Research article Topical Sections

A catalogue of human secreted proteins and its implications

  • Received: 27 September 2016 Accepted: 17 November 2016 Published: 24 November 2016
  • Under both normal and pathological conditions, cells secrete variety of proteins through classical and non-classical secretory pathways into the extracellular space. Majority of these proteins represent pathophysiology of the cell from which it is secreted. Recently, though more than 92% of the protein coding genes has been mapped by human proteome map project, but number of those proteins that constitutes secretome of the cell still remains elusive. Secreted proteins or the secretome can be accessible in bodily fluids and hence are considered as potential biomarkers to discriminate between healthy and diseased individuals. In order to facilitate the biomarker discovery and to further aid clinicians and scientists working in these arenas, we have compiled and catalogued secreted proteins from the human proteome using integrated bioinformatics approach. In this study, nearly 14% of the human proteome is likely to be secreted through classical and non-classical secretory pathways. Out of which, ~38% of these secreted proteins were found in extracellular vesicles including exosomes and shedding microvesicles. Among these secreted proteins, 94% were detected in human bodily fluids including blood, plasma, serum, saliva, semen, tear and urine. We anticipate that this high confidence list of secreted proteins could serve as a compendium of candidate biomarkers. In addition, the catalogue may provide functional insights in understanding the molecular mechanisms involved in various physiological and pathophysiological conditions of the cell.

    Citation: Shivakumar Keerthikumar. A catalogue of human secreted proteins and its implications[J]. AIMS Biophysics, 2016, 3(4): 563-570. doi: 10.3934/biophy.2016.4.563

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  • Under both normal and pathological conditions, cells secrete variety of proteins through classical and non-classical secretory pathways into the extracellular space. Majority of these proteins represent pathophysiology of the cell from which it is secreted. Recently, though more than 92% of the protein coding genes has been mapped by human proteome map project, but number of those proteins that constitutes secretome of the cell still remains elusive. Secreted proteins or the secretome can be accessible in bodily fluids and hence are considered as potential biomarkers to discriminate between healthy and diseased individuals. In order to facilitate the biomarker discovery and to further aid clinicians and scientists working in these arenas, we have compiled and catalogued secreted proteins from the human proteome using integrated bioinformatics approach. In this study, nearly 14% of the human proteome is likely to be secreted through classical and non-classical secretory pathways. Out of which, ~38% of these secreted proteins were found in extracellular vesicles including exosomes and shedding microvesicles. Among these secreted proteins, 94% were detected in human bodily fluids including blood, plasma, serum, saliva, semen, tear and urine. We anticipate that this high confidence list of secreted proteins could serve as a compendium of candidate biomarkers. In addition, the catalogue may provide functional insights in understanding the molecular mechanisms involved in various physiological and pathophysiological conditions of the cell.


    The model of micropolar fluids which respond to micro-rotational motions and spin inertia was first introduced by Eringen [16] in 1966. The mathematical theory of micropolar fluids has been developing in two directions. One explores incompressible and the other compressible flows. For more physical background, we can refer to [2], [3], [39]. In this paper we consider the compressible cylinder symmetric flow of the isotropic, viscous and heat-conducting micropolar fluid which is in the thermodynamical sense perfect and polytropic. The mathematical model of the described fluid is stated for example in the book of G. Lukaszewicz [32] and reads

    ˙ρ=ρv, (1)
    ρ˙v=T+ρf, (2)
    ρjI˙w=C+Tx+ρg, (3)
    ρ˙E=˙q+T:v+C:ωTxw, (4)
    Tij=(p+λvk,k)δi,j+μ(vj,ivi,j)2μrεmijwm, (5)
    C=c0wk,kδi,j+cd(wi,j+wj,i)+ca(wj,iwi,j), (6)
    q=kθθ, (7)
    p=Rρθ, (8)
    E=cvθ. (9)

    with notation:

    ρmassdensityvvelocitywmicrorotationvelocityEinternalenergydensityθabsolutetemperatureTstresstensorcouplestresstensorqheatfluxdensityvectorfbodyforcedensitygbodycoupledensityppressurejImicroinertiadensity(apositiveconstant)kθheatconductioncoefficienμr,c0,caandcdcoefficientsofmicroviscosityRspecificgasconstantcvspecificheat(positiveconstant)Txaxialvector((Tx)i=εijkTjk)εijkLeviCivitasymbolδijKroneckerdelta.

    Equations (1)-(4) are, respectively, local forms of the conservation laws for the mass, momentum, momentum moment and energy. Equations (5)-(6) are constitutive equations for the micropolar continuum. Equation (7) is the Fourier law and equations (8)-(9) present the assumptions that our fluid is perfect and polytropic. On account of the Clausius-Duhamel inequalities, they must have the following properties:

    μ0,3λ+2μ0,μr>0, (10)
    cd0,3c0+2cd0,|cdca|cd+ca. (11)

    For simplicity reasons, we assume that

    f=g=0.

