|,|ι||γ|t2ΓβvLΓγvL2tΓιvL+β+γ+ι=α|β|,|γ||ι|t2ΓβvLtΓγvLΓιvL2)dtT0t2E32κ(t)dt.

    To consider the fifth term on the right-hand side of (3.9), we separate two cases to consider it. For the case xR, by (2.20), we have

    β+γ=α,|β||α|i,j=1,2,3,ij|T0R3ξ(s)fk(rΓαv+Γαvr)Δ1(2ΓβviΓγvj)dxdt|β+γ=α,|β||α|i,j=1,2,3,ijT0r14r12(|rΓαv+|Γαv|r)L2tr2ΓβviΓγvjL2dtβ+γ=α|γ|<|β|<|α|T0t1r14r12(|rΓαv+|Γαv|r)L2tr2ΓβvL2tΓγvLdt+β+γ=α|β||γ|T0t1r14r12(|rΓαv+|Γαv|r)L2(rtr(1ξ(s))2ΓβvL+t22ΓβvL(R))ΓγvL2dtsup0tTE12κ(t)LEκ(T)+T0t2E32κ(t)dt.

    To consider the case xRc, we use the integration by parts to get

    β+γ=α,|β||α|i,j=1,2,3,ij|T0R3[1ξ(s)]fk(rΓαv+Γαvr)Δ1(2ΓβviΓγvj)dxdt|β+γ=α,|β||α|i,j=1,2,3,ij|T0R3[[1ξ(s)]fk(rΓαv+Γαvr)]Δ1(2ΓβviΓγvj)dxdt|β+γ=α,|β||α|i,j=1,2,3,ijT0t1|rΓαv|+|Γαv|rL2Δ1(2ΓβviΓγvj)L2dt+β+γ=α,|β||α|i,j=1,2,3,ijT0ΓαvL22ΓβviΓγvjL2dt. (3.10)

    For the first term on the right-hand side of (3.10), we have

    \begin{align*} & \sum\limits_{\beta+\gamma = \alpha, |\beta|\neq|\alpha|\atop i, j = 1, 2, 3, i\neq j}\int_0^T \langle t\rangle^{-1}\Big\||\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big\|_{L^2}\Big\| \Delta^{-1}\nabla\cdot\Big(\partial^2\Gamma^\beta {\mathbf{v}}_i\partial\Gamma^\gamma {\mathbf{v}} _j \Big)\Big\|_{L^2}\; dt\notag\\ \lesssim\;& \sum\limits_{\beta+\gamma = \alpha\atop |\beta|\leq|\gamma| }\int_0^T \langle t\rangle^{-\frac43}\Big\||\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big\|_{L^2 } \langle t\rangle^\frac13\| \partial^2\Gamma^\beta {\mathbf{v}}\|_{L^\infty}^\frac13\|\partial^2\Gamma^\beta {\mathbf{v}}\|_{L^2}^\frac23\|\partial\Gamma^\gamma {\mathbf{v}}\|_{L^2 }\;dt\notag\\ &+\sum\limits_{\beta+\gamma = \alpha\atop |\gamma| < |\beta| < |\alpha| }\int_0^T \langle t\rangle^{-\frac43}\Big\||\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big\|_{L^2 }\| \partial^2\Gamma^\beta {\mathbf{v}}\|_{L^2}\|\partial\Gamma^\gamma {\mathbf{v}}\|_{L^2}^\frac23 \langle t\rangle^\frac13\|\partial\Gamma^\gamma {\mathbf{v}}\|_{L^\infty}^\frac13\; dt\notag\\ \lesssim\;&\int_0^T\langle t\rangle^{-\frac43}\mathcal{E}_\kappa^\frac32(t)\; dt. \end{align*}

