OPTIMAL DECAY TO THE NON-ISENTROPIC COMPRESSIBLE MICROPOLAR FLUIDS

. In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic compressible micropo- lar ﬂuid. For the initial data near the given equilibrium we prove the global well-posedness of classical solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. Moreover, it turns out that the density, the velocity and the temperature tend to the corresponding equi- librium state with rate (1+ t ) − 3 / 4 in L 2 norm and the micro-rotational velocity tends to the equilibrium state with the faster rate (1+ t ) − 5 / 4 in L 2 norm. The proof is based on the detailed analysis of the Green function and time-weighted energy estimates.

Due to its importance in mathematics and physics, there is a lot of literature devoted to the mathematical theory of the micropolar fluid system. For the compressible equations of the micropolar fluids, Mujaković has made a series of efforts for this model in one dimensional space or with spherical symmetry in three dimensional space. The authors considered the local-in-time existence and uniqueness in [15], the global existence in [14] and regularity in [16] of the solutions to an initialboundary value problem with homogenous boundary conditions of the compressible one-dimensional micropolar fluid system respectively. Recently, Liu-Zhang [13] derived the long-time behavior of the solutions to the compressible micropolar fluids with external force and [12] proved the optimal decay to the isentropic compressible micropolar fluids. Then, Wang-Wu [23] obtained the pointwise estimates of solutions to the isentropic compressible micropolar fluids through the analysis of the Green function. However, for the non-isentropic micropolar system, the related results are few. Very recently, Huang-Liu-Zhang [7] generalized the isentropic case to the non-isentropic case with additional efforts by taking care of the temperature equation and obtained the global solutions with general initial data and vacuum. Based on these works, in this article, we will focus on the optimal long-time behavior to the non-isentropic micropolar system. Now, the main results concerning the global existence and large time behavior of solutions to the Cauchy problem (1.1)-(1.3) are stated as follows: Theorem 1.1. For any real number s ≥ 4. There exists small enough δ 1 > 0 and C > 0, such that H s . Moreover, if we further assume that for small enough δ 2 > 0, then, there exists a C 1 > 0 such that the solutions [ρ, u, w, θ] satisfy for any t ≥ 0.
Remark 1.1. Here, the algebraic decay rate (1.4) is optimal in sense that this rate coincides with that of the corresponding linear system.
The main strategy to prove the time decay rates stated in Theorem 1.1 relies on the Green function G of the linearized viscous non-isentropic compressible micropolar fluids. In fact, the solution to the linearized homogenous system can be written as the sum of the fluid part and the electromagnetic part in the form of  The decomposition is quite useful in dealing with complex linearized system containing curl. We can refer [1,3,4,5,23] for the detailed spectrum analysis by using a similar decomposition. Notice that the two terminologies, fluid part and electromagnetic part have been used in [1,3,5]. With the help of the above decomposition, we give explicit representations of solutions to the two eigenvalue problems by learning their Green function G 1 , G 2 respectively. The Fourier transform of G 1 and G 2 have different behaviors over different frequency domains. For the fluid part For the electromagnetic part with two properly chosen constants 0 < 1 K < ∞. Thus, from the above estimates, it is well known the solution over the high frequency domain decays exponentially while over the low frequency domain it decays polynomially with the rate of the heat kernel. It should be emphasized that those time decay rates are based on the elementary Lyapunov property of the linearized system, which implies (1.5) However, it will fail to yield the ones for the nonlinear solution given in Theorem 1.1.