    In this paper, we consider the three dimensional case of (1)-(9) with the assumption of cylindrical symmetry, and we study the problem with homogeneous boundary conditions as in [13]:

    t=0:(ρ,v,w,θ)(r,0)=(ρ0(r),v0(r),w0(r),θ0(r)),rG, (12)
    v|G=0,w|G=0,θr|G=0,t>0, (13)

    where G={(x1,x2,x3)R3,0<a<r<b<+,x3R,r=x21+x22} is the spatial domain of our problem and v=(υ1,υ2,υ3), w=(w1,w2,w3) denote the velocity vector and microrotation velocity respectively. In the following work we give the mathematical model with cylindrical symmetry, first in the Eulerian description, which is then transformed to the Lagrangian description. The reduced system of the three-dimensional equations in the Eulerian coordinate is now of the form [11] and [22]:

    ρt+v1ρr+ρ(v1r+v1r)=0, (14)
    ρ(v1t+v1v1r)=Rr(ρθ)+(λ+2μ)r(v1r+v1r)+ρv22r, (15)
    ρ(v2t+v1v2r)=(μ+μr)r(v2r+v2r)ρv1v2r2μrw3r, (16)
    ρ(v3t+v1v3r)=(μ+μr)(2v3r2+1rv3r)+2μr(w2r+w2r), (17)
    ρjI(w1t+v1w1r)=(c0+2cd)r(w1r+w1r)+ρjIw2v2r4μrw1, (18)
    ρjI(w2t+v1w2r)=(ca+cd)r(w2r+w2r)ρjIw1v2r2μrv3r4μrw2, (19)
    ρjI(w3t+v1w3r)=(ca+cd)(2w3r2+1rw3r)+2μr(v2r+v2r)4μrw3, (20)
    cVρ(θt+v1θr)=κ(2θr2+1rθr)Rρθ(v1r+v1r)+(λ+2μ)(v1r+v1r)24μv1rv1r+(μ+μr)(v2r+v2r)24μv2rv2r+(μ+μr)(v3r)2+(c0+2cd)(w1r+w1r)24cdw1rw1r+(cd+ca)(w2r+w2r)24cdw2rw2r+(cd+ca)(w3r)2+4μrw21+4μrw22+4μrw23+4μrw2v3r4μr(v2r+v2r)w3. (21)

    To analyze the system and draw the desired results, it is convenient to transform the system (14)-(21) to Lagrangian coordinates. The Eulerian coordinates (r,t) are connected to the Lagrangian coordinates (ξ,t) by the relation

    r(ξ,t)=r0(ξ)+t0˜v1(ξ,τ)dτ, (22)

    where ˜v1(ξ,t)=v1(r(ξ,t),t) and

    r0(ξ)=η1(ξ),η(r)=rasρ0(s)ds,rG. (23)

    From (14) and it follows that

    t(rasρ(s,t)ds)=0,

    which implies

    rasρ(s,t)ds=r0asρ0(s)ds=ξ. (24)

    Now, we have ξΩ=[0,L], where

    L=basρ(s,t)ds=basρ0(s)ds,t0. (25)

    Moreover, differentiating (24) with respect to ξ yields

    rξ=1r(ξ,t)ρ(r(ξ,t),t). (26)

    Let us introduce the temporary notation ˜ϕ(ξ,t) for ϕ(r(ξ,t),t), we obtain

    ˜ϕ(ξ,t)t=ϕ(r,t)t+v1ϕ(r,t)r, (27)
    ˜ϕ(ξ,t)ξ=ϕ(r,t)rr(ξ,t)ξ=1rρ(r,t)ϕ(r,t)r, (28)
    ρξ(r˜ϕ(ξ,t))=ϕr+ϕr. (29)

    Hereafter, without danger of confusion, we will write (x,t) instead of (ξ,t) and omit . Subscripts t and x will denote the (partial) derivatives with respect to t and x, respectively, and we will use u=1ρ to denote the specific volume. Thus, by (27)-(29), system (14)-(21) can be rewritten using the new variables (x,t),xΩ,t0 as follows:

    ut=(rv1)x, (30)
    v1t=r[(λ+2μ)(rv1)xuRθu]x+v22r, (31)
    v2t=(μ+μr)r((rv2)xu)xv1v2r2μrrw3x, (32)
    v3t=(μ+μr)r((rv3)xu)x+(μ+μr)uv3r2+2μr(rw2)x, (33)
    jIw1t=(c0+2cd)r((rw1)xu)x+jIw2v2r4μruw1, (34)
    jIw2t=(cd+ca)r((rw2)xu)xjIw1v2r2μrrv3x4μruw2, (35)
    jIw3t=(cd+ca)r((rw3)xu)x+(cd+ca)uw3r2+2μr(rv2)x4μruw3, (36)
    cVθt=κ(r2θxu)x+1u[(λ+2μ)(rv1)xRθ](rv1)x+(μ+μr)(rv2)2xu+(c0+2cd)(rw1)2xu+(cd+ca)(rw2)2xu+(μ+μr)r2v23xu+(cd+ca)r2w23xu2μ(v21+v22)x2cd(w21+w22)x+4μr(uw21+uw22+uw23)+4μrrw2v3x4μrw3(rv2)x, (37)

