GLOBAL REGULARITY FOR THE 2D MICROPOLAR EQUATIONS WITH FRACTIONAL DISSIPATION

. Micropolar equations, modeling micropolar ﬂuid ﬂows, consist of coupled equations obeyed by the evolution of the velocity u and that of the mi- crorotation w . This paper focuses on the two-dimensional micropolar equations with the fractional dissipation ( − ∆) α u and ( − ∆) β w , where 0 < α,β < 1. The goal here is the global regularity of the fractional micropolar equations with minimal fractional dissipation. Recent eﬀorts have resolved the two borderline cases α = 1, β = 0 and α = 0, β = 1. However, the situation for the general critical case α + β = 1 with 0 < α < 1 is far more complex and the global regularity appears to be out of reach. When the dissipation is split among the equations, the dissipation is no longer as eﬃcient as in the borderline cases and diﬀerent ranges of α and β require diﬀerent estimates and tools. We aim at the 2010 Mathematics Subject Classiﬁcation. Primary: 35Q35, 35B65, 76A10; 76B03. subcritical case α + β > 1 and divide α ∈ (0 , 1) into ﬁve sub-intervals to seek the best estimates so that we can impose the minimal requirements on α and β . The proof of the global regularity relies on the introduction of combined quantities, sharp lower bounds for the fractional dissipation and delicate upper bounds for the nonlinearity and associated commutators.

Here and in what follows, In addition to their applications in engineering and physics, the micropolar equations are also mathematically significant due to their special structures. The wellposedness problem on the micropolar equations and closely related equations such as the magneto-micropolar equations have attracted considerable attention recently and very interesting results have been established ( [4,9,11,10,15,22,30,31,33]). Generally speaking, the global regularity problem for the micropolar equations is easier than that for the corresponding incompressible magnetohydrodynamic equations and harder than that for the corresponding incompressible Boussinesq equations.
Recent efforts are focused on the 2D micropolar equations with partial dissipation. When there is full dissipation, the global well-posedness problem on (1.2) is easy and can be solved similarly as that for the 2D Navier-Stokes equations (see, e.g., [5,8,23,28]). When there is only partial dissipation, the global existence and regularity problem can be difficult. Due to recent efforts, the global regularity for several partial dissipation cases have been resolved. In [11] Dong and Zhang obtained the global regularity of (1.2) without the micro-rotation viscosity, namely γ = 0. For (1.2) with ν = 0, γ > 0, κ > 0 and κ = γ, Xue obtained the global well-posedness in the frame work of Besov spaces [30]. Very recently, Dong, Li and Wu [10] proved the global well-posedness of (1.2) with only angular viscosity dissipation. [10] makes use of the maximal regularity of the heat operator and introduces a combined quantity to obtain the desired global bounds. In addition, [10] also obtains explicit decay rates of the solutions to this partially dissipated system. This paper aims at the global existence and regularity of classical solutions to the 2D micropolar equations with fractional dissipation        ∂ t u + u · ∇u + (ν + κ)Λ 2α u − 2κ∇ × w + ∇π = 0, where 0 < α, β < 1 and Λ = (−∆) 1/2 denotes the Zygmund operator, defined via the Fourier transform Λ α f (ξ) = |ξ| α f (ξ). Clearly, (1.3) generalizes (1.2) and reduces to (1.2) when α = β = 1. Mathematically (1.3) has an advantage over (1.2) in the sense that (1.3) allows the study of a family of equations simultaneously. Our attempt is to establish the global regularity of (1.3) with the minimal amount of dissipation, namely for smallest α, β ∈ (0, 1). As aforementioned, the two endpoint cases, α = 1 and β = 0, and α = 0 and β = 1 have previously been resolved in [11] and [10], respectively. The global regularity for the general critical case when 0 < α, β < 1 and α + β = 1 appears to be extremely challenging.
