Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behaviour and optimal decay estimates of the solutions as t→∞ .

Here φ is called the communication weight, measuring the strength of the alignment interactions. It is naturally assumed to be a non-negative and radially decreasing function.
The system (1.1) is known as the Euler-alignment system. It is the macroscopic representation of the celebrated Cucker-Smale model [9]   Ẋ an M -agent interacting system that describes the collective motions in animal flocks. Ha and Tadmor [19] formally derive (1.1) from (1.3) through a kinetic equation (1.4) Hydrodynamic limiting systems (1.1) with different type of pressures (1.2) can be rigorous derived from (1.4), including the pressure-less dynamics (κ = 0) [18], isothermal pressure (κ > 0, γ = 1) [23], and others [17,34]. The global well-posedness theory for the pressureless Euler-alignment system (1.1) with P ≡ 0 has been established in [41] for the case when the communication weight is bounded and Lipschitz. A critical threshold phenomenon was discovered: global regularity depends on initial data. A sharp threshold condition is obtained in [3] for the system in 1D with the help of an auxiliary quantity G = ∂ x u + φ * ρ that satisfies the continuity equation. The theory has been extended to the case when the communication weight is weakly singular: unbounded but integrable [43]. For higher dimensions, sharp results are only available for radial [44] and uni-directional [26] data, due to the lack of the auxiliary quantity, see also [21].
Another interesting type of communication weights are strongly singular near the origin, with a prototype taking the following form . (1.5) It is evident that for α ∈ (0, 2), the singular alignment D(u, ρ) with such a weight φ α can be expressed as a commutator form related to the fractional Laplace operator Λ α = (−∆) α 2 (see Definition 2.4): (1. 6) This nonlocal dissipation has an intriguing regularization effect to the solutions. Global regularity is obtained for any non-viscous smooth initial data for the system in the one-dimensional torus by Shvydkoy and Tadmor [37] for 1 α < 2, and by Do et al [16] for 0 < α < 1 (see also [39]). The results are then applied to general singular alignment interactions [24], as well as taking into account the misalignment effect [31]. For the multi-dimensional case, global well-posedness are only known for small initial data around an equilibrium state. See the work of Shvydkoy [36] for smooth initial data (ρ 0 , u 0 ) ∈ H N +4 (T N ) × H N +3+α (T N ), and Danchin et al [14] for small initial data that lie in critical Besov space (ρ 0 , u 0 ) ∈Ḃ 1 N,1 (R N ) ×Ḃ 2−α N,1 (R N ), subject to additional regularity assumptions. Global regularity for general large initial data remains a challenging open problem.
The global well-posedness theory for the Euler-alignment system (1.1) with pressure is much less understood compared with the pressure-less system. When the communication weight φ is bounded and Lipschitz, Y.-P. Choi [7] proved global regularity of the system with isothermal pressure, for small smooth initial data in the periodic domain T N . A similar result was obtained in [46] for the system with isentropic pressures (κ > 0, γ > 1).
The main focus of this paper is on the Euler-alignment system with pressure and with a strongly singular communication weight (1.5). The goal is to understand the interplay between the pressure (1.2) and the nonlocal regularization from the alignment (1.6).
In 1D periodic domain T, Constantin, Drivas and Shvydkoy [8] proved the global existence of smooth solutions for the system with an additional local dissipation term of the form (1.7). They make use of the auxiliary quantity to build a hierarchy of entropies. The result does not require a smallness assumption, but is limited to one dimension.
For the system in T N , Chen, Tan and Tong [5] established the global well-posedness for smooth initial data with a smallness assumption. The result is partially extended to the whole space R N . However, an additional linear damping term is required to obtain the desired result.
We would like to comment that most global well-posedness results in the literature on the Euleralignment system (1.1) with strongly singular alignment (1.6) are on the periodic domain T N . One important reason is that solutions can lose regularity when vacuum arises [42,1]. It is easier to obtain a priori positive lower bound on the density under periodic setup, as mass cannot diffuse to infinity. Additional analytical treatments are required to guarantee no vacuum formations for the system in the whole space R N .
1.2. The barotropic compressible Navier-Stokes system. To study the Euler-alignment system (1.1) in R N , we shall mention a very related system. If we replace the dissipation term D(u, ρ) by D loc (u) = µ 1 ∆u + µ 2 ∇ div u, µ 1 > 0, µ 1 + µ 2 > 0, (1.7) the system (1.1) becomes the classical barotropic compressible Navier-Stokes system, which has been intensely studied in the recent decades. Serrin [35] and Nash [32] established the local existence and uniqueness of smooth non-vacuous solutions. One can also see Solonnikov [40] and Valli [47] for the local well-poseness of strong solutions with Sobolev regularities. Matsumura and Nishida [29,30] proved the global existence and uniqueness of strong solutions provided that initial data (ρ 0 , u 0 ) is a small perturbation of constant non-vacuous state (ρ, 0) in three dimension, and under an additional L 1 -smallness of the initial perturbation, they showed the following optimal decay estimate (ρ −ρ, u)(t) L 2 (1 + t) − 3 4 , ∀ t > 0. (1.8) Later, noting that the barotropic compressible Navier-Stokes system is invariant under the transformation ρ(t, x) → ρ(λ 2 t, λx), u(t, x) → λu(λ 2 t, λx), λ > 0, with a modification of the pressure P → λ 2 P , Danchin [10] proved the global existence and uniqueness of strong solution in the framework of critical L 2 -based Besov space with initial data close to a stable equilibrium. More precisely, under the following smallness condition in critical Besov spaces (see Definition 2.1) the barotropic compressible Navier-Stokes system has a global unique solution.
Furthermore, for small perturbation of non-vacuous equilibrium (ρ, 0) in L p -type Besov norms, Charve and Danchin [4] and Chen, Miao and Zhang [6] independently constructed the global unique strong solution in the framework of critical L p -based Besov spaces. One can also see [20] for a simpler proof of the same result by using a good unknown called the effective velocity. Concerning the large-time behavior of the above obtained global strong solutions, Okita [33] considered the N 3 dimension and established the optimal time-decay estimate of global solutions in the critical L 2 -framework with an additional smallness condition on ρ 0 −ρ Ḃ0 [13] gave an another description of the time-decay estimate as above with N 2. Danchin and Xu [15] showed that under an additional smallness condition of low frequencies (see (2.2) for the definition of norm · ℓḂ the L p norm of the global critical solutions constructed in [4,6,20] decays like t −N ( 1 p − 1 4 ) for t → +∞ (exactly as (1.8) with p = 2, N = 3). One can see Xu [49] for a different low-frequency smallness assumption to get the same time-decay estimate. Recently, Xin and Xu [48] replaced the smallness condition (1.9) with a mild assumption like (ρ 0 −ρ, u 0 ) ℓḂ −s 0 2,∞ < ∞ and obtained the optimal time-decay estimate in the general critical L p -framework.