    together with the initial and boundary conditions

    u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),θ(x,0)=θ0(x),xΩ, (38)
    v(0,t)=v(L,t)=w(0,t)=w(L,t)=0,θx(0,t)=θx(L,t)=0,t>0. (39)

    A a result of (22) and (26), we can conclude that r(x,t) is determined by

    rt(x,t)=v1(x,t),r(x,t)rx(x,t)=u(x,t), (40)
    r0(x)=(a2+2x0u0(y)dy)12. (41)

    It is easy to see from (40) and (41) that the following is satisfied:

    rt(x,t)=υ1(x,t),r(x,t)rx(x,t)=u(x,t). (42)

    Let us mention some related results in this direction. When w=0, it reduces to be classical Navier-Stokes equations, which provide a suitable model to motion of several important fluids, such as water, oil, air, etc., the existence and asymptotic behavior of Navier-Stokes equations has been regarded as an important problem in the fluid of dynamics, and has been receiving much attention for many researchers (see[1,4,19,17,18,20,28,32,30,31] and references therein). Among them, Fujita and Kato [19] obtained the global well-posedness for small initial data and the local well-posedness for any initial data in Hs(Rn) with sn21. Kato [28] improved results have been established in Ln(Rn). Recently, Lei and Lin [30] proved global well-posedness result in the space χ1. Li and Liang [29] proved large time behavior for one dimensional compressible Navier-Stokes equations in unbounded domains with large data.

    For the micropolar fluids case (i.e., w0), compared with the classical Navier-Stokes equations, the angular velocity w in this model brings benefit and trouble. Benefit is the damping term -vw can provides extra regularity of w, while the term vw2 is bad, it increases the nonlinearity of the system. In the one dimensional case, Mujaković made a series of efforts in studying the local-in-time existence and uniqueness, the global existence and regularity of solutions to an initial-boundary value problem with both homogenous [33,34,35] and non-homogenous boundary conditions [36,37,38] respectively. Later, Huang and Nie [25] proved the exponential stability. Recently, the global attractor of this system has been established in [27]. Besides, we would also like to refer to the works in [5,14,15] for the 1D micropolar fluid model.

    In the three dimensional case, for the spherical symmetric model of described micropolar fluid in a bounded annular domain, the local existence, uniqueness, global existence and the large time behavior and regularity of the solution has been proved in [6,7,8,9,10], and the exponential stability and regularity of the spherically symmetric solutions with large initial data has been established in [24,23]. Recently, for the spherical symmetric model of described micropolar fluid in an exterior unbounded domain, we proved the large time behavior for spherically symmetric flow of viscous polytropic gas with large initial data in [26]. In the case of cylinder symmetry, which model described micropolar fluid in a bounded domain with two coxial cylinders that present the solid thermoinsulated walls, Dražić and Mujaković [11] established the local existence of generalized solutions, then they proved global existence [12] and the uniqueness [13], Huang and Dražić [21,22] studied the large time behavior of the cylindrically symmetric with small initial data, but the regularity is open. Besides, we would like to mention the work on the global wellposedness of the three-dimensional magnetohydrodynamic equations, Wang and Wang [41] obtained the global existence results for classical 3-D MHD (α=1). Wang and Qin [40] obtained global wellposedness and analyticity results to 3-D generalized magnetohydrodynamic equations. Later, Ye [42] obtained the global existence results for classical 3-D GMHD (12α1).

    As mentioned above, the regularity and exponential stability of generalized (global) solutions in H2(Ω) has never been studied for system (14)-(21) with boundary conditions (12) and initial conditions (13). Therefore, we shall continue the work by Huang and Dražić [22] and establish the regularity and exponential stability of solutions with small initial data.

