REMARK ON EXPONENTIAL DECAY-IN-TIME OF GLOBAL STRONG SOLUTIONS TO 3D INHOMOGENEOUS INCOMPRESSIBLE MICROPOLAR EQUATIONS

. This paper addresses the Cauchy problem of the three-dimensional inhomogeneous incompressible micropolar equations. We prove the global ex- istence and exponential decay-in-time of strong solution with vacuum over the whole space R 3 provided that the initial data are suﬃciently small. The initial vacuum is allowed.

More recently, Zhang-Zhu [34] proved the global well-posedness of strong and classical solutions for the 3D inhomogeneous incompressible micropolar equations with vacuum provided that the initial data are sufficiently small. Therefore, it is also interesting to study the large time behavior of the global solution obtained in [34]. This is the main aim of the present paper. More precisely, our main result reads as follows.

Remark 1.
We remark that if the initial data (ρ 0 , u 0 , w 0 ) satisfy some additional regularity and the corresponding compatibility conditions, namely, , then the global strong solution obtained in Theorem 1.1 becomes a classical one away from the initial time t = 0. Moreover, the exponential decay-in-time as in (3) still holds true.
2. The proof of Theorem 1.1. This section is devoted to the proof of Theorem 1.1. In this paper, we shall use the convention that C denotes a generic constant which may change from line to line. Before proving our main result, we recall the following classical Gronwall inequality, which will be used frequently. For the convenience of the reader, we present it here.
, β(t) and γ(t) be non-negative functions. In addition, β(t) and γ(t) are two integrable functions over [a, b]. If the following differential inequality holds d dt In particular, for β(t) = 0, it holds We begin with the local existence and uniqueness theorem of strong solutions whose proof can be performed by using a semi-Galerkin's scheme (see [5,18,27,33,20] for example). Consequently, it suffices to establish some necessary a priori bounds for smooth solutions to the micropolar system (1) to extend the local strong solution guaranteed by Lemma 2.2. To start, the following lemma concerns the basic energy estimates. Lemma 2.3. Under the assumptions of Theorem 1.1, the corresponding solution (ρ, w) of the micropolar system (1) admits the following bounds for any t ≥ 0

ZHUAN YE
where the positive constant C 1 depends only on µ 1 , µ 2 and ξ.
Proof. First, the non-negativeness of ρ is a direct consequence of the maximum principle and ρ 0 ≥ 0. We multiply the equation (1) 1 by |ρ| p−2 ρ, integrate it over R 3 and use ∇ · u = 0 to conclude d dt This implies (4). Multiplying the equations (1) 2,3 by (u, w) and integrating the resultant, we obtain 1 2 Notice the following facts where C 1 > 0 is an absolute constant. As a result, we deduce where Integrating (6) in time yields (5). This completes the proof of Lemma 2.3.
Next we will establish the time-independent estimates on the L ∞ (0, T ;Ḣ 1 (R 3 ))norm of u and w provided that E 0 is sufficiently small.
where X(t) is given by

EXPONENTIAL DECAY-IN-TIME OF MICROPOLAR EQUATIONS 6729
If there exists a sufficiently small absolute constant > 0, independent of initial data such that where C 0 depends only on Proof. Multiplying the equations (1) 2,3 by (∂ t u, ∂ t w), integrating by parts and using where ρu · ∇w · ∂ t w dx.

Direct computations yields
Inserting (11) into (10) gives In view of the Gagliardo-Nirenberg inequality, it yields
The following estimates play a key role in deriving the higher order estimates of the solutions.
Proof. Using (1) 1 , we derive As a result, applying ∂ t to the equations (1) 2,3 gives Multiplying (24), (25) by ∂ t u, ∂ t w respectively, and integrating it over R 3 , we obtain According to the embedding inequalities and the Young inequality, we conclude By the same arguments, we have . The last term can be easily estimated by

Inserting all the above estimates into (26) and taking δ suitably small imply
By means of (9), we deduce that for any t ∈ [0, 1] which together with (27) and the Gronwall inequality yield for any t ∈ [0, 1] Thanks to (16) and (22), it gives

EXPONENTIAL DECAY-IN-TIME OF MICROPOLAR EQUATIONS 6735
We also deduce from (27) that We resort to (9) to get that for any t ≥ 0 Therefore, we apply the Gronwall inequality to (29) to deduce that for any t ≥ 1 where we have used (28). Keeping in mind (16) and (22), one derives Consequently, we conclude the proof of Lemma 2.5.
The following estimates play an important role in proving the uniqueness and the higher regularity of the solutions. Lemma 2.6. Under the assumptions of Theorem 1.1, the corresponding solution (ρ, w) of the micropolar system (1) admits the following bounds for any t ≥ 0 where C 0 depends only on µ 1 , Proof. For any 2 < p < 6, we have

ZHUAN YE
Applying the L p -estimate to (15), we derive Now it gives for 3 < p < 6 that With the above bounds in hand, one may deduce ≤ C 0 and for any t ≥ 1 Consequently, we get for any t ≥ 0 Applying the L p -estimate to (1) 3 , we have By the same arguments, it allows us to show that As ρ satisfies ∂ t ρ + u · ∇ρ = 0, we have by differentiating it with respect to x i , i = 1, 2, 3 Thanks to ∇ · u = 0, we get by direct computations d dt It follows from the Gronwall inequality and (32) that Noticing the following facts This ends the proof of Lemma 2.6.
Proof of Theorem 1.1. The global existence of strong solutions follows by the local strong solution (see Lemma 2.2) and the estimates of the above Lemmas 2.3-2.6. To explain the ideas clearly, in the next we will present a formal argument which can be rigorous by appropriate regularization. Firstly, we show the time continuity of the solution, namely ρu, ρw ∈ C([0, T ]; L 2 (R 3 )).
By the standard arguments (see, e.g., [19,21,27]), one may derive the continuity of (33) and (34) away from the initial time t = 0. Consequently, it suffices to show the continuity of (33) and (34) at the initial time t = 0. Since ∂ t ρ = −u · ∇ρ, we have where in the last line we have used (8) and (31). By the Hölder inequality, one has This implies that ρ continuous at the original time and satisfies the initial condition ρ| t=0 = ρ 0 , which further gives (33). We next show (34). In fact, we have that where we have used (8) and (31) again. Using the Hölder inequality yields which implies that ρu continuous at the original time and satisfies the initial condition ρu| t=0 = ρ 0 u 0 . By the same argument, we can show that ρw continuous at the original time and satisfies the initial condition ρw| t=0 = ρ 0 w 0 . To complete the proof, we still need to prove the uniqueness. To this end, consider two solutions (ρ, u, w) and ( ρ, u, w) of the system (1), emanating from the same initial data, and fulfilling the properties of Theorem 1.1. Then, it is not difficult to check that Multiplying (36) by u − u and (37) by w − w, then integrating them the result over It is not hard to check that Consequently, one obtains To close (38), it suffices to estimate ρ − ρ L 3 2 . To this end, we derive from (35) that d dt (ρ − ρ)(t) Denoting , we therefore deduce from (33), (38) and (39) that We thus obtain the desired estimates (41). Thanks to (34), one gets Combining (40), (41) and (42), we derive by using the variant of Gronwall inequality (see [20,Lemma 2.5]) that √ ρ(u − u)(t) L 2 + √ ρ(w − w)(t) L 2 + (ρ − ρ)(t) proving the uniqueness part of Theorem 1.1. Therefore, we complete the proof of Theorem 1.1.