When α + β = 1, the dissipation is not sufficient in controlling the nonlinearity and standard energy estimates do not yield the desired global a priori bounds on the solutions. Due to the presence of the linear derivative terms ∇ × w and ∇ × u in (1.3), we need α + β > 1 even in the proof of the global L 2 -bound for the solution. It does not appear to be possible to bound the nonlinear terms when we estimate the Sobolev norms of the solutions in the critical or supercritical case α + β ≤ 1. This paper focuses on the subcritical case α + β > 1, but we intend to get as close as possible to the critical case α + β = 1. We are able to prove the following global existence and regularity result for (1.3).
Even though Theorem 1.1 requires α + β > 1, we have made serious efforts towards the critical case α+β = 1. We divide α ∈ (0, 1) into five different subranges to seek the best estimates so that we can impose the minimal requirements on α and β. As we can tell from (1.4), α + β is close to the critical case when either α is close to 0 or close to 1. Figure 1 below depicts the regions of α and β for which the global regularity is established in Theorem 1.1.

Figure 1. Regularity region
We briefly summarize the main challenge for each subrange and explain what we have done to achieve the global regularity. Here and in what follows, we set the viscosity coefficients ν = κ = γ = 1 for simplicity. In order to prove Theorem 1.1, we need global a priori bounds on the solutions in sufficiently functional settings. More precisely, if we can show, for any T > 0, then Theorem 1.1 would follow from a more or less standard procedure. For any α ∈ (0, 1) and α + β > 1, the L 2 -norm of (u, w) is globally bounded (see Proposition   GLOBAL REGULARITY FOR THE 2D MICROPOLAR EQUATIONS   4137 2.2). The next natural step is to obtain a global H 1 -bound for (u, w). We invoke the equation of the vorticity Ω ≡ ∇ × u, For 0 < α < 3 4 , we need to estimate Ω L 2 and Λ 2β−1 w L 2 simultaneously in order to bound the coupled terms. The index 2β − 1 is chosen to minimize the requirement on β, which turns out to be More regular global bound can be obtained for w, which, due to α + 2β > 2, implies ∇w ∈ L 1 t L ∞ x for all t > 0. However,, it appears impossible to derive (1.5) from the vorticity equation (1.6) due to the presence of the term ∆w. We overcome this difficulty by considering the combined quantity The equation of Γ eliminates the term ∆w from the vorticity equation and makes it possible to estimate the L q -norm of Γ. In fact, by making use of sharp lower bounds for the dissipative term and suitable commutator estimates, we are able to obtain the global bound for Γ L q for q satisfying Due to the regularity of w, we obtain a global bound for Ω L q as a consequence. By further assuming we are able to show that which, especially, implies (1.5 and Ω ∈ L 2 t H 1+α , which yield the desired bound in (1.5). We remark that the estimates here actually hold for any α ∈ (0, 1) and β ≥ 3 2 −α. We restrict α to the range 3 4 ≤ α ≤ 7 8 in order to minimize the assumption on β.
For 7 8 ≤ α < 1, it appears very difficult to obtain any global bounds beyond the L 2 -norm for (u, w). The strategy here is to work with another combined quantity The advantage of the G-equation is that it removes ∆w from the vorticity equation. For α and β satisfying β ≥ 5(1 − α), (1.9) we are able to establish the global L 2 bound for G, for any t > 0, This global bound serves as an adequate preparation for the following global L qbound for w, for any 2 ≤ q < 2β 1−α , whereḂ 2β q q,q denotes a homogenerous Besov space. More information on Besov spaces are provided in the appendix. Making use of this L q bound and further assuming that α and β satisfy we obtain a global bound for Ω L 2 and ∇w L 2 . To achieve (1.5), we further bound ∇Ω L 2 and ∆w L 2 . (1.4) for 7 8 ≤ α < 1 is a combination of (1.9) and (1.10).