    Here we study the problem (30)-(37) on the spatial domain Ω. We introduce the space

    H1+={(u,v,w,θ)H1(0,L)×H1(0,L)×H1(0,L)×H1(0,L):v(0,t)=v(L,T)=0,w(0,t)=w(L,t)=0}, (43)
    H2+={(u,v,w,θ)H2(0,L)×H2(0,L)×H2(0,L)×H2(0,L):v(0,t)=v(L,T)=0,w(0,t)=w(L,t)=0,θx(0,t)=θx(L,t)=0}, (44)

    which becomes the metric space equipped with the metrics induced from the usual norms. In this paper we will denote by Lˉp,1ˉp+,Wm,ˉp,mN,H1=W1,2,H10=W1,20 denote the usual (Sobolev) spaces on [0,1]. In addition, B denotes the norm in the space B, we also put =L2. Subscripts t and x denote the (partial) derivatives with respect to t and x, respectively. We use Ci(i=1,2) to denote the generic positive constant depending only on Hi norm of initial datum (u0,v0,w0,θ0), minx[0,L]u0(x) and minx[0,L]θ0(x), but independent of variable t.

    We assume that the initial data have the following properties

    E0=L0(12|v0|2+jI2|w0|2+cVθ0)dx,u0,θ0m, (45)

    where m is a positive constant.

    Now, we are in a position to state our main result.

    Theorem 1.1. Suppose that initial (u0,v0,w0,θ0)H2+ and (45) hold, there exists a constant α0=α0(C1)>0, such that if E0α0, problem (30)-(39) has a unique generalized global solution (u(t),v(t),w(t),θ(t))H2+ verifying that for any t>0,

    uuL([0,+),H2(Ω))L2([0,+),H2(Ω)),vL([0,+),H2(Ω))L2([0,+),H3(Ω)),wL([0,+),H2(Ω))L2([0,+),H3(Ω)),θθL([0,+),H2(Ω))L2([0,+),H3(Ω)). (46)

    Moreover, there exists a positive constant γ2=γ2(C2) such that for any fixed γ(0,γ2] and for any t>0, the following estimate holds

    uu,v,w,θθC2eγt (47)

    where u=1LL0u0(x)dx, θ=1cVLL0(12|v0(x)|2+jI2|w0(x)|2+cVθ0(x))dx, r=(a2+2ux)1/2.

    In this section, we shall complete the proof of Theorem 1.1. The global existence of cylindrically symmetric solutions for system (30)- (39) was proved in [22], we shall continue the work and prove the regularity and exponential of the solution. We begin with the following Lemma.

    Lemma 2.1. (See [24] and [22]) If (u0,v0,w0,θ0)H1+ and E0α0 are true, there exists a unique global weak solution (u,v,w,θ)H1+ to the problem (30)-(39) satisfies the following estimates

    0<ar(x,t)b,(x,t)Ω×[0,+),0<C11u(x,t),θ(x,t)C1(x,t)Ω×[0,),uu2H1+v12H1+v22H1+v32H1+w12H1+w22H1
    +w32H1+θθ2H1+t0(uu2H1+v12H2+v22H2+v32H2+w12H2+w22H2+w32H2+θθ2H2+v1t2+v2t2+v3t2+w1t2+w2t2+w3t2+θt2)dsC1. (48)

    Moreover, there exists constant γ1=γ1(C1)>0, for any fixed γ(0,γ1] and t>0

    eγt(uu2H1+v12H1+v22H1+v32H1+w12H1+w22H1+w32H1+θθ2H1)+t0eγs(ux2+v1x2H1+v2x2H1+v3x2H1+w1x2H1+w2x2H1+w3x2H1+θx2H1+v1t2+v2t2+v3t2+w1t2+w2t2+w3t2+θt2)dsC1, (49)

    where u,θ and r are the same as in Theorem 1.1.

    Lemma 2.2. Under the assumptions of Theorem 1.1, the following estimates hold for any t>0:

    v1t(t)2+v1xx(t)2+v1x(t)2L+t0v1tx(s)2dsC2, (50)
    v2t2+v2xx2+v2x2L+t0v2tx2dsC2, (51)
    v3t2+v3xx2+v3x(t)2L+t0v3tx2dsC2. (52)

    Proof. Differentiating (31) with respect to t, multiplying the resulting equation by v1t in L2(0,L), using an integration by parts, we have

    12ddtv1t2=L0[r((λ+2μ)(rv1)xuRθu)x]tv1tdx+L0(v22r)tv1tdx(λ+2μ)L0r2v1tx2udx+C1v1t(v1x+ux+v1xux+θx+v1xx+v2t+v22)+C1v1tx(v1t+v21x+θ1t+v1x+v1xv1)C11v1tx2+C1(v1x2H1+v1t2+v2t2+θt2+ux2+θx2). (53)

    Integrating (53) with respect to t over [0,t] (t>0), and using Lemma 2.1, there holds

    v1t2+t0v1tx2dsC2. (54)

    Moreover, integrating (31) with respect to x, and using Lemma 2.1 and Young's inequality, we obtain

    v1xxC1(ux+θx+v1t+uxv1xL+v22)12v1xx+C1(ux+θx+vt+vx). (55)

    Now the above facts along with the Gagliardo-Nirenberg interpolation inequality yields

    v1xLC1v1x12v1xx12+C1v1x.