1.3.
Main result: global well-posedness. In this paper, we consider the Euler-alignment system (1.1) in R N , with power-law type pressure (1.2) and strongly singular alignment interactions (1.6) with 1 < α < 2. We mainly study the global well-posedness of the system with initial data (ρ 0 , u 0 ) around the non-vacuous equilibrium (ρ ≡ 1, u ≡ 0), with minimal regularity assumptions on the initial data.
Analogous to the study of the barotropic compressible Navier-Stokes system, we observe that the system (1.1) is invariant under the transformation with a modification of the pressure P → λ 2α−2 P . Therefore, we shall aim to solve the Euleralignment system (1.1) in the critical function space which is invariant with respect to the transform (1.10). Obviously, the homogeneous Besov spaceḂ of initial data (ρ 0 − 1, u 0 ) is a suitable space that is scaling critical. However, spectral analysis of the linearized equation of system (1.1) (see (3.3) below) indicates that the regularity ρ ∈Ḃ N 2 2,1 is not enough to control the pressure term, as it is not invariant under the scaling (1.10). Instead, we work on a hybrid Besov space 2,1 (see Definition 2.1). This approach is pioneered by Danchin [10] on the barotropic compressible Navier-Stokes system.
then the Euler-alignment system (1.1) has a global unique solution (ρ, u) such that