LVQIAO LIU AND LAN ZHANG
Next, we point out some relationships of time rates between the linearized system and nonlinear ones. From Corollary 2.3 the solution [ρ, u, w, θ] to the linearized homogeneous system corresponding to system (2.3) decays as for any t ≥ 0, provided that [ρ 0 −1,u 0 , w 0 ,θ 0 −1] belongs to L 1 ∩H s for properly large N.
Moreover, a general approach for obtaining the optimal time decay of solutions in L p space with p ≥ 2 in any space dimension was developed by Kawashima [10] and Shizuta-Kawashima [18]. Precisely, the main tool used by Kawashima [10] is the Fourier analysis applied to the linearized homogeneous system, and the key part in the proof is to construct some compensation function to capture the dissipation of the hyperbolic component in the solution and then obtain an estimate on the Fourier transformÛ of the solution U as (1.5).
The rest of the paper is organized as follows. In Section 2, we reformulate the Cauchy problem on the non-isentropic micropolar fluid system around the constant steady state [1, 0, 0, 1], and study the decay structure of the linearized homogeneous system by the Fourier energy method. In Section 3, we focus on the spectral analysis of the linearized system by two parts. The fist part is for the fluid part, the second one for the electromagnetic part. In Section 4, we first prove the global existence of solutions by the energy method, and then show the time asymptotic rate of solutions around the constant states.
Notation. Throughout this paper, the norms in the Sobolev Spaces H s R 3 and L 2 R 3 are denoted respectively by · H s and · for integer N ≥ 1. ·, · denotes the inner product in L 2 R 3 ; and we set [A, B] X = A X + B X ; To simplify the notation, by A ∼ B we mean there exists a positive constant C, such that 1 C B ≤ A ≤ CB. Moreover, C denotes a general constant which may vary in different estimates.
2. Decay property of linearized system. In this section, we shall establish the time-decay estimates of solutions to the linearized system by using the Fourier energy method. The main motivation to present this part is to understand the linear dissipative structure of such complex system based on the direct energy method.
here the nonhomogeneous source terms S i , (i = 1, 2, 3, 4) are defined as The associated initial data is given by In this section, we still use U = [n, u, w, m] to denote the solutions to the following linearized homogeneous system By applying the Fourier energy method to the Cauchy problem (2.3) and (2.4), we show that there exists a time-frequency Lyapunov functional which is equivalent with |Û (t, ξ)| 2 and moreover its dissipation rate can also be characterized by the functional itself. The main result will be given as follows.
Theorem 2.1. Let U (t,x), t > 0, x ∈ R 3 be a well-defined solution to the system (2.3) and (2.4), then there exists a time-frequency Lyapunov functional E(Û (t,ξ)) with for some suitable positive constant c > 0. In particular, holds for any t > 0 and ξ ∈ R 3 .
Theorem 2.1 exactly shows the dissipative structure of the linearized Micropolar fluids system (1.1). It plays a key role in the study of the nonlinear asymptotic stability of the constant steady state under small perturbations.
2.3. Time-decay property. As in [3,4] it is now a standard procedure to derive from Theorem 1.1 the L p − L q time decay property of solutions to the linearized system (2.3). Here, we consider it by obtaining the following lemma.
holds for any t ≥ 0 and ξ ∈ R 3 .
where [·] − denotes the integer part of the argument.
Based on the pointwise estimate of Lemma 2.2, it is also straightforward to obtain the L p − L q time-decay property to the Cauchy problem (2.3) − (2.4). By applying Theorem 2.1 together with Lemma 2.2 to the linearized stystem (2.3), we have Proposition 2.1. Let 1 ≤ p, s ≤ 2 ≤ q ≤ ∞, |α| ≥ 0, and let j ≥ 0 be an integer. Let U = [n, u, w, m] satisfy the system (2.3) for all t > 0, x ∈ R 3 . Then, the solution of the linearized homogeneous system satisfies for any t ≥ 0. For the linearized system (1.1) , Proposition 2.1 describes the unified time decay property of the full solution. But, it can not be directly applied to the nonlinear system (4.1)-(4.2) to obtain the time decay rates of the solution. Therefore, we turn to the study of the Green function of the linearized stystem.
3. Green function. In fact, as in [12,23] the linearized micropolar fluid system (2.6) can be written as two decoupled subsystems which govern the time evolution of n, ∇ · u, ∇ · w and ∇ × u, ∇ × w respectively. We decompose the solution to (2.3) − (2.4) into two parts in the form of where u , u ⊥ are defined by and likewise for w , w ⊥ . For brevity, the first part on the right of (3.1) is called the fluid part and the second part is called the electromagnetic part, and we also write We now derive the equations of U and U ⊥ , respectively. Taking the divergence of 3) 3 and noticing ∇ · u = ∇ · u , we see that the fluid part U satisfies The initial data is given by n, u , w , m t=0 = n 0 , u 0 , w 0 , m 0 .
Applying −∆ −1 ∇× to the above two equations and noticing ∇ × u = ∇ × u ⊥ , we see that electromagnetic part U ⊥ satisfies with initial data where σ = (2(c 0 + c d ) − c a ) and I is identity matric.