    Combined (54) and (55) to arrive at

    v1t2+v1xx2+v1x2L+t0v1tx2dsC2.

    Similarly, differentiating (32) with respect to t, multiplying the resulting equations by v2t, and then integrating by parts, we obtain

    12ddtv2t2=(μ+μr)L0[r((rv2)xu)x]tv2tdxL0(v1v2r)tv2tdx2μL0(rw3x)tv2t(μ+μr)L0r2v22xtudx+C1v2t(v2xx+v2x+ux+v2xv1x+v2t+w3t)+C1v2xt(v2t+v2xv1x+v2xv1+v1xv2+w3t)C11v2tx2+C1(v2x2H1+v2t2+ux2+w3t2). (56)

    We integrate (56) with respect to t, and use Lemma 2.1 to have

    v2t2+t0v2tx2dsC2. (57)

    Furthermore, integrating (32) with respect to x, and using the same way, we obtain

    v2xxC1(ux+v2t+w3x+uxv2xL)12v2xx+C1(ux+v2t+v2x+w3x). (58)

    We use the Gagliardo-Nirenberg interpolation inequality to give

    v2xLC1v2x12v2xx12+C1v2x.

    Combining with (57)-(58), we arrive at

    v2t2+v2xx2+v2x2L+t0v2tx2dsC2.

    Likewise, differentiating (33) with respect to t, multiplying by v3t and integrating by parts, we have

    12ddtv3t2=(μ+μr)L0[r((rv3)xu)x]tv3tdx+(μ+μr)L0uv3r2v3tdx(μ+μr)L0r2v23xtudx+C1v3t(v3x+ux+v3xux+v3xx)+C1v3tx(v3t+v3x+w2+w2t)C1v3tx2+C1(v3t2H1+v3t2+ux2+w2t2). (59)

    Integrating (59) with respect to t over [0,t] (t>0), and using Lemma 2.1 to have

    v3t2+t0v3tx2dsC2. (60)

    Similarly argument, we integrate (33) with respect to x, and apply Young's inequality and Lemma 2.1 to yield

    v3xxC1(v3t+ux+uxv3xL)12v3xx+C1(ux+v3t+v3x). (61)

    Here, we have from the Gagliardo-Nirenberg interpolation inequality that

    v3xLC1v3x12v3xx12+C1v3x,

    which, combined with (60)-(61), gives

    v3t2+v3xx2+v3x2L+t0v3tx2dsC2.

    Thus, we complete the proof.

    Lemma 2.3. The following estimates hold true for any t>0

    w1t2+w1xx2+w1x2L+t0w1tx2dsC2, (62)
    w2t2+w2xx2+w2x2L+t0w2tx2dsC2, (63)
    w3t2+w3xx2+w3x2L+t0w3tx2dsC2. (64)

    Proof. Differentiating (34) with respect to t, multiplying the resulting identity by w1t and integrating by parts, we obtain

    12jIddtw1t2=(c0+2cd)L0[r((rw1)xu)x]tw1tdx+jIL0(w2v2r)tw1tdx4μrL0(uw1)tw1tdx(c0+2cd)L0r2w21txudx+C1w1t(w1x+w1xx+w1xux+ux+w1xv1x+w2t+v2t)+c1w1tx(w1t+w1xv1x+w1x+w1v1x)C11w1tx2+C1(w1x2H1+w1t2+ux2+w2t2+v2t2+v1x2). (65)

    Integrating (65) with respect to t over [0,t] (t>0) and integrating (34) with respect to x, and using Lemma 2.1, respectively, we arrive at

    w1t2+t0w1tx2dsC2, (66)
    w1xxC1(w1t+u+w1xLux+w2v2+w1u)12w1xx+C1(w1t+w1x+ux). (67)

    We can use the Gagliardo-Nirenberg interpolation inequality to yield

    w1xLC1w1x12w1xx12+C1w1x,

    which together with (66)-(67) further implies that

    w1t2+w1xx2+w1x2L+t0w1tx2dsC2.

    Similarly, differentiating (35) with respect to t, multiplying by w2t and applying integration by parts to find

    12jIddtw2t2=(ca+cd)L0[r((rw2)xu)x]tw2tdxjIL0(w1v2r)tw2tdx2μrL0(rv3x)tw2tdx4μrL0(uw2)tw2tdx(ca+cd)L0r2w2xtudx+C1w2t(w2x+w2xx+w2ux+w2xux+w2v1x+w2xv1x+w2v1x+v3x+v3xt)+C1w2tx(w2t+w2v1x+w2xv1+w1tv2+w1v2t)C11w2tx2+C1(w2x2H1+ux2+v1x2+w2x2+v3x2+v3tx2+w2t2+w1t2+v2t2). (68)

    Integrating (68) with respect to t over [0,t] (t>0), and using Lemma 2.1, we arrive at

    w2t2+t0w2tx2dsC2. (69)

    We integrate (35) with respect to x, and apply Lemma 2.1 and Young's inequality to obtain

    w2xxC1(w2t+w2+w2ux+w2xux+w1v2+v3x+w2u)C1(w2t+ux+w2xLux+v3x+w1+v2+u+w2)12w2xx+C1(ux+w2t+w2x+v3x). (70)

    Similar argument, we deduce from the Gagliardo-Nirenberg interpolation inequality that

    w2xLC1w2x12w2xx12+C1w2x.