12)
and then the above constructed solution also belongs to the corresponding space, i.e., When N = 1, we are able to establish a priori estimates of the Euler-alignment system as in Section 3. However, some technical estimates fail when constructing the solution e.g. (4.6), (4.8), as well as the uniqueness argument. This is due to the roughness of initial data that we consider. If we further assume ∂ x ρ 0 ∈ B 3 2 −α, 1 2 (R) and ∂ x u 0 ∈Ḃ 3 2 −α 2,1 (R), Theorem 1.1 can be easily extended to N = 1. Remark 1.3. For less singular communication weight with 0 < α 1, we observe a different spectral structure in the linearized equation (3.3). Hence, the global well-posedness result is expected to be different. This case will be discussed in a separate work.
1.4. Asymptotic behavior. Next, we turn our attention to the asymptotic behavior of the Euleralignment system (1.1). The system inherits a remarkable flocking phenomenon from the Cucker-Smale model (1.3). For the pressure-less system with bounded Lipschitz communications, Tadmor and Tan [41] showed that for any smooth subcritical initial data, the solution converges (in appropriate sense) to a traveling wave profile There are two ingredients of flocking. First, the support of density ρ stays bounded in all time, i.e., if ρ 0 is compactly supported, then ρ ∞ is compactly supported as well. Another ingredient is the velocity alignment. Here,ū represents the average velocity. It is determined by initial data, thanks to the conservation of momentum. Without loss of generality, we assumeū = 0 throughout the paper. See [16,38] for discussions on flocking for pressure-less Euler-alignment system with strongly singular alignment interactions (1.6) on T. Additional geometric structures of the limiting profile ρ ∞ is investigated in [27,25]. When pressure is presented, the asymptotic density profile is known to be a constant ρ ∞ (x) ≡ρ. For simplicity, we setρ = 1. The asymptotic flocking behavior is proved in [45] for bounded alignment interactions, and [5] for strongly singular alignment interactions. Both results considered the periodic domain T N .
To our best knowledge, most existing results on asymptotic behaviors for the Euler-alignment system with singular alignment interactions are on the periodic domains. The decay rates in (1.15) are exponentially in time. For the system in R N , we do not expect the decay rate to be exponential. Rather, analogous to the heat equation, the diffusion leads to a polynomial rate of decay in time.
Our next result is concerned with the asymptotic behavior of the Euler-alignment system (1.1) in R N . Theorem 1.2 (Asymptotic behavior). Let N 2 and 1 < α < 2. Assume (ρ, u) is a global solution of the Euler-alignment system (1.1) that satisfies (1.12) and (1.16) Then for every 0 < s < 1 − 1 α we have where the constant C > 0 depends on the norms of (ρ − 1, u) in (1.16). Besides, we have If we assume, in addition,  (1.16). In the existing literature on the barotropic compressible Navier-Stokes system [15,33,49] and so on, smallness assumptions on the solution (ρ − 1, u) are required to obtain the decay estimates. We adopt a different approach that greatly relaxes the assumptions compared with the aforementioned work. Let us point out that our decay estimate (1.18) requires s 0 < N 2 . The endpoint s 0 = N 2 is not captured by our approach on a basic lack of paraproduct in endpointḂ −N/2 2,1 . Under additional smallness conditions on initial data, the decay estimate can be proved in an alternative way, analogous to [15,33,49].
1.5. Outline of the paper. The paper is organized as follows. In Section 2, we introduce the definition of hybrid Besov space and fractional Laplace operator, and present some useful auxiliary lemmas. In Section 3, we reformulate our system to (3.2), establish the a priori estimates for a linearized system (3.3) and the nonlinear system (3.2). Section 4 are devoted to the proof of global existence and uniqueness for our system (1.1). In Section 5, we present the proof of large time behavior stated in Theorem 1.2.