    We use (2.5) to estimate the second term on the right-hand side of (3.10) by

    \begin{align*} &\sum\limits_{\beta+\gamma = \alpha, |\beta|\neq|\alpha|\atop i, j = 1, 2, 3, i\neq j}\int_0^T \|\nabla\cdot\Gamma^\alpha {\mathbf{v}}\|_{L^2} \|\partial^2\Gamma^\beta {\mathbf{v}}_i\partial\Gamma^\gamma {\mathbf{v}} _j \|_{L^2}\;dt\notag\\ \lesssim&\sum\limits_{\beta+\gamma = \alpha, \tilde\beta+\tilde\gamma +\tilde\iota = \alpha\atop |\beta|\leq|\gamma|, |\tilde\gamma|, |\tilde\iota| |\leq|\tilde\beta|}\int_0^T\langle t\rangle^{-2} \|\partial\Gamma^{\tilde\beta}{\mathbf{v}}\|_{L^2}\langle t\rangle\|\partial\Gamma^{\tilde\gamma} {\mathbf{v}}\|_{L^\infty} (1+\|\partial\Gamma^{\tilde\iota} {\mathbf{v}}\|_{L^\infty})\langle t\rangle\|\partial^2\Gamma^\beta {\mathbf{v}}\|_{L^\infty} \|\partial\Gamma^\gamma {\mathbf{v}}\|_{L^2}\; dt\notag\\ &+\sum\limits_{\beta+\gamma = \alpha, \tilde\beta+\tilde\gamma +\tilde\iota = \alpha\atop |\gamma| < |\beta| < |\alpha|, |\tilde\gamma|, |\tilde\iota|\leq|\tilde\beta|}\int_0^T\langle t\rangle^{-2} \|\partial\Gamma^{\tilde\beta}{\mathbf{v}}\|_{L^2}\langle t\rangle\|\partial\Gamma^{\tilde\gamma} {\mathbf{v}}\|_{L^\infty} (1+\|\partial\Gamma^{\tilde\iota} {\mathbf{v}}\|_{L^\infty})\|\partial^2\Gamma^\beta {\mathbf{v}}\|_{L^2} \langle t\rangle \|\partial\Gamma^\gamma {\mathbf{v}}\|_{L^\infty}\; dt\notag\\ \lesssim\;&\int_0^T\langle t\rangle^{-2}\mathcal{E}_\kappa^2(t)\; dt. \end{align*}

    For the last term on the right-hand side of (3.9), it follows from the Sobolev embedding inequality, (2.6), and (2.19), that

    \begin{align*} &\sum\limits_{\beta+\gamma = \alpha, |\beta|\neq|\alpha| \atop i, j = 1, 2, 3, i\neq j}\int_0^T\int_{\mathbb{R}^3} \Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\partial\eta\partial^2\Gamma^\beta {\mathbf{v}}_i\partial\Gamma^\gamma {\mathbf{v}}_j \Big)\Big|\; dxdt\notag\\ \lesssim\;&\sum\limits_{\beta+\gamma = \alpha\atop |\gamma| < |\beta| < |\alpha|}\int_0^T\langle t\rangle^{-2}\Big\|| \partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big\|_{L^2}\|\langle r\rangle\nabla\eta\|_{L^3}\|\langle t-r\rangle\partial^2\Gamma^\beta {\mathbf{v}}\|_{L^2} \langle t\rangle\|\partial\Gamma^\gamma {\mathbf{v}}\|_{L^\infty}\;dt \notag\\ &+\sum\limits_{\beta+\gamma = \alpha\atop |\beta|\leq|\gamma|}\int_0^T\langle t\rangle^{-\frac32}\Big\|| \partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big\|_{L^2}\|\partial\Gamma^\gamma {\mathbf{v}}\|_{L^2}\Big(\| \nabla\eta\|_{L^3(\mathcal{R})}\langle t\rangle^\frac32\| \partial^2\Gamma^\beta {\mathbf{v}}\|_{L^\infty(\mathcal{R})} \notag\\ & +\|\langle r\rangle(1-\xi(s))\nabla\eta\|_{L^3}\|\langle r\rangle(1-\xi(s))\partial^2\Gamma^\beta {\mathbf{v}}\|_{L^\infty}\Big) \; dt\notag\\ \lesssim\;&\int_0^T\langle t\rangle^{-\frac32}\mathcal{E}_\kappa^\frac32(t)\; dt. \end{align*}

    To solve the second term on the right-hand side of (3.8), we rewrite it as follows

    \begin{align} &\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \partial_t^2\Gamma^\alpha{\mathbf{v}} \Big)\Big|\; dxdt\\ \lesssim\;&\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \partial_t^2\nabla\Delta^{-1}\nabla\cdot\Gamma^\alpha{\mathbf{v}} \Big)\Big|\; dxdt\\ &+\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \partial_t^2\nabla^\bot\Delta^{-1}\nabla^\bot\cdot\Gamma^\alpha{\mathbf{v}} \Big)\Big|\; dxdt. \end{align} (3.11)