As aforementioned, once the global bound in (1.5) is established, Theorem 1.1 can then be established following standard approaches. The rest of this paper is divided into four sections and one appendix. Each one of the sections is devoted to establishing the global a priori bounds for one of the three cases described above. Section 5 outlines the proof of Theorem 1.1. The appendix provides the definitions and related facts concerning the Besov spaces. In addition, we also supply the details on several notations and simple facts used the regular sections.
As aforementioned, the proof of Theorem 2.1 relies on suitable global a priori bounds for the solutions. This section focuses on the necessary global a priori bounds. These bounds are proved in Proposition 2.2, Proposition 2.6 and Proposition 2.7.
where C is a constant depending on the indices s, r, p 1 , q 1 , p 2 and q 2 .
The following lemma can be found in [20, p.614].
Lemma 2.4. Let 0 < s < 1 and 1 < p < ∞. Then The following lemma generalizes the Kato-Ponce inequality, which requires m to be an integer (see, e.g., [20]). This lemma extends it to any real number m ≥ 2. For the convenience of the readers, we provide a proof for this lemma.
Proof of Lemma 2.5. It is easy to see that, for 0 < s < σ and p,p ∈ [1, ∞] 2 , the Besov space B σ p,p is embedded in the Bessel potential space L p s (see, e.g., [16, where B σ p,p andḂ σ p,p denote the standard inhomogenerous and homogeneous Besov spaces, respectively. Besov spaces and their properties are provided in the appendix. A short proof for (2.7) is also given in the appendix. Settingp = p and invoking the equivalence definition ofḂ σ p,p in (A.3), we have By the Hölder inequality, and Hölder's inequality, This completes the proof of (2.6).
We remark that, if we replace the Bessel potential space norm by the norm of the Sobolev-Slobodeckij space W s,p , the proof of Lemma 2.5 then implies In fact, (2.8) follows and combined with the rest of the proof for Lemma 2.5. The definition of W s,p and some embedding properties are given in the appendix. Here we also want to remark that unfortunately, it is not clear whether the term f B s q, p of (2.8) can be replaced by f W s,q .
Proof of Proposition 2.2. Taking the L 2 inner product of (1.3) with (u, w), we find where we have used the condition α + β > 1 in the last inequality as well as the following facts, due to ∇ · u = 0, Applying Gronwall inequality gives, for 0 < t < ∞, which is (2.2). To prove (2.3) and (2.4), we apply ∇× to the first equation of (1.3) to obtain the vorticity equation where we have used −∇ × ∇ × w = ∆w. Taking the inner product of (2.10) with Ω and the inner product of third equation of (1.3) with Λ 2(2β−1) w leads to where we have used the facts To estimate I 1 , we integrate by parts and apply Hölder's inequality, the Gagliardo-Nirenberg inequality and Young's inequality to obtain where we have used (2.1), To estimate I 2 , we employ Hölder's inequality and Sobolev's inequality and invoke Lemma 2.3 to obtain where the indices are given by and they are so chosen to fulfill the requirements of the Sobolev inequalities, Inserting the bounds for I 1 and I 2 in(2.11) yields which is (2.3). We now prove (2.4). Taking the inner product of third equation of (1.3) with Λ 2(α+β) w yields Similar to the estimates for I 2 , we have, after applying Hölder's inequality, Sobolev's imbedding inequality and Lemma 2.3, where we have used that α and β satisfy Inserting the bounds for J 1 and J 2 in (2.12) and applying Gronwall's inequality, we have the global existence and regularity then follows. It does not appear plausible to prove (2.13) directly via (2.10) when 0 < α < 1 2 . Due to the term ∆w in (2.10), we need Λ 2 w ∈ L 1 t L ∞ , which is unavailable at this moment. To overcome this difficulty, we work with the combined quantity which, together with (2.10), yields the equation for Γ, (2.14) Although (2.14) appears to be more complex than (2.10), it eliminates the most regularity demanding term ∆w and allows us to derive the L q bounds of Γ, which is crucial to derive the Ω ∈ L 1 t L ∞ . More precisely, we prove the following proposition. Proposition 2.6. Consider (1.3) with α and β satisfying (2.1). Assume (u 0 , w 0 ) satisfies the conditions of Theorem 1.1 and let (u, w) be the corresponding solution. Then, for q satisfying (u, w) obeys the following global bounds, where C > 0 depends only on t and (u 0 , w 0 ) H s .