LVQIAO LIU AND LAN ZHANG
2. when ε ≤ |ξ| ≤ K, λ i has the following spectrum gap property: 3. when |ξ| > K, λ i has the following expansion: Here ν = (2µ + λ) and all a j , b j , c j , d j , e j are real constants.
As [3], it is now a standard procedure to derive from Lemma 3.4 the L p − L q time decay property of the fluid part .
for any t ≥ 0, where C = C(m, p, r, q) and 3 1
for any t ≥ 0, where C = C(m, p, r, q) and [3( 1 r − 1 q )] + is defined in Definition 2.3. Now, by combining two representations of the Green functions to the fluid part and the electromagnetic part obtained in Proposition 3.2 and Proposition 3.3 respectively, we obtain Theorem 3.6. Let 1 ≤ p, r ≤ 2 ≤ q ≤ ∞ and let m ≥ 0 be an integer. Suppose that [n, u, w, m] is the solution to the Cauchy problem (2.1)-(2.4) Then U = [n, u, w, m] satisfies the following time-decay property: for any t ≥ 0, where C = C(k, p, r, q) and [3( 1 r − 1 q )] + is defined in Definition 2.3. For later use, we need the following result which is an immediate corollary from Theorem 3.6, Corollary 1. Under the assumption of Theorem 3.6, there exists a constant C > 0, such that for any t ≥ 0.
4. Nonlinear asymptotic stability. 6.1. Global existence. First of all, still using the notation [n, u, w, m] for simplicity, let us write the reformulated nonlinear system (1.1) and (1.2) as 4.1. Global existence. In this subsection, we will show that there exists a unique global-in-time solution to the stystem (1.1)-(1.2). We first obtain the uniform a priori estimates.
The following lemma in [20] is useful for the forthcoming estimates.
Lemma 4.1. There exists a constant C > 0 such that for any f, g, h ∈ H s R 3 and any multi-index α with 1 ≤ |α| ≤ N . Lemma 4.2 (Moser-type inequality). For functions f, g ∈ H s ∩ L ∞ , and |α| ≤ N, we have with the constant depends only on N .
Then, the global existence of the reformulated Cauchy problem (2.1)-(2.2) with small smooth initial data can be stated as follows. and To prove Theorem 4.3, it suffices to prove the following uniform-in-time a priori estimate. Define where η is a constant to be properly chosen later. Notice that, since the constant η is small enough, one has As long as the above estimate is proved, Theorem 4.3 follows in the standard way by combining it with the local-in-time existence and uniqueness as well as the continuity argument. Therefore, in what follows we prove (4.3) only, and other details are omitted for simplicity.
Proof. We will establish zero-order energy estimates. Multiplying (4.1)-(4.2) by n, u, w, m respectively, then taking integration and summation, we finally get 1 2 Here, we follow the similar argument in [12]. Let 1 ≤ |α| ≤ N. Applying differentiation ∂ |α| to (4.1) yields where and [A, B] denotes the commutator (AB − BA) for two operators A and B. Multiplying the first equation of (4.4) by ∂ |α| n, and taking integration in x give 1 2 Multiplying the second equation of (4.4) by ∂ |α| u, and taking integration in x give 1 2 Multiplying the third equation of (4.4) by ∂ |α| w, and taking integration in x give Multiplying the fourth equation of (4.4) by ∂ |α| m, and taking integration in x give Taking the summation of (4.5) − (4.8), we get |∇∂ |α| u| 2 n + 1 dx + (µ + λ − µ r ) |div∂ |α| u| 2 n + 1 dx Noticing the similarity of the quadratically nonlinear terms in f 1 , f 2 , f 3 and f 4 , we first give useful estimates which will play an important role in our calculation. By using Leibniz formula, Lemma 4.1 and Lemma 4.2, we can get Direct calculation implies, Hence Then, by putting the above estimates together for properly chosen constants 0 < η 1, one has The proof of Theorem 4.3 is complete.

4.2.
Large-time behavior. From now, we suppose that all conditions in Theorem 4.3 hold and U = [n, u, w, m] is the obtained solution to the Cauchy problem (4.1) and (4.2) . In this subsection we devote ourselves to proving the time decay rate of the full energy U (t) 2 H s or equivalently E N (U (t)). For that purpose, define X(t) = sup 0<τ <t (1 + s)  Proof. Under the smallness of U 0 N , (4.3) implies that d dt E N (U (t)) + cD N (U (t)) ≤ 0, (4.10) for any t ≥ 0. We now apply the time-weighted energy estimate and iteration to the Lyapunov inequality (4.10). Let |α| ≥ 0. Multiplying (4.10) by (1 + t) 3 2 + and taking integration over [0, t] give (1 + t) 3 2 + E N (U (t)) + c It is straightforward to verify [S 1 , S 2 , S 3 , S 4 ] L 1 ∩L 2 ≤ CE N (U ).
Corollary 2. If U 0 L 1 ∩H s is sufficiently small, then Proof. This can be proved by using Lemma 4.4 and (4.12). We omit the details.