    In view of (2.5), we formulate the first term on the right-hand side of (3.11) by

    \begin{align} &\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \partial_t^2\nabla\Delta^{-1}\nabla\cdot\Gamma^\alpha{\mathbf{v}} \Big)\Big|\; dxdt\\ \lesssim\;&\sum\limits_{\beta+\gamma = \alpha\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \partial_t^2\nabla\Delta^{-1}\big(\partial_l\Gamma^\beta {\mathbf{v}}_i\partial_m\Gamma^\gamma {\mathbf{v}}_j\big)\Big)\Big|\; dxdt\\ &+ \sum\limits_{\beta+\gamma+\iota = \alpha\atop i, j, l = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1} \Big(\nabla\eta\cdot \partial_t^2\nabla\Delta^{-1} \big(\partial_i\Gamma^\beta{\mathbf{v}}_1\partial_j\Gamma^\gamma {\mathbf{v}}_2\partial_l\Gamma^\iota {\mathbf{v}}_3\big) \Big)\Big|\; dxdt. \end{align} (3.12)

    Since the second term has analogous estimates to the first term of (3.12), it suffices to concentrate on the first term. We observe that

    \begin{align} &\sum\limits_{\beta+\gamma = \alpha\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \partial_t^2\nabla\Delta^{-1}\big(\partial_l\Gamma^\beta {\mathbf{v}}_i\partial_m\Gamma^\gamma {\mathbf{v}}_j\big)\Big)\Big|\; dxdt\\ \lesssim\;&\sum\limits_{\beta+\gamma = \alpha \atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}\big(\partial_t^2\partial_l\Gamma^\beta {\mathbf{v}}_i\partial_m\Gamma^\gamma {\mathbf{v}}_j\big)\Big) \Big|\; dxdt\\ &+\sum\limits_{\beta+\gamma = \alpha \atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}(\partial_t\partial_l\Gamma^\beta {\mathbf{v}}_i\partial_t\partial_m\Gamma^\gamma {\mathbf{v}}_j)\Big) \Big|\; dxdt\\ &+\sum\limits_{\beta+\gamma = \alpha \atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}(\partial_l\Gamma^\beta {\mathbf{v}}_i\partial_t^2\partial_m\Gamma^\gamma {\mathbf{v}}_j)\Big) \Big|\; dxdt. \end{align} (3.13)

    Here, we restrict our analysis to the first term on the right-hand side of (3.13), as the remaining two terms have similar estimates.

    \begin{align} &\sum\limits_{\beta+\gamma = \alpha \atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}\big(\partial_t^2\partial_l\Gamma^\beta {\mathbf{v}}_i\partial_m\Gamma^\gamma {\mathbf{v}}_j\big)\Big) \Big|\; dxdt\\ \lesssim\;&\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}\big(\partial_t^2\Gamma^\beta{\mathbf{v}}_i\partial_l\partial_m \Gamma^\gamma {\mathbf{v}}_j\big)\Big) \Big|\; dxdt\\ &+\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}\partial_l\big(\partial_t^2\Gamma^\beta {\mathbf{v}}_i\partial_m\Gamma^\gamma {\mathbf{v}}_j\big)\Big) \Big|\; dxdt\\ &+\sum\limits_{\beta+\gamma = \alpha, |\beta|\leq |\gamma|\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}\big(\partial_t^2\partial_l\Gamma^\beta {\mathbf{v}}_i\partial_m\Gamma^\gamma {\mathbf{v}}_j\big)\Big)\Big|\; dxdt. \end{align} (3.14)

    For the first term on the right-hand side of (3.14), we deduce from (2.4) that

    \begin{align} & \sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}\big(\partial_t^2\Gamma^\beta{\mathbf{v}}_i\partial_l\partial_m\Gamma^\gamma {\mathbf{v}}_j\big)\Big) \Big|\; dxdt\\ \lesssim\;& \sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1}\big(\rho^{-1}\Delta\Gamma^\beta {\mathbf{v}}_i\partial_l\partial_m \Gamma^\gamma {\mathbf{v}}_j\big)\Big)\Big|\; dxdt\\ &+ \sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1} \big(\rho^{-1} N_i^\beta\partial_l\partial_m\Gamma^\gamma {\mathbf{v}}_j\big)\Big) \Big|\; dxdt. \end{align} (3.15)