When α < 3 4 , the requirement on β in (3.1) is more than those in (2.1) and thus Theorem 2.1 is sharper for α < 3 4 . Similarly, as we shall see in the coming section, for α > 7 8 , Theorem 4.1 in the subsequent section is stronger, Theorem 3.1 is significant only for α in the range between 3 4 and 7 8 . As explained in the previous section, it suffices to provide the necessary global a priori bounds.
The following proposition establishes the global bound for (∇u, ∇w) L 1 t L ∞ x , which is sufficient for the proof of Theorem 3.1. Proposition 3.2. Consider (1.3) with α and β satisfying (3.1). Assume (u 0 , w 0 ) satisfies the conditions of Theorem 1.1. Then the corresponding solution (u, w) of (1.3) obeys, for any 0 < t < ∞,

3) and (3.4) imply that
Proof. We first remark that the global L 2 -bound in (2.2) remains valid since it only requires α + β > 1. To show the global bound in (3.2), we take the L 2 inner product of (2.10) with Ω and L 2 inner product of third equation of (1.3) with Λw to obtain 1 2 For the conciseness of our presentation, attention is focused on the case β = 3 2 − α. The case β > 3 2 − α is even simpler. Noting that α + β = 3 2 and 0 < 1 − α < β, we have, by the interpolation inequality, where we have used the following facts By the divergence-free condition of u, we will show Thanks to the following variant version of Lemma 2.3 (its proof is the same one as for Lemma 2.3) and Sobolev's inequality, it ensures that Inserting the estimates for M 1 and M 2 in (3.6) and applying Gronwall's inequality, we obtain (3.2). To prove (3.3), we take the L 2 inner product of third equation of (1.3) with Λ 3 w to obtain Again, due to α + β = 3 2 ,
4. The case when 7 8 ≤ α < 1. This section focuses on the case when 7 8 ≤ α < 1. We prove theorem 1.1 for this range of α. More precisely, the following theorem holds.
One of the main difficulties to prove Theorem 4.1 is that direct energy estimates on (1.3) do not yield the desired global bounds on the derivatives of u and w. To overcome this difficulty, we consider the combined quantity (4. 2) The following proposition establishes that G L 2 admits a global bound.
In order to prove this proposition, we need the following commutator type estimates involving the fractional Laplacian operator. The following lemma is taken from [32]. Similar commutator estimates have been used previously (see, e.g., [17]).
We are now ready to prove Proposition 4.2.

Gronwall's inequality then implies
This completes the proof of Proposition 4.2.
The global bound for G in the previous proposition serves as a bridge to the global bounds on w and Ω. The following lemma controls the L q -norm of w.
we have, for any 0 < t < ∞, Proof. Multiplying the third equation of (1.3) by |w| q−2 w, integrating over R 2 and using the divergence-free condition, we obtain 1 q where, in the last line above, we have used Ω = G + Λ 2−2α w. As in (2.17), the following lower bound holds where C 0 = C 0 (β, q) > 0. On one hand, for q ≤ 2 1−α , we obtain by the Hölder inequality . On the other hand, for q > 2 1−α , we have where we have used the simple fact that 2βq (α+2β−1)q+2 < 2. Therefore, for any q ≥ 2, the first term in (4.8) can be bounded by To bound the second term in (4.8), we use Lemma 2.5. For q satisfying (4.6), we choose 0 < s < σ < 1 satisfying By Hölder's inequality and Lemma 2.5, where we have used the embeddings, due to (4.11), This explains why we need to restrict q to the range in (4.6). Combining (4.10) and (4.12), we obtain . Gronwall's inequality, together with Proposition 4.2, yields which is (4.7).