    Combining with (69)-(70), we have

    w2t2+w2xx2+w2x2L+t0w2tx2dsC2.

    Differentiating (36) with respect to t, multiplying the resulting equation by w3t and using integrating by parts, we obtain

    12jIddtw3t2=(ca+cd)L0[r((rw3)xu)x]tw3tdx+(ca+cd)L0(uw3r2)tw3tdx+2μrL0[(rv2)x]tw3tdx4μrL0(uw3)tw3tdx
    (ca+cd)L0r2w23txudx+C1w3t(w3xx+w3xux+ux+v1x+w3t+w3xv1x+v2t+v2x+v2xt)+C1w3tx(w3v1x+w3t+w3x+w3v1x)C11w3tx2+C1(w3x2H1+ux2+w3t2+v1x2+v2t2+v2x2+v2tx2). (71)

    Integrating (71) with respect to t over [0,t] (t>0), we can obtain

    w3t2+t0w3tx2dsC2. (72)

    Next, integrating (36) with respect to x, and then applying the same way to yield

    w3xxC1(w3t+ux+w3xLux(t))12w3xx+C1(w3t+ux+w3x), (73)

    where we have used the following simple Gagliardo-Nirenberg interpolation inequality

    w3xLC1w3x12w2xx12+C1w3x.

    By (72) and (73), we can get

    w3t2+w3xx2+w3x2L+t0w3tx2dsC2.

    The proof is complete.

    Lemma 2.4. Under the assumptions of Theorem 1.1, the following estimate holds for any t>0:

    θt2+θxx2+θx2L+t0θtx2dsC2, (74)

    Proof. Differentiating (37) with respect to t, multiplying by θt, and using integration by parts, we get

    12cvddtθt2=κL0(r2θxu)xtθtdx+L0[1u[(λ+2μ)(rv1)xRθ](rv1)x]tθtdx+(μ+μr)L0((rv2)2xu)tθtdx+(c0+2cd)L0((rw1)2xu)tθtdx+(ca+cd)L0((rw2)2xu)tθtdx+(μ+μr)L0(r2v23xu)tθtdx+(ca+cd)L0(r2w23xu)tθtdx2μrL0(v21+v22)xtθtdx2cdL0(w21+w22)xtθtdx+4μrL0(uw21+uw22+uw23)tθtdx+4μrL0(rw2v3x)tθtdx4μrL0(w3(rv2)x)tθtdx
    C11θxt2+C1(θx2+v1x2H1+v1xt2+v2x2H1+v2xt2+v3x2H1+v3xt2+w1x2H1+w1xt2+w2x2H1+w2xt2+w3x2H1+w3xt2+ux+v1t2+v2t2+v3t2+w1t2+w2t2+w3t2+θt2),

    which, together with (50)-(52), (62)-(64) and Lemma 2.1, gives

    θt2+θxx2+t0θtx2dsC2. (75)

    By virtue of the Gagliardo-Nirenberg interpolation inequality and (75), we can obtain (74). The proof is complete.

    Lemma 2.5. Under the assumptions of Theorem 1.1, the following estimates hold for any t>0:

    uxx2+t0uxx2dsC2, (76)
    t0(v1xxx2+v2xxx2+v3xxx2+w1xxx2+w2xxx2+w3xxx2+θxxx2)dsC2. (77)

    Proof. Differentiating (31) with respect to x, we have

    v1tx=[r((λ+2μ)(rv1)xuRθu)x]x+(v22r)x=uv1tr2uv22r2+(λ+2μ)r(rv1)xxxu(λ+2μ)r(rv1)xxuxu2(λ+2μ)r(rv1)xxuxu2(λ+2μ)r(rv1)xuxxu2+2(λ+2μ)r(rv1)xu2xu3rR(λ+2μ)θxxu+(λ+2μ)rRθxuxu2+(λ+2μ)rRθxuxu2+(λ+2μ)rRθuxxu22(λ+2μ)rRθu2xu3+2v2v2xrv22ur3.