    We formulate the first term on the right-hand side of (3.15) as follows

    \begin{align} &\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m, n = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla\Big(\eta \nabla\Delta^{-1}\partial_n\big(\rho^{-1} \partial_n\Gamma^\beta {\mathbf{v}}_i\partial_l\partial_m \Gamma^\gamma {\mathbf{v}}_j\big)\Big) \Big|\; dxdt\\ &+\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m, n = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\partial_n \Big( \rho^{-1}\eta \partial_n\Gamma^\beta {\mathbf{v}}_i\partial_l\partial_m \Gamma^\gamma {\mathbf{v}}_j \Big) \Big|\; dxdt\\ &+\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m, n = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla\Big(\eta \nabla\Delta^{-1} \big[ \partial_n\Gamma^\beta {\mathbf{v}}_i\partial_n\big(\rho^{-1}\partial_l\partial_m \Gamma^\gamma{\mathbf{v}}_j\big)\big]\Big)\Big|\; dxdt\\ &+\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m, n = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1} \Big(\rho^{-1}\partial_n\eta \partial_n\Gamma^\beta {\mathbf{v}}_i\partial_l\partial_m \Gamma^\gamma {\mathbf{v}}_j\Big) \Big|\; dxdt\\ &+\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m, n = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1} \Big(\eta \partial_n\Gamma^\beta {\mathbf{v}}_i\partial_n\big(\rho^{-1}\partial_l \partial_m \Gamma^\gamma{\mathbf{v}}_j\big)\Big)\Big|\; dxdt. \end{align} (3.16)

    The first three terms on the right-hand side of (3.16) can be estimated by

    \begin{align*} &\sum\limits_{ \beta+\gamma = \alpha\atop |\gamma| < |\beta| }\int_0^T\langle t\rangle^{-1}\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle \eta \|_{L^\infty}\|\nabla\Gamma^\beta {\mathbf{v}}\|_{L^2}\langle t\rangle\|\nabla^2\Gamma^\gamma{\mathbf{v}}\|_{L^\infty}\; dt\notag\\ &+\sum\limits_{ \beta+\gamma = \alpha\atop |\gamma| < |\beta| }\int_0^T\langle t\rangle^{-1}\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle\eta\|_{L^3}\|\nabla\Gamma^\beta {\mathbf{v}}\|_{L^2}\langle t\rangle\|\nabla\big(\rho^{-1}\nabla^2\Gamma^\gamma {\mathbf{v}}\big)\|_{L^\infty}\;dt\notag\\ \lesssim\;& \delta L\mathcal{E}_\kappa(T)+\int_0^T \langle t\rangle^{-2}\mathcal{E}_\kappa^2(t)\; dt. \end{align*}

    We utilize (2.19) to solve the last two terms on the right-hand side of (3.16) by

    \begin{align*} &\sum\limits_{ \beta+\gamma = \alpha\atop |\gamma| < |\beta| }\int_0^T\langle t\rangle^{-\frac32}\Big\||\partial_r\Gamma^\beta {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big\|_{L^2}\|\nabla\Gamma^\beta {\mathbf{v}}\|_{L^2}\Big(\|\langle r\rangle(1-\xi(s))\nabla\eta\|_{L^3}\|\langle r\rangle(1-\xi(s))\nabla^2\Gamma^\gamma{\mathbf{v}}\|_{L^\infty}\notag\\ &+\| \nabla\eta\|_{L^3(\mathcal{R})}\langle t\rangle^{\frac32} \|\nabla^2\Gamma^\gamma{\mathbf{v}}\|_{L^\infty(\mathcal{R})}+\|\langle r\rangle(1-\xi(s))\eta\|_{L^3}\|\langle r\rangle(1-\xi(s))\nabla\big(\rho^{-1}\nabla^2\Gamma^\gamma {\mathbf{v}}\big)\|_{L^\infty}\notag\\ &+\|\eta\|_{L^3(\mathcal{R})}\langle t\rangle^{\frac32}\| \nabla\big(\rho^{-1}\nabla^2 \Gamma^\gamma{\mathbf{v}}_j\big)\|_{L^\infty(\mathcal{R})}\Big)\; dt\notag\\ \lesssim\;&\int_0^T \langle t\rangle^{-\frac32}\mathcal{E}_\kappa^\frac32(t)\; dt. \end{align*}

    Along the same line, the second term on the right-hand side of (3.15) can be handled by