We now show that, for any 0 < t < ∞, ∇u, ∇w ∈ L 1 t L ∞ , which allows us to establish the desired global regularity.
Proposition 4.5. Assume (u 0 , w 0 ) satisfies the conditions of Theorem 1.1 and let (u, w) be the corresponding solution of (1.3) with α and β satisfying (4.1). Then, for any 0 < t < ∞, In particular, (4.14) implies ∇u, ∇w ∈ L 1 t L ∞ . Proof. Taking the L 2 inner product of (2.10) with Ω and the L 2 inner product of third equation of (1.3) with Λ 2 w, we have Due to β ≤ 1, by Sobolev's inequality and Young's inequality, Due to G = Ω − Λ 2−2α w and the Biot-Savart law ∇u = ∇∇ ⊥ ∆ −1 Ω, we write Correspondingly, J 2 can be written into two parts, J 21 and J 22 can be bounded as follows. By Hölder's inequality, the boundedness of Riesz transforms and Sobolev's inequality, where we have used the embedding inequalities, due to α + β > 1, 1−α to be specified later and let q 0 = q0 q0−1 be its dual index. By the duality of the Besov spaces and Sobolev's inequality, we have where q 0 ∈ 1−β α+β−1 , 2β 1−α and l, λ, θ are given by We note that, in the third inequality above, we used the norm equivalence B s p,p ≈ W s,p , as explained in the appendix. In addition, we invoked (2.8) in the fourth inequality above. Combining all the estimates above, we have We explain that, when α and β satisfy (4.1), we have In fact, (4.1) implies Therefore, we can choose q 0 < 2β 1−α such that Then (4.15) follows from Proposition 4.4. Gronwall's inequality then implies the desired bound (4.13).
In order to prove (4.14), we take the L 2 inner product of (2.10) with Λ 2 Ω and the L 2 inner product of the third equation of (1.3) with Λ 4 w to obtain Due to α + β > 1 and β ≤ 1, we have by using Lemma 2.3, the Gagliardo-Nirenberg inequality and Young's inequality where we have used the fact 2(1−α) 2β+α−1 < 2. These estimates combined with Gronwall's inequality then allow us to obtain (4.14). This completes the proof of Proposition 4.5. where F and F −1 denote the Fourier and inverse Fourier transforms, respectively, and χ B(0,n) denotes the characteristic function on the ball B(0, n). Consider the approximate equations of (1.3) ∂ t u n + 2J n Λ 2α u n = 2J n P ∇ × w n − J n P (J n u n · ∇J n u n ), ∇ · u n = 0, ∂ t w n + J n Λ 2β w n + 4J n w n = 2J n ∇ × u n − J n (J n u n · ∇J n w n ), u n (x, 0) = J n u 0 , w n (x, 0) = J n w 0 , where P denotes the standard projection onto divergence-free vector fields. The standard Picard type theorem ensures that, for some T n > 0, there exists a unique local solution (u n , ω n ) on [0, T n ) in the functional setting {f ∈ L 2 (R 2 ) : supp F(f ) ⊂ B(0, n)}. Due to J 2 n = J n and P J n = J n P , it is easy to see that (J n u n , J n w n ) is also a solution. The uniqueness of such local solutions implies u n = J n u n , w n = J n w n . Therefore, (5.1) becomes        ∂ t u n + 2Λ 2α u n = 2P ∇ × w n − J n P (u n · ∇u n ), ∇ · u n = 0, ∂ t w n + Λ 2β w n + 4w n = 2∇ × u n − J n (u n · ∇w n ), u n (x, 0) = J n u 0 , w n (x, 0) = J n w 0 .

