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.
1.
Introduction
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
with notation:
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:
For simplicity reasons, we assume that
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]:
where G={(x1,x2,x3)∈R3,0<a<r<b<+∞,x3∈R,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]:
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
where ˜v1(ξ,t)=v1(r(ξ,t),t) and
From (14) and it follows that
which implies
Now, we have ξ∈Ω=[0,L], where
Moreover, differentiating (24) with respect to ξ yields
Let us introduce the temporary notation ˜ϕ(ξ,t) for ϕ(r(ξ,t),t), we obtain
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),x∈Ω,t≥0 as follows:
together with the initial and boundary conditions
A a result of (22) and (26), we can conclude that r(x,t) is determined by
It is easy to see from (40) and (41) that the following is satisfied:
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 s≥n2−1. Kato [28] improved results have been established in Ln(Rn). Recently, Lei and Lin [30] proved global well-posedness result in the space χ−1. 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., w≠0), 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 (12≤α≤1).
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
which becomes the metric space equipped with the metrics induced from the usual norms. In this paper we will denote by Lˉp,1≤ˉp≤+∞,Wm,ˉp,m∈N,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
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 E0≤α0, problem (30)-(39) has a unique generalized global solution (u(t),v(t),w(t),θ(t))∈H2+ verifying that for any t>0,
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
where u∗=1L∫L0u0(x)dx, θ∗=1cVL∫L0(12|v0(x)|2+jI2|w0(x)|2+cVθ0(x))dx, r∗=(a2+2u∗x)1/2.
2.
Proof of Theorem 1.1
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 E0≤α0 are true, there exists a unique global weak solution (u,v,w,θ)∈H1+ to the problem (30)-(39) satisfies the following estimates
Moreover, there exists constant γ1=γ1(C1)>0, for any fixed γ∈(0,γ1] and ∀t>0
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:
Proof. Differentiating (31) with respect to t, multiplying the resulting equation by v1t in L2(0,L), using an integration by parts, we have
Integrating (53) with respect to t over [0,t] (t>0), and using Lemma 2.1, there holds
Moreover, integrating (31) with respect to x, and using Lemma 2.1 and Young's inequality, we obtain
Now the above facts along with the Gagliardo-Nirenberg interpolation inequality yields
Combined (54) and (55) to arrive at
Similarly, differentiating (32) with respect to t, multiplying the resulting equations by v2t, and then integrating by parts, we obtain
We integrate (56) with respect to t, and use Lemma 2.1 to have
Furthermore, integrating (32) with respect to x, and using the same way, we obtain
We use the Gagliardo-Nirenberg interpolation inequality to give
Combining with (57)-(58), we arrive at
Likewise, differentiating (33) with respect to t, multiplying by v3t and integrating by parts, we have
Integrating (59) with respect to t over [0,t] (t>0), and using Lemma 2.1 to have
Similarly argument, we integrate (33) with respect to x, and apply Young's inequality and Lemma 2.1 to yield
Here, we have from the Gagliardo-Nirenberg interpolation inequality that
which, combined with (60)-(61), gives
Thus, we complete the proof.
Lemma 2.3. The following estimates hold true for any t>0
Proof. Differentiating (34) with respect to t, multiplying the resulting identity by w1t and integrating by parts, we obtain
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
We can use the Gagliardo-Nirenberg interpolation inequality to yield
which together with (66)-(67) further implies that
Similarly, differentiating (35) with respect to t, multiplying by w2t and applying integration by parts to find
Integrating (68) with respect to t over [0,t] (t>0), and using Lemma 2.1, we arrive at
We integrate (35) with respect to x, and apply Lemma 2.1 and Young's inequality to obtain
Similar argument, we deduce from the Gagliardo-Nirenberg interpolation inequality that
Combining with (69)-(70), we have
Differentiating (36) with respect to t, multiplying the resulting equation by w3t and using integrating by parts, we obtain
Integrating (71) with respect to t over [0,t] (t>0), we can obtain
Next, integrating (36) with respect to x, and then applying the same way to yield
where we have used the following simple Gagliardo-Nirenberg interpolation inequality
By (72) and (73), we can get
The proof is complete.
Lemma 2.4. Under the assumptions of Theorem 1.1, the following estimate holds for any t>0:
Proof. Differentiating (37) with respect to t, multiplying by θt, and using integration by parts, we get
which, together with (50)-(52), (62)-(64) and Lemma 2.1, gives
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:
Proof. Differentiating (31) with respect to x, we have
By virtue of (30) (utxx=(rv1)xxx), we can rewrite the above equation as
where
Multiplying (78) by uxxu, and using Young's inequality, by virtue of (50)-(52), (62)-(64), (74) and Lemma 2.1, we can obtain
where
Integrating (79) with respect to t, together with (50)-(52), (62)-(64), (74) and Lemma 2.1 that
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
which implies
or
Differentiating (32) with respect to x, using Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality to find
thus
or
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
thus
or
We can differentiate (34) with respect to x, combine Lemma 2.1 and the Gagliaro-Nirenberg inequality to get
Then
or
Differentiating (35) with respect to x, using Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality to yield
thus
or
Likewise, we differentiate (36) with respect to x, apply Lemma 2.1 and the Gagliaro-Nirenberg inequality to arrive at
Then we have
or
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
or
According to the estimate above, we have
or
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
Proof. We multiply (53) by eγt, integrate the resulting over [0,t], and then apply Lemma 2.1 and Lemma 2.2 to deduce
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
On the other hand, multiplying (79) by eγt, integrating over [0,t] to have
Picking γ2=min{14C1,γ1} such that for any fixed γ∈(0,γ2], using Lemmas 2.1 - 2.2 and (96)-(97), we obtain
By (31)-(37), (81)-(87), (96)-(97) and (99), we have
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.
Acknowledgments
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).