    \begin{align*} &\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \nabla\Delta^{-1} \big(\rho^{-1} N_i^\beta\partial_l\partial_m \Gamma^\gamma{\mathbf{v}}_j\big)\Big) \Big|\; dxdt\notag\\ \lesssim\;&\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla\Big(\eta\cdot \nabla\Delta^{-1} \big(\rho^{-1} N_i^\beta\partial_l\partial_m \Gamma^\gamma {\mathbf{v}}_j\big)\Big) \Big|\; dxdt\notag\\ & +\sum\limits_{ \beta+\gamma = \alpha, |\gamma| < |\beta|\atop i, j, l, m = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3}\Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1} \Big(\rho^{-1}\eta N_i^\beta\partial_l \partial_m \Gamma^\gamma {\mathbf{v}}_j \Big) \Big|\; dxdt\notag\\ \lesssim\;&\sum\limits_{ \beta+\gamma = \alpha\atop |\gamma| < |\beta| }\int_0^T\langle t\rangle^{-1}\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle \eta\|_{L^3}\|N^\beta\|_{L^2}\langle t\rangle\|\nabla^2\Gamma^\gamma{\mathbf{v}}\|_{L^\infty}\; dt\notag\\ &+\sum\limits_{ \beta+\gamma = \alpha\atop |\gamma| < |\beta| }\int_0^T\langle t\rangle^{-2}\Big\||\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big\|_{L^2}\|N^\beta\|_{L^2}\|\langle r\rangle(1-\xi(s)) \eta\|_{L^3}\|\langle r\rangle(1-\xi(s))\nabla^2\Gamma^\gamma{\mathbf{v}}\|_{L^\infty}\; dt \notag\\ &+\sum\limits_{ \beta+\gamma = \alpha\atop |\gamma| < |\beta| }\int_0^T\langle t\rangle^{-\frac32}\Big\||\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big\|_{L^2}\|N^\beta\|_{L^2}\| \eta\|_{L^3(\mathcal{R})}\langle t\rangle^{\frac32}\| \nabla^2\Gamma^\gamma{\mathbf{v}}\|_{L^\infty(\mathcal{R})}\; dt\notag\\ \lesssim\;& \delta L\mathcal{E}_\kappa(T)+\int_0^T \langle t\rangle^{-\frac32}\mathcal{E}_\kappa^2(t)\; dt. \end{align*}

    The same estimates hold for the last two terms on the right-hand side of (3.14).

    Applying (2.4), we formulate the second term on the right-hand side of (3.11) as follows

    \begin{align} &\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\Big(\nabla\eta\cdot \partial_t^2\nabla^\bot\Delta^{-1}\nabla^\bot\cdot\Gamma^\alpha{\mathbf{v}} \Big)\Big|\; dxdt\\ = \;&\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta\cdot \partial_t^2\Delta^{-1}\nabla^\bot\cdot\Gamma^\alpha{\mathbf{v}} \Big)\Big|\; dxdt\\ \lesssim\;&\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \nabla^\bot\cdot\Gamma^\alpha {\mathbf{v}}\Big)\Big|\; dxdt\\ &+\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot\big[(\nabla X)^{-T}\nabla\Gamma^\alpha p\big]\Big)\Big|\; dxdt\\ &+\sum\limits_{\beta+\gamma = \alpha}\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big(\Gamma^\gamma \eta \partial_t^2\Gamma^\beta {\mathbf{v}}\big)\Big)\Big|\; dxdt\\ &+\sum\limits_{\beta+\gamma = \alpha\atop |\beta|\neq|\alpha|}\int_0^T\int_{\mathbb{R}^3} \Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big[C_\alpha^\beta(\nabla X)^{-T}(\nabla\Gamma^\gamma {\mathbf{v}})^T\\ &\cdot\big(\partial_t^2\Gamma^\beta {\mathbf{v}}-\Delta\Gamma^\beta {\mathbf{v}}+\sum\limits_{\iota_1+\iota_2 = \beta}C_{\beta}^{\iota_1}\Gamma^{\iota_2}\eta\partial_t^2\Gamma^{\iota_1}{\mathbf{v}} \big)\big]\Big)\Big|\; dxdt. \end{align} (3.17)

    The first term on the right-hand side of (3.17) is solved by

    \begin{align*} &\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \nabla^\bot\cdot\Gamma^\alpha {\mathbf{v}}\Big)\Big|\; dxdt\notag\\ \lesssim\;&\int_0^T\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle^2\nabla\eta \|_{L^\infty}\big\|r^{-\frac14}\langle r\rangle^{-\frac12}|\nabla\Gamma^\alpha {\mathbf{v}}|\big\|_{L^2}\; dt\notag\\ \lesssim\;&\delta L\mathcal{E}_\kappa(T). \end{align*}