Preliminary
This section includes some basic analytical tools needed in this paper. We first introduce the concept of Besov spaces and some properties. Then we recall the definition of fractional Laplacian and the Kato-Ponce type commutator estimates, particularly in Besov spaces.
2.1. Besov spaces and some related estimates. We first introduce the Littlewood-Paley decomposition. One can choose a nonnegative radial function ϕ ∈ C ∞ c (R N ) be supported in the annulus {ξ ∈ R N : 3 4 |ξ| 8 3 } such that (e.g. see [2]) where ϕ j (ξ) = ϕ(2 −j ξ). We define the localization operator: Now we present the definition of (homogeneous) Besov space and its hybrid type.
Let j 0 ∈ Z, and we define the hybrid Besov space B s 1 ,s 2 = B s 1 ,s 2 (R N ) as the set of all u ∈ S ′ h (R N ) such that By restricting the norm ofḂ s p,r to the low or high frequency parts of tempered distributions, we also get that, for some j 0 ∈ Z, u ℓḂ s p,r := {2 js ∆ j u L p } j j 0 ℓ r , and u hḂ s p,r The following Bony's paraproduct decomposition is very useful in the proof.
, uv has the Bony's paraproduct decomposition: where We have the following product estimates in the hybrid Besov space B s 1 ,s 2 .
(2) The proof of (2.5) is quite analogous with that of (2.4) and [2, Theorem 2.52], and we omit the details.
(4) When s 1 = s 2 > 0, (2.7) is guaranteed by the classical inequality in [2, Corollary 2.54]. For the general case 0 < s 1 < s 2 , we use Bony's decomposition (2.3), and noting that for every j ∈ Z, and Hence, collecting the above estimates leads to (2.7). While for (2.8), it can be easily obtained by arguing as the deduction in (2.10) and (2.11).
In the analysis of asymptotic behavior, the following weighted paraproduct and reminder estimates play an important role.
(1) By virtue of the spectral support property, we see that (2.15) Plugging the definition ofṠ j ′ −1 f to (2.15), we deduce that for every r 1 0, (2) The inequality (2.13) for every r 2 ∈ R can be deduced in the same manner.
(3) By using the spectral property of the dyadic operators, we get where we have used the fact that 2 jβ C for every j j 0 . Consequently, the desired inequality (2.14) directly follows.
Let us state a continuity result for the composition in Besov space (for the proof one can see Proposition 1.5.13, Corollary 1.4.9 of [11] and Proposition A.3 of [15]).
Let I be an open interval of R, and let f : Then the function f (u) belongs toḂ s p,r (R N ), and there exists a constant C = C(s, I, J, N ) such that where [s] denotes the integer part of s.
Remark 2.1. After some trivial modification, the above composition inequalities can be adapted to the hybrid Besov space B s 1 ,s 2 with 0 < s 1 , s 2 N 2 . The result below is useful in the existence part.
The following three lemmas are concerned with the commutator estimates.
Lemma 2.5. Let α > 1. Then the following inequality holds true: Proof of Lemma 2.5. Taking advantage of the spectrum support property, we see that
Proof of Lemma 2.9. For every p < p 2 ∞, by taking α 1 = 1 and thus the desired estimate (2.23) follows from the triangle inequality and Hölder's inequality. Similarly, by taking α 1 = α−1 and α 2 = 1 and switching u, v in (2.21), and using the Calderón-Zygmund theorem of singular integral operator, we obtain which leads to (2.23) in the case p 2 = p. Now, we present a useful commutator estimate in Besov space as follows. (

2.24)
Proof of Lemma 2.10. We here apply Lemma 2.9 to show (2.24). Using Bony's decomposition (2.3) leads to In light of Lemma 2.9 and the spectrum support property of dyadic operators, we get Similarly, we have For the term Π 3 , we do not need to use the commutator structure, and we infer that for every s > − N 2 , Combining these inequalities (2.25)-(2.26) completes the proof of this lemma.
As a direct consequence of Lemma 2.10 with s = N 2 + 1 − r, s 1 = r 1 − α and s 2 = r − r 1 , we have the following commutator estimate.

