A BLOW-UP CRITERION OF STRONG SOLUTIONS TO TWO-DIMENSIONAL NONHOMOGENEOUS MICROPOLAR FLUID EQUATIONS WITH VACUUM

. We deal with the Cauchy problem of nonhomogeneous micropolar ﬂuid equations with zero density at inﬁnity in the entire space R 2 . We show that for the initial density allowing vacuum, the strong solution exists globally if a weighted density is bounded from above. It should be noted that our blow-up criterion is independent of micro-rotational velocity.

The system (1.2) is supplemented with the initial condition (ρ, ρu, ρw)(x, 0) = (ρ 0 , ρ 0 u 0 , ρ 0 w 0 )(x), x ∈ R 2 , (1. 3) and the far field behavior (ρ, u, w)(x, t) → (0, 0, 0), as |x| → +∞. (1.4) The micropolar fluid equations, which were suggested and introduced by Eringen in 1960s (see [7]), are a significant step toward generalization of the Navier-Stokes equations. It is a type of fluids which exhibits micro-rotational effects and microrotational inertia, and can be viewed as a non-Newtonian fluid. Physically micropolar fluid may represent fluids that consist of rigid, randomly oriented (or spherical particles) suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena that appear in a large number of complex fluids such as the suspensions, animal blood, liquid crystals which cannot be characterized appropriately by the Navier-Stokes system, and that it is important to the scientists working with the hydrodynamic-fluid problems and phenomena. For more background and applications, we refer to [8,16] and references therein.
When ρ is constant, which means the fluid is homogeneous, the micropolar fluid equations have been extensively studied. In particular, many authors investigated global existence and regularity of 2D homogeneous micropolar fluid equations with partial dissipation. Dong et al. [4] studied global regularity and large time behavior of solutions to the 2D micropolar fluid equations with only angular viscosity dissipation. Later on, Dong et al. [5] proved global regularity of the two-dimensional fractional micropolar fluid equations with minimal fractional dissipation. Recently, by imposing natural boundary conditions and minimal regularity assumptions on the initial data, the authors [11] established the global existence and uniqueness of solutions to the 2D micropolar fluid equations with only angular velocity dissipation in a smooth bounded domain. On the other hand, by Fourier localization method, Chen and Miao [3] constructed global solutions of the 3D micropolar fluid system in Fourier-Besov spaces with a class of highly oscillating initial data of Cannone's type, while Zhu [22] investigated ill-posedness of 3D Cauchy problem in Fourier-Besov spaces. For other studies of homogeneous micropolar fluid equations, please refer to [2,6,9,15,18] and references therein.
When ρ is not constant, the system (1.2) is so-called nonhomogeneous micropolar fluid equations. For the initial density allowing vacuum states, Lukaszewicz [16,Chapter 3] proved the local existence of weak solutions for three-dimensional initial boundary value problem. Imposing a compatibility condition on the initial data, Zhang and Zhu [20] showed the global existence of strong solution in R 3 under some smallness condition. Later on, Ye [19] improved their result by removing the compatibility condition and furthermore obtained exponential decay of strong solution. Recently, by some weighted energy estimates, Zhong [21] showed the local existence of strong solutions to the Cauchy problem of (1.2) in R 2 . In this paper, we will investigate the structure of possible singularities of strong solutions obtained in [21].
Before stating our main result, we first explain the notations and conventions used throughout this paper. For r > 0, set For 1 ≤ p ≤ ∞ and integer k ≥ 0, the standard Sobolev spaces are denoted by: and (ρ, u, P, w) satisfies both (1.2) almost everywhere in R 2 ×(0, T ) and (1.3) almost everywhere in R 2 . Herex and η 0 is a positive number.
Without loss of generality, we assume that the initial density ρ 0 satisfies which implies that there exists a positive constant N 0 such that Our main result can be stated as follows: Theorem 1.1. In addition to (1.6) and (1.7), assume that the initial data (ρ 0 ≥ 0, u 0 , w 0 ) satisfies for any given numbers a > 1 and q > 2, Let (ρ, u, P, w) be a strong solution to the problem (1.2)-(1.4). If T * < ∞ is the maximal time of existence for that solution, then for any δ > 0, we have Remark 1.1. The local existence of an unique strong solution with initial data as in Theorem 1.1 was established in [21]. Hence, the maximal time T * is well-defined.
Remark 1.2. The conclusion in Theorem 1.1 is somewhat surprising since the criterion (1.9) is independent of micro-rotational velocity. It indicates that the mechanism of blowup of nonhomogeneous micropolar fluid equations is similar to the nonhomogeneous Navier-Stokes equations [13, Theorem 1.2] and does not depend on further sophistication of the equation (1.2) 3 . It is worth mentioning that our Theorem 1.1 holds for arbitrary a > 1 which is in sharp contrast to Liang [13] where a ∈ (1, 2) is required.
The rest of the paper is organized as follows: In Section 2, we collect some elementary facts and inequalities which will be needed in later analysis. Sections 3 is devoted to the proof of Theorem 1.1.

Preliminaries.
In this section, we will recall some known facts and elementary inequalities which will be used frequently later.
there exists some generic constant C > 0 which may depend on p, r, and s such that The following weighted L m bounds for elements of the Hilbert (2.1) The combination of Lemma 2.2 and the Poincaré inequality yields the following useful results on weighted bounds, whose proof can be found in [12,Lemma 2.4].
for positive constants M 1 , M 2 , and N 1 ≥ 1. Then for ε > 0 and η > 0, there is a positive constant C depending only on ε, η, M 1 , M 2 , and Finally, we state a critical Sobolev inequality of logarithmic type, which is originally due to Brézis-Wainger [1]. The reader can refer to [13, Lemma 2.5] for the proof.
Lemma 2.4. Assume that f ∈ L 2 (s, t; D 1,2 ∩ W 1,q (R 2 )) with some q > 2 and 0 ≤ s < t ≤ ∞, then there is a constant C > 0 independent of s and t such that 3. Proof of Theorem 1.1. Let (ρ, u, P, w) be a strong solution described in Theorem 1.1. Suppose that (1.9) were false, that is, there exists a constant M 0 > 0 such that We begin with the following standard energy estimate for (ρ, u, P, w) and the estimate on the L p -norm of the density.