EXPONENTIAL STABILITY AND REGULARITY OF COMPRESSIBLE VISCOUS MICROPOLAR FLUID WITH CYLINDER SYMMETRY

. This paper is concerned with three-dimensional compressible viscous and heat-conducting micropolar ﬂuid in the domain to the subset of R 3 bounded with two coaxial cylinders that present the solid thermoinsulated walls, being in a thermodynamical sense perfect and polytropic. We prove that the regularity and the exponential stability in H 2 .

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 readsρ with notation: ρ − mass density v − velocity w − microrotation velocity E − internal energy density θ − absolute temperature T − stress tensor C − couple stress tensor q − heat f lux density vector f − body f orce density g − body couple density p − pressure j I − microinertia density (a positive constant) k θ − heat conduction coef f icien µ r , c 0 , c a and c d − coef f icients of microviscosity R − specif ic gas constant c v − specif ic heat (positive constant) 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]: t = 0 : (ρ, v, w, θ)(r, 0) = (ρ 0 (r), v 0 (r), w 0 (r), θ 0 (r)), r ∈ G, (12) v| ∂G = 0, w| ∂G = 0, ∂θ ∂r ∂G = 0, t > 0, (13) where is the spatial domain of our problem and v = (υ 1 , υ 2 , υ 3 ), w = (w 1 , w 2 , w 3 ) 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 whereṽ 1 (ξ, t) = v 1 (r(ξ, t), t) and From (14) and it follows that Moreover, differentiating (24) with respect to ξ yields ∂r ∂ξ = 1 r(ξ, t)ρ(r(ξ, t), t) .
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 H s (R n ) with s ≥ n 2 − 1. Kato [28] improved results have been established in L n (R n ). 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 vw 2 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 ( 1 2 ≤ α ≤ 1). As mentioned above, the regularity and exponential stability of generalized (global) solutions in H 2 (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 Lp, denote the usual (Sobolev) spaces on [0, 1]. In addition, · B denotes the norm in the space B, we also put · = · L 2 . Subscripts t and x denote the (partial) derivatives with respect to t and x, respectively. We use C i (i = 1, 2) to denote the generic positive constant depending only on H i norm of initial datum 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.

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 2.2.
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 v 1t in L 2 (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 v 2t , and then integrating by parts, we obtain We integrate (56) with respect to t, and use Lemma 2 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 v 3t 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 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 w 1t 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 w 2t 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 w 3t 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 θ t , 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.