A priori estimates
This section is devoted to establishing a priori estimates for the Euler-alignment system (1.1), with isothermal or isentropic pressures (1.2), and strongly singular alignment interactions (1.6).
Let us begin with rewriting the system into a more treatable form by introducing a new quantity It is easy to see that Consequently, (σ, u) satisfies the following coupled system where λ := √ κγ and µ := 1 c α,N . In the following, we first study paralinearized equations of (3.2), and obtain a priori estimates in a hybrid Besov space. The analysis is inspired by the work of Danchin [10] (see also [4,6]). We then calculate some nonlinear estimates and obtain a priori local/global uniform estimates for the system (3.2).
3.1. A priori estimates for the paralinearized equations. In this subsection we study the where v is a given vector field of R N , and F , G are given source terms. We denote The system (3.3) contains the major linear structures of (3.2). A priori energy estimates of (3.3) are stated as follows.
We decompose u into two parts u = −∇Λ −1 d + Pu. where Now we estimate the compressible part (∆ j σ,∆ j d) and the incompressible part P∆ j u respectively. For the sake of simplicity, we denote (f |g) = R d f (x) · g(x)dx in the sequel.
Step 1: the compressible part. Taking the L 2 inner product of the first two equations of (3.6) respectively and using the integration by parts give that and In order to get the smoothing effect of σ, we consider the cross term (Λ α−1∆ j σ|∆ j d). Applying the operator Λ α−1 to the first equation of (3.6) yields Together with the second equation of (3.6), we get To proceed, we work with low frequencies and high frequencies separately. Define (3.14) For low frequencies j j 0 , namely we set where δ > 0 is a suitable small constant, e.g. δ = 1 300 . It is easy to check that (3.16) Gathering the equations (3.10), (3.11) and (3.13), we see that Y j satisfies For the terms on the right-hand side of equality (3.17), using Hölder's inequality, we get and thanks to the integration by parts and Lemma 2.5, we find For the terms on the left-hand side of (3.17), by virtue of Hölder's and Young's inequalities, we have that for all j j 0 , Choosing δ small enough (e.g. δ = 1 300 ) and using (3.15), we obtain Thus the equation (3.17) can be rewritten as whereμ := δµ 8 and we have used the following estimate Now we consider the high frequency case j > j 0 , namely We define another energy, still denoted by Y j , as follows From (3.12) we can gain the following L 2 -estimate Noting that 2 λ µ (∆ j d|Λ α−1∆ j σ) 3 2 λ 2 and Y j satisfies For the left-hand side of the above equality, applying (3.20) we have On the other hand, for the terms on the right-hand side, owing to Hölder's and Bernstein's inequalities, we infer that and using the integration by parts and Corollary 2.5, Similarly as obtaining inequality (3.18), we also see that Hence, by lettingν := λ 2 4µ and gathering the above estimates, we have the following inequality Combining the estimates on both low and high frequencies (3.19) and (3.22), we obtain Integrating this with respect to the time variable and using (3.16), (3.21), we obtain (3.23) In the above estimate, the smoothing effect of d in the high-frequency can be improved. Indeed, taking the L 2 -inner product with d in the second equation of (3.6), we deduce that and integrating on the time variable leads to Inserting the above inequality into (3.23) yields (3.24) Step 2: the incompressible part. By taking the L 2 -estimate of equation (3.6) 3 , the incompressible part satisfies which implies that Integrating with respect to the time variable, we have (3.26) Step 3: the a priori estimate for (σ, u). We need to combine the compressible and incompressible estimates to show the a priori estimate of (σ, u). Multiplying 2 js on both sides of (3.24) and (3.26), taking the ℓ 1 -norm with regard to j, and noting that u Ḃs 2,1 d Ḃs Recalling that f j , g j and g j are given by (3.7)-(3.9), and by virtue of Lemma 2.6, we see that and j∈Z 2 js ( g j L 2 + g j L 2 ) C G Ḃs With the help of Gronwall's inequality, we conclude the proof of (3.4).

3.2.
A priori estimates for the nonlinear system. Let us turn our discussion to the nonlinear system (3.2). It can be viewed as (3.3) with v = u and and We state the following a priori estimate on X(T ), assuming X 0 is sufficiently small. The uniform bound on X(T ) will play an important role in the global well-posedness theory for the system (3.2).
Proof of Proposition 3.2. Apply Proposition 3.1 with s = N 2 + 1 − α and get . Now, we estimate the source terms F and G. It follows from Lemma 2.1 that and Taking advantage of Corollary 2.11 and Lemma 2.3, we obtain (The endpoint corresponds to vacuum ρ = 0, which needs to be avoided). Therefore, we may assume e.g.
to make sure C is a universal constant. (3.36) can be enforced by an appropriately chosen smallness condition.
We can further choose a smaller ε 0 to get a smaller X(T ) when needed. In particular, since B N 2 +1−α, N 2 ֒→ L ∞ , we can choose ε small enough to guarantee (3.36).
Next, we present an improved a priori estimate on X(T ) for some positive time T , without a smallness assumption on u 0 . This allows us to obtain a stronger local well-posedness result (see Theorem 4.2). This method has been used on the barotropic compressible Navier-Stokes system, see e.g.