    By virtue of (30) (utxx=(rv1)xxx), we can rewrite the above equation as

    (λ+2μ)t(uxxu)+(λ+2μ)Rθuxxu2=r1v1tx+E(x,t), (78)

    where

    E(x,t)=r3uv1t+r3uv22+2(λ+2μ)(rv1)xxuxu22(λ+2μ)(rv1)xu2xu3+R(λ+2μ)θxxu2(λ+2μ)Rθxuxu2+2(λ+2μ)Rθu2xu32r2v2v2x+r4v22u.

    Multiplying (78) by uxxu, and using Young's inequality, by virtue of (50)-(52), (62)-(64), (74) and Lemma 2.1, we can obtain

    ddtuxxu2C11uxxu2+C1E(x,t)2, (79)

    where

    E(x,t)212C1uxxu2+C1(v1t2+v1x2H1+ux2+θx2H1+v2x2). (80)

    Integrating (79) with respect to t, together with (50)-(52), (62)-(64), (74) and Lemma 2.1 that

    ux2+L0uxx2dsC2.

    On the other hand, differentiating (31) with respect to x, we have from Lemma 2.1 and (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality that

    v1tx=uv1tr2uv22r2+(λ+2μ)r(rv1)xxxu2(λ+2μ)r(rv1)xxuxu2(λ+2μ)r(rv1)xuxxu2+2(λ+2μ)r(rv1)xu2xu3rR(λ+2)θxxu+2(λ+2μ)rRθxuxu2+(λ+2μ)rRθuxxu22(λ+2μ)rRθu2xu3+2v2v2xrv22ur3,

    which implies

    v1txC1(v1xH2+uxH1+θxH1+v2x),

    or

    v1xxxC1(v1xH1+v1tx+uxH1+θxH1+v2x).

    Differentiating (32) with respect to x, using Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality to find

    v2tx=(μ+μr)rx(rv2)xxu(μ+μr)rx(rv2)xuxu2+(μ+μr)r(rv2)xxxu3(μ+μr)r(rv2)xxuxu2(μ+μr)r(rv2)xxuxu2(μ+μr)r(rv2)xuxxu2+2(μ+μr)r(rv2)xu2xu3(v2xv1+v2v1x)rv2v1rxr22μr(rxw3x+rw3xx),

    thus

    v2txC1(v2xH2+uxH1+v1x+w3xH1),

    or

    v2xxxC1(v2xH1+v2tx+uxH1+w3xH1+v1x).

    Similarly, differentiating (33) with respect to x, and applying Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality to deduce

    v3tx=(μ+μr)rx(rv3)xxu(μ+μr)rx(rv3)xuxu2+(μ+μr)r(rv3)xxxu(rv3)xxuxu2(μ+μr)r[(rv3)xxux+(rv3)xuxx]u22(rv3)xu2xuu4+(μ+μr)uxv3+uv3xr2(μ+μr)2uv3rxr3+2μr(rxxw2+rxw2x+rxw2x+rw2xx),

    thus

    v3txC1(v3xH2+uxH1+w2xH1),

    or

    v3xxxC1(v3xH1+v3tx+uxH1+w2xH1).

    We can differentiate (34) with respect to x, combine Lemma 2.1 and the Gagliaro-Nirenberg inequality to get

    jIw1tx=(c0+2cd)rx(rw1)xxu(c0+2cd)rx(rw1)xuxu2+(c0+2cd)r(rw1)xxxu(rw1)xxux)xxu2(c0+2cd)r[(rw1)xxux+(rw1)xuxx]u22(rw1)xu2xuu4+jIw2xv2+w2v2xrjIw2v2rxr24μr(uxw1+uw1x).

    Then

    w1txC1(w1xH2+uxH1+v2x+w2x),

    or

    w1xxxC1(w1xH1+w1tx+uxH1w2x+v2x).

    Differentiating (35) with respect to x, using Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality to yield

    jIw2tx=(ca+cd)rx(rw2)xxu(ca+cd)rx(rw2)xuxu2+(ca+cd)r(rw2)xxxu(rw2)xxuxu2jIw1xv2+w1v2xr(ca+cd)r[(rw2)xxux+(rw2)xuxx]u22(rw2)xu2xuu4+jIw1v2rxr22μr(rxw3xx+rv3xx)4μr(uxw2+uw2x),

    thus

    w2txC1(w2xH2+uxH1+v2x+w1x+w3x+v3xH1),

    or

    w2xxxC1(w2xH1+w2tx+uxH1+w1x+v2x+w3x+v3xH1).

    Likewise, we differentiate (36) with respect to x, apply Lemma 2.1 and the Gagliaro-Nirenberg inequality to arrive at

    jIw3xt=(ca+cd)rx(rw3)xxu(ca+cd)rx(rw3)xuxu2+(ca+cd)r(rw3)xxxu(rw3)xxuxu2(ca+cd)r(rw3)xxux+(rw3)xuxxu2+(ca+cd)r2(rw3)xu2xu3+(ca+cd)uxw3+uw3xr22(ca+cd)u2w3r4+2μr(rxxv2+rxv2x+rxv2x+rv2xx)4μr(uxw3+uw3x).