    The calculations in (2.12) imply that

    \begin{align} &\|\nabla\Gamma^\alpha p\|_{L^2}+\|\rho\partial_t^2 \Gamma^\alpha {\mathbf{v}}-\Delta \Gamma^\alpha {\mathbf{v}}\|_{L^2 }\\ \lesssim\;&\sum\limits_{\beta+\gamma = \alpha}\langle t\rangle^{-1}\|\langle r\rangle\langle t-r\rangle|\partial^2\Gamma^\beta {\mathbf{v}}||\partial\Gamma^\gamma {\mathbf{v}}|\|_{L^2}+\sum\limits_{\beta+\gamma+\iota = \alpha}\langle t\rangle^{-1}\|\langle r\rangle\langle t-r\rangle|\partial^2\Gamma^\beta {\mathbf{v}}||\partial\Gamma^\gamma {\mathbf{v}}||\partial\Gamma^\iota{\mathbf{v}}|\|_{L^2}\\ &+\sum\limits_{\beta+\gamma = \alpha\atop |\beta|\neq|\alpha|}\langle t\rangle^{-1}\|\langle r\rangle\langle t-r\rangle|\Gamma^\gamma\eta||\partial_t^2\Gamma^\beta {\mathbf{v}}|\|_{L^2}+\sum\limits_{1\leq \iota\leq 3}\|\langle r\rangle\nabla^\iota\eta \|_{L^\infty}\big\| r^{-\frac14}\langle r\rangle^{-\frac12}|\nabla\Gamma^\alpha {\mathbf{v}}|\big\|_{L^2}\\ &+ \sum\limits_{\beta+\gamma = \alpha\atop |\gamma| < |\beta| < |\alpha|} \langle t\rangle^{-1}\| \langle r\rangle \nabla\Gamma^\gamma {\mathbf{v}}\|_{L^\infty }\big\|\langle t-r\rangle(\partial_t^2\Gamma^\beta {\mathbf{v}}-\Delta \Gamma^\beta {\mathbf{v}} + \sum\limits_{\iota_1+\iota_2 = \beta}C_\beta^{\iota_1}\Gamma^{\iota_2}\eta\partial_t^2\Gamma^{\iota_1} {\mathbf{v}})\big\|_{L^2 } \\ & + \sum\limits_{\beta+\gamma = \alpha\atop |\beta|\leq|\gamma|} \langle t\rangle^{-1} \|\nabla\Gamma^\gamma {\mathbf{v}}\|_{L^2 }\big\| \langle r\rangle \langle t-r\rangle( \partial_t^2\Gamma^\beta {\mathbf{v}}-\Delta \Gamma^\beta {\mathbf{v}} + \sum\limits_{\iota_1+\iota_2 = \beta}C_\beta^{\iota_1}\Gamma^{\iota_2}\eta\partial_t^2\Gamma^{\iota_1} {\mathbf{v}})\big\|_{L^\infty }. \end{align} (3.18)

    Substituting (3.18) into the second term on the right-hand side of (3.17), we have

    \begin{align} &\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot[(\nabla X)^{-T}\nabla\Gamma^\alpha p]\Big)\Big|\; dxdt\\ \lesssim\;&\sum\limits_{i, j, l, m, n = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3} f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot\big(\partial_l {\mathbf{v}}_i\partial_m {\mathbf{v}}_j\partial_n\Gamma^\alpha p\big)\Big)\; dxdt\\ &+\sum\limits_{i, j, l = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3} f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot\big( \partial_j {\mathbf{v}}_i\partial_l\Gamma^\alpha p\big)\Big)\; dxdt\\ \lesssim\;& \int_0^T\langle t\rangle^{-1}\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle \nabla\eta\|_{L^\infty}\langle t\rangle\|\nabla{\mathbf{v}}\|_{L^\infty} \|\nabla{\mathbf{v}}\|_{L^3}\|\nabla\Gamma^\alpha p\|_{L^2}\; dt \\ &+ \int_0^T \Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle \nabla\eta\|_{L^\infty}\|\nabla{\mathbf{v}}\|_{L^3}\|\nabla\Gamma^\alpha p\|_{L^2}\; dt\\ \lesssim\;& \delta L\mathcal{E}_\kappa(T)+\int_0^T \langle t\rangle^{-2}\mathcal{E}_\kappa^\frac32(t)\; dt. \end{align} (3.19)

    For the third term on the right-hand side of (3.17), we have

    \begin{align} &\sum\limits_{\beta+\gamma = \alpha}\int_0^T\int_{\mathbb{R}^3} \Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big(\Gamma^\gamma \eta \partial_t^2\Gamma^\beta {\mathbf{v}}\big)\Big)\Big|\; dxdt\\ \lesssim\;&\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot\big (\eta \partial_t^2\Gamma^\alpha{\mathbf{v}}\big)\Big)\Big|\; dxdt\\ &+\sum\limits_{\beta+\gamma = \alpha\atop |\beta|\neq|\alpha|}\int_0^T\int_{\mathbb{R}^3} \Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big(\Gamma^\gamma \eta \partial_t^2\Gamma^\beta {\mathbf{v}}\big)\Big)\Big|\; dxdt. \end{align} (3.20)