Proof of Proposition 3.3.
We split u into u L + u, with u L satisfying It is obvious that u L = e −µtΛ α u 0 , where e −µtΛ α = F −1 (e −µt|ξ| α ) * is the fractional heat semigroup operator. By using Bernstein's inequality and [22, Proposition 2.2], we have and .
According to (3.30) of Proposition 3.1 with s = N 2 + 1 − α, we infer that Hölder's inequality implies Now we calculate the terms involving F and G. By virtue of Hölder's inequality and the interpolation, we get div u L L 1 It follows from Lemma 2.1 that σ div u and

From (3.35) we get
Plugging the estimates above into (3.41), we have with C 1 , C 2 > 0. Since lim t→0 U L (t) = 0, by letting T > 0 small enough, we infer that for every t ∈ [0, T ], 8C 1 C 2 and C = 4C 1 , a similar continuity argument as in Proposition 3.2 yields

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. As a byproduct, we also include the local well-posedness result.
We will mainly prove the following global well-posedess result to the system (3.2).
The local solution is constructed through a standard approximation by the paralinearized equations (3.3) and applying a priori estimates to pass to the limit. We sketch the proof in below, with special attention to the nonlocal alignment term that needs a careful treatment.
The assumption (4.3) only requires smallness on σ 0 , thanks to Proposition 3.3. It is possible to obtain local well-posedness without such condition (see [12] on the compressible Navier-Stokes system). Since our global well-posedness result requires a stronger smallness assumption (4.1), we do not make an effort to remove this smallness condition.
Proof of Theorem 4.2: existence. We first construct the following approximate system: where J n is the Friedrichs projector defined by and u n =ū n + u n L with u n L satisfying ∂ t u n L + µΛ α u n L = 0, u n L | t=0 = J n u 0 , and F n (σ n ,ū n ) = −λJ n divū n − λJ n div u n L − J n (u n · ∇σ n ) − (γ − 1)J n (σ n div u n ), In addition, we define the setL 2 n := {f ∈ L 2 (R N ); supp f ⊂ C n }. It is easy to see that the map (σ n ,ū n ) → F n (σ n ,ū n ), G n (σ n ,ū n ) is locally Lipschitz continuous and also continuous with respect to t in (L 2 n ) N +1 . Via the Cauchy-Lipschitz theorem, there is a unique local solution (σ n ,ū n ) ∈ C 1 ([0, T * n ); (L 2 n ) N +1 ) to (4.4). Since J n is uniformly bounded from L 2 to L 2 , from (3.39) and (3.40), we see that u n L is bounded uniformly in n, and satisfies Arguing as obtaining the a priori estimate in Proposition 3.3, under the assumption (4.3) with small η > 0, we infer that (σ n , u n ) is uniformly-in-n bounded and satisfies that for some T > 0, where X 0 is given by (3.33). As a result, we have uniformly in n. The interpolation also implies that u n ∈ L r T (Ḃ Due to the fact that {(σ n ,ũ n )} n∈N is a Cauchy sequence in the considered space, there exists functions (σ,ũ) such thatσ n →σ in L ∞ 2,1 ). Now denote byσ n := σ n −σ n for every n ∈ N. In order to gain more time-continuity information, we consider (∂ tσ n , ∂ tū n ). Note that ∂ tσ n = −λJ n div u n − λ divũ n − J n (u n · ∇σ n ) − (γ − 1)J n (σ n div u n ), and by virtue of the above uniform estimates and Lemma (2.1), we have Thus,σ n is uniformly bounded in Meanwhile, recalling the equation ofū n in (4.4), and with the help of Lemma 2.1 and Corollary 2.11, we obtain . Via Fatou's property for Besov space (see [2,Thm. 2.25]) and the uniform estimates of (φσ n , φū n ), we moreover get 2,1 ). (4.10) Now, we pass to the limit in the approximate system (4.4). We here only deal with the term J n Λ α u n h(σ n ) −u n Λ α h(σ n ) , since the remaining terms can be treated by the standard process. Let φ ∈ C ∞ c (R N ) be any fixed function with supp φ ⊂ B(0, R). Recalling that ϕ is the function introduced in (2.1), there exists a bump function χ ∈ C ∞ c (R N ) supported in the ball B(0, 4 3 ) such that χ(x) + j 0 ϕ j (x) = 1 for every x ∈ R N (see [2]). Clearly, χ ≡ 1 in B(0, 3 4 ), and we can choose k ∈ N large enough so that χ k (·) = χ(2 −k ·) ≡ 1 in B(0, 2R) and it also holds that χ k (x) + j k ϕ j (x) = 1 for every x ∈ R N . We write . Similarly as (3.35), we know that Λ α (u n h(σ n )) − u n Λ α (h(σ n )) has a uniformly-in-n bound in ). Then by virtue of the dual property and the smoothness of φ, we can show that lim n→∞ I n 1 = 0.
Next we consider I n 3 . Noting that ϕ j h(σ n ) = ϕ j h(χ j+2 σ n ) for every j ∈ N, and by using the integral formula of Λ α given in (2.20) and the support property, we infer that where in the last line we have used the uniform (in j and n) estimates For any ǫ > 0, Lemma 2.4 and (4.5), (4.10) ensure that with C > 0 independent of j and n, then there exists a large number J ∈ N so that On the other hand, the above established convergence result guarantees that there exists n 0 = n 0 (α, N, J, ǫ) ∈ N so that for any n n 0 , Thus for any ǫ > 0, there exists n 0 ∈ N so that for any n n 0 , Similarly, for the remaining term in I n 3 we have Arguing as above, we can deduce that there exists a constant n 1 ∈ N so that for any n n 1 , Hence, combining (4.11) with (4.12), for any ǫ > 0 we have that |I n 3 | ǫ for every n max{n 0 , n 1 }, which implies lim n→∞ I n 3 = 0. Therefore, gathering the above convergence result of I n 1 -I n 3 , we conclude the convergence of the term J n Λ α u n h(σ n ) − u n Λ α h(σ n ) .
Finally, we show the time continuity property of (σ, u). Note that σ ∈ L ∞ T ( B ).