    Then we have

    w3txC1(w3xH2+uxH1+v2xH1),

    or

    w3xxxC1(w3xH1+w3tx+ux(t)H1+v2xH1).

    We can differentiate (37) with respect to x, and easily deduce from Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality that

    θtxC2(θxH2+uxH1+v1xH1+v2xH1+v3xH1+w1xH1+w2xH1+w3xH1),

    or

    θxxxC2(θxH1+θxt+uxH1+v1xH1+v2xH1+v3xH1+w1xH1+w2xH1+w3xH1).

    According to the estimate above, we have

    v1txC2(v1xH2+uxH1+θxH1+v2x), (81)
    v2txC2(v2xH2+uxH1+v1x+w3xH1), (82)
    v3txC2(v3xH2+uxH1+w2xH1), (83)
    w1txC2(w1xH2+uxH1+v2x+w2x), (84)
    w2txC2(w2xH2+uxH1+v2x+w1x+w3x+v3xH1), (85)
    w3txC2(w3xH2+uxH1+v2xH1), (86)
    θtxC2(θxH2+uxH1+v1xH1+v2xH1+v3xH1+w1xH1+w2xH1+w3xH1), (87)

    or

    v1xxxC2(v1xH1+v1tx+uxH1+θxH1+v2x), (88)
    v2xxxC2(v2xH1+v2tx+uxH1+w3xH1+v1x), (89)
    v3xxxC2(v3xH1+v3tx+uxH1+w2xH1), (90)
    w1xxxC2(w1xH1+w1tx+uxH1w2x+v2x), (91)
    w2xxxC2(w2xH1+w2tx+uxH1+w1x+v2x+w3x+v3xH1), (92)
    w3xxxC2(w3xH1+w3tx+uxH1+v2xH1), (93)
    θxxxC2(θxH1+θxt+uxH1+v1xH1+v2xH1+v3xH1+w1xH1+w2xH1+w3xH1). (94)

    By (50)-(52), (62)-(64), (74), (76), (81)-(87) and Lemma 2.1, we get (77). The proof is complete.

    Lemma 2.6. Under assumptions of Theorem 1.1, there exists a positive constant γ2=γ2(C2) such that for any fixed γ(0,γ2], the following estimate holds for any t>0

    eγt(v1xx2+v2xx2+v3xx2+w1xx2+w2xx2+w3xx2+uxx2+θxx2+v1t2+v2t2+v3t2+w1t2+w2t2+w3t2+θt2)+t0eγs(v1x2H2+v2x2H2+v3x2H2+w1x2H2+w2x2H2+w3x2H2+θx2H2+ux2H1+v1tx2+v2tx2+v3tx2+w1tx2+w2tx2+w3tx2+θtx2)dsC1. (95)

    Proof. We multiply (53) by eγt, integrate the resulting over [0,t], and then apply Lemma 2.1 and Lemma 2.2 to deduce

    eγtv1t2+t0eγsv1tx2dsC2+C1t0eγs(v1x2H1+v1t2+v2t2+θt2+ux2+θx2)dsC2. (96)

    Multiplying (56), (59), (65), (68), (71) and (75) by eγt, and adding them together, by virtue of Lemmas 2.1-2.2, we can conclude

    eγt(v2t2+v3t2+w1t2+w2t2+w3t2+θt2)+t0eγs(v1tx2+v2tx2+v3tx2+w1tx2+w2tx2+w3tx2+θtx2)dsC2. (97)

    On the other hand, multiplying (79) by eγt, integrating over [0,t] to have

    eγtuxxu2+12C1t0eγsuxxu2dsC2+γt0eγsuxxu2+C1t0eγs(v1x2H1+ux2+v1t2+θx2H1+v2x2)ds. (98)

    Picking γ2=min{14C1,γ1} such that for any fixed γ(0,γ2], using Lemmas 2.1 - 2.2 and (96)-(97), we obtain

    eγtuxx2+t0eγsuxx2dsC2. (99)

    By (31)-(37), (81)-(87), (96)-(97) and (99), we have

    eγt(v1xx2+v2xx2+v3xx2+v1xx2+w1xx2+w2xx2+w3xx2+θxx2)+t0eγs(v1xx2+v2xx(t)2+v3xx2+w1xx2+w2xx2+w3xx2+θxx2)C2,

    which, combined with (96)-(97) and (99), yields (95). The proof is complete.

    Proof of Theorem 1.1. According to Lemmas 2.2-2.6, Theorem 1.1 is complete.

    The authors are grateful to the referees for their helpful suggestions which improved the presentation of this paper. This research was supported by the NSFC (No. 11871212 and No. 11501199) and Key Research Projects in Colleges of Henan Province (No. 20A110026).

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