    We use (2.4) to formulate the first term on the right-hand side of (3.20) as follows

    \begin{align} &\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot\big( \rho^{-1}\eta \Delta \Gamma^\alpha{\mathbf{v}}\big)\Big)\Big|\; dxdt\\ &+\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big(\rho^{-1}\eta (\nabla X)^{-T}\nabla\Gamma^\alpha p\big)\Big)\Big|\; dxdt\\ &+\sum\limits_{\beta+\gamma = \alpha\atop |\beta|\neq|\alpha|}\int_0^T\int_{\mathbb{R}^3} \Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big( \rho^{-1}\eta \Gamma^\gamma \eta \partial_t^2\Gamma^\beta {\mathbf{v}}\big)\Big)\Big|\; dxdt\\ &+\sum\limits_{\beta+\gamma = \alpha\atop |\beta|\neq|\alpha|}\int_0^T\int_{\mathbb{R}^3} \Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big[ \rho^{-1}\eta\big(\nabla X)^{-T}(\nabla\Gamma^\gamma {\mathbf{v}}\big)^T \\ &\cdot\big(\partial_t^2\Gamma^\beta {\mathbf{v}}-\Delta \Gamma^\beta {\mathbf{v}}+\sum\limits_{\iota_1+\iota_2 = \iota}C_\beta^{\iota_1}\Gamma^{\iota_2}\eta \partial_t^2\Gamma^{\iota_1}{\mathbf{v}}\big)\big]\Big)\Big|\; dxdt \end{align} (3.21)

    For the first term on the right-hand side of (3.21), we have

    \begin{align*} &\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big( \rho^{-1}\eta \Delta \Gamma^\alpha{\mathbf{v}}\big)\Big)\Big|\; dxdt\notag\\ \lesssim\;&\sum\limits_{i = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3} f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot\partial_i \big( \rho^{-1}\eta \partial_i \Gamma^\alpha{\mathbf{v}}\big)\Big)\; dxdt\notag\\ &+\sum\limits_{i = 1, 2, 3}\int_0^T\int_{\mathbb{R}^3} f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot\big[ \partial_i ( \rho^{-1}\eta ) \partial_i \Gamma^\alpha{\mathbf{v}}\big]\Big)\; dxdt\notag\\ \lesssim\;&\int_0^T\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle \nabla\eta\|_{L^\infty}\Big(\|\langle r\rangle\eta\|_{L^\infty}+\|\langle r\rangle\nabla(\rho^{-1}\eta)\|_{L^3}\Big)\notag\\ &\cdot\big\|r^{-\frac14}\langle r\rangle^{-\frac12}|\nabla\Gamma^\alpha {\mathbf{v}}|\big\|_{L^2}\; dt\notag\\ \lesssim \;&\delta L\mathcal{E}_\kappa(T). \end{align*}

    The second term on the right-hand side of (3.21) can be solved as (3.19). We employ the analogous method utilized for the first two terms on the right-hand side of (3.4) to solve the last two terms on the right-hand side of (3.21).

    For the second term on the right-hand side of (3.20), we have

    \begin{align*} &\sum\limits_{\beta+\gamma = \alpha\atop |\beta|\neq|\alpha|}\int_0^T\int_{\mathbb{R}^3} \Big|f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla^\bot\cdot\Big(\nabla\eta \Delta^{-1}\nabla^\bot\cdot \big(\Gamma^\gamma \eta \partial_t^2\Gamma^\beta {\mathbf{v}}\big)\Big)\Big|\; dxdt\notag\\ \lesssim\;&\sum\limits_{\beta+\gamma = \alpha\atop |\gamma| < |\beta| < |\alpha|}\int_0^T\langle t\rangle^{-1}\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle \nabla\eta\|_{L^\infty}\|\langle r\rangle\Gamma^\gamma\eta\|_{L^3}\|\langle t-r\rangle\partial_t^2\Gamma^\beta{\mathbf{v}}\|_{L^2}\; dt\notag\\ &+\sum\limits_{\beta+\gamma = \alpha\atop |\beta|\leq|\gamma|}\int_0^T\langle t\rangle^{-1}\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}|}+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\|\langle r\rangle \nabla\eta\|_{L^\infty}\|\langle r\rangle\Gamma^\gamma\eta\|_{L^2}\|\langle t-r\rangle\partial_t^2\Gamma^\beta{\mathbf{v}}\|_{L^3}\;dt\notag\\ \lesssim\;&\delta L\mathcal{E}_\kappa(T)+\delta\int_0^T \langle t\rangle^{-2}\mathcal{E}_\kappa(t)\;dt. \end{align*}