(5.2)
A basic L 2 energy estimate implies (u n , w n ) of (5.2) satisfies where C is independent of n. Therefore, the local solution can be extended into a global one, by the standard Picard Extension Theorem (see, e.g., [23]). Next we show that (u n , w n ) admits a uniform global bound in H s (R 2 ) with (s > 2). Following the proofs of Propositions 2.7, 3.2 and 4.5, we obtain, for any t > 0, and S 0 denotes its dual. S 0 can be identified as S 0 = S /S ⊥ 0 = S /P, where P denotes the space of multinomials. We also recall the standard Fourier transform and the inverse Fourier transform, To introduce the Littlewood-Paley decomposition, we write for each j ∈ Z The Littlewood-Paley decomposition asserts the existence of a sequence of functions Therefore, for a general function ψ ∈ S, we have In addition, if ψ ∈ S 0 , then ∞ j=−∞ Φ j (ξ) ψ(ξ) = ψ(ξ) for any ξ ∈ R d .
That is, for ψ ∈ S 0 , in the sense of weak- * topology of S 0 . For notational convenience, we definė We now choose Ψ ∈ S such that Then, for any ψ ∈ S, Ψ * ψ + in S for any f ∈ S . We set if j = 0, 1, 2, · · · . (A.2) For notational convenience, we write ∆ j for∆ j when there is no confusion. They are different for j ≤ −1. As provided below, the homogeneous Besov spaces are defined in terms of∆ j while the inhomogeneous Besov spaces are defined in ∆ j . Besides the Fourier localization operators ∆ j , the partial sum S j is also a useful notation. For an integer j, where ∆ k is given by (A.2). For any f ∈ S , the Fourier transform of S j f is supported on the ball of radius 2 j and S j f f in S .
In addition, for two tempered distributions u and v, we also recall the notion of paraproducts and Bony's decomposition, see e.g. [1], In addition, the notation ∆ k , defined by ∆ k = ∆ k−1 + ∆ k + ∆ k+1 , is also useful.
Definition A.1. For s ∈ R and 1 ≤ p, q ≤ ∞, the homogeneous Besov spaceḂ s p,q consists of f ∈ S 0 satisfying f Ḃs p,q ≡ 2 js ∆ j f L p l q < ∞.
An equivalent norm of the the homogeneous Besov spaceḂ s p,q with s ∈ (0, 1) is given by f Ḃs Definition A.2. The inhomogeneous Besov space B s p,q with 1 ≤ p, q ≤ ∞ and s ∈ R consists of functions f ∈ S satisfying f B s p,q ≡ 2 js ∆ j f L p l q < ∞. Many frequently used function spaces are special cases of Besov spaces. The following proposition lists some useful equivalence and embedding relations.  For any non-integer s > 0, the Hölder space C s is equivalent to B s ∞,∞ . Bernstein's inequalities are useful tools in dealing with Fourier localized functions. These inequalities trade integrability for derivatives. The following proposition provides Bernstein type inequalities for fractional derivatives. The upper bounds also hold when the fractional operators are replaced by partial derivatives.
We now provide the proof of (2.7). By Proposition A.4, Finally we provide the definition of Sobolev-Slobodeckij space W s,p . Let us assume s ≥ 0 and 1 ≤ p ≤ ∞. When s ≥ 0 is an integer, the Sobolev norm is standard, namely When s > 0 is a fraction, the norm in W s,p is given by We remark that, except for p = 2, W s,p with this norm is different from the most frequently used definition of Sobolev spaces of fractional order, or the Bessel potential space L p s (or sometimes denoted by W s,p or H s p ) (see, e.g., [16,Chapter 1.3.1], [26, p.13]). The norm in L p s is given by f L p s = f L p + Λ s f L p . W s,p is closely related to Besov spaces (see, e.g., [26,29]). In fact, W s,p ≈ B s p,p → L p s , 1 < p ≤ 2; L p s → B s p,p ≈ W s,p , 2 ≤ p < ∞.