    For the last term on the right-hand side of (3.17), by Lemmas 2 and 6, we have

    \begin{align*} &\int_0^T\int_{\mathbb{R}^3}\Big| f_k\Big(\partial_r\Gamma^\alpha {\mathbf{v}}+\frac{\Gamma^\alpha {\mathbf{v}}}{r}\Big)\cdot \nabla\Delta^{-1}\nabla_i^\bot\Big(\nabla_i\eta \Delta^{-1}\nabla^\bot\cdot \big[C_\alpha^\beta(\nabla X)^{-T}(\nabla\Gamma^\gamma {\mathbf{v}})^T\notag\\ &\cdot\big(\partial_t^2\Gamma^\beta {\mathbf{v}}-\Delta\Gamma^\beta {\mathbf{v}}+\sum\limits_{\iota_1+\iota_2 = \beta}C_{\beta}^{\iota_1}\Gamma^{\iota_2}\eta\partial_t^2\Gamma^{\iota_1}{\mathbf{v}} \big)\big]\Big)\Big|\; dxdt\notag\\ \lesssim\;&\sum\limits_{\beta+\gamma = \alpha\atop |\gamma| < |\beta| < |\alpha|}\int_0^T\langle t\rangle^{-1}\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\| \langle r\rangle\nabla\eta\|_{L^\infty}\| \nabla\Gamma^\gamma {\mathbf{v}}\|_{L^3}\notag\\ &\cdot\big\|\langle t\rangle\big(\partial_t^2\Gamma^\beta {\mathbf{v}}-\Delta\Gamma^\beta {\mathbf{v}}+\sum\limits_{\iota_1+\iota_2 = \beta}C_{\beta}^{\iota_1}\Gamma^{\iota_2}\eta\partial_t^2\Gamma^{\iota_1}{\mathbf{v}}\big) \big\|_{L^2}\; dt\notag\\ &+ \sum\limits_{\beta+\gamma = \alpha\atop |\beta|\leq|\gamma|}\int_0^T\langle t\rangle^{-1}\Big\|r^{-\frac14}\langle r\rangle^{-\frac12}\Big(|\partial_r\Gamma^\alpha {\mathbf{v}}|+\frac{|\Gamma^\alpha {\mathbf{v}}|}{r}\Big)\Big\|_{L^2}\| \langle r\rangle\nabla\eta\|_{L^\infty}\| \nabla\Gamma^\gamma {\mathbf{v}}\|_{L^2}\notag\\ &\cdot\big\|\langle t\rangle\big(\partial_t^2\Gamma^\beta {\mathbf{v}}-\Delta\Gamma^\beta {\mathbf{v}}+\sum\limits_{\iota_1+\iota_2 = \beta}C_{\beta}^{\iota_1}\Gamma^{\iota_2}\eta\partial_t^2\Gamma^{\iota_1}{\mathbf{v}}\big) \big\|_{L^3}\; dt\notag\\ \lesssim\;&\delta L\mathcal{E}_\kappa(T)+\int_0^T\langle t\rangle^{-2}\mathcal{E}_\kappa^\frac32(t)\; dt. \end{align*}

    Combining all the estimates, we conclude that

    \begin{align*} & \sup\limits_{0\leq t\leq T}\int_{\mathbb{R}^3}|\partial \Gamma^\alpha{\mathbf{v}}|^2(t)\; dx+L \mathcal{E}_\kappa(T)\notag\\ \leq\;&C_0\int_{\mathbb{R}^3}|\partial\Gamma^\alpha {\mathbf{v}}|^2 (0)\;dx+C C_0\big(\delta +\sup\limits_{0\leq t\leq T}\mathcal{E}_\kappa^\frac12(t)\big) L\mathcal{E}_\kappa(T) +C\int_0^T\langle t\rangle^{-\frac43}\mathcal{E}_\kappa^\frac32(t)\; dt +C\delta\int_0^T\langle t\rangle^{-\frac43}\mathcal{E}_\kappa (t)\; dt, \end{align*}

    where C > 0 is some positive constant. By the smallness of \delta , \mathcal{E}_\kappa(t) , and the standard continuity method, we arrive at the main result.

    All authors contributed equally.

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

    The work of the first author was partially supported by the National Natural Science Foundation of China under Grants 12401278. The work of the second author was partially supported by the RFS grant and GRF grants from the Research Grants Council (Project Nos. PolyU 11302021, 11310822, and 11302523). The authors would like to thank the research center for nonlinear analysis at PolyU for the opportunity of discussions and encouragement.

    The authors declare there is no conflict of interest.

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