4.2.
Uniqueness. We continue with a uniqueness argument for our constructed solution.

Applying (2.17) in Lemma 2.3, we get
Taking advantage of Lemma 2.1, Corollary 2.11 and the above inequality, we infer that .
By setting η > 0 in (4.13) be small enough so that C σ 1 1 4C , we gather the above estimates to get Gronwall's inequality guarantees that δσ In other words, σ 1 ≡ σ 2 and u 1 ≡ u 2 on [0, T ] × R N .

4.3.
Global existence. Now we are ready to prove Theorem 4.1. Under the smallness assumption on X 0 in (4.1), Proposition 3.2 ensures the smallness of X(T ). Hence, we can extend the solution using the local existence result in Theorem 4.2, viewing T as the initial time. Repeating the process, we obtain a global solution.
Proof of Theorem 4.1. Take ε ′ = min{ε 0 , η/C * }, where (ε 0 , C * ) are the constants in Proposition 3.2 and η is the constant in Theorem 4.2. Let T * be the maximal existence time of the solution (σ, u) to (3.2), namely We will show T * = ∞ by contradiction. Suppose T * is finite. A direct application of Proposition 3.2 yields The continuity of X then implies X(T * ) η. We then apply Theorem 4.2 to the system (3.2) initiated at time T . There exists a time T > 0 such that (σ, u) exists in [T * , T * + T ]. This contradicts the definition of T * .

4.4.
Propagation of smooth initial data. In this subsection, we show that if the initial data is smoother (also known as subcritical), the solution will inherit the initial regularity.
Proof of Proposition 4.3. Let F and G be defined as in (3.31). The system (3.2) can be seen as in the form of (3.3) with v replaced by u. Then according to Proposition 3.1, we have dτ . Owing to Lemmas 2.1 and 2.10, and using (2.18) and (4.2), we infer that , and . Since ε ′ is small enough, we obtain the desired estimate (4.14).

Asymptotic behavior: proof of Theorem 1.2
This section aims at proving Theorem 1.2. We shall mainly prove the following asymptotic behavior of the global solution (σ, u) for system (3.2).
where C depends on the norms of (σ, u) in (5.1). Besides, we have If we assume, in addition, Our task remains to prove Proposition 5.1. Let us denote Recalling the estimates (3.19), (3.22) and (3.25), we have whereμ := δµ 8 , µ h := min(ν2 j 0 (2−α) , µ2 j 0 α ) and f j , g j , g j are defined by (3.7)-(3.9) with v = u, and F = −(γ − 1)σ div u − T ∇σ · u − R(u, ∇σ), Note that X j for low-frequency part j j 0 has a dissipation effect analogous to the fractional heat operator, while for high-frequency part j > j 0 it has a damping effect. Thus one may expect that (σ, u) altogether will have a polynomial decay by developing the dissipation/damping effect. In the sequel we will treat the low-frequency part and high-frequency part separately to show the desired decay estimates. Before proceeding forward, we introduce the following notations: for −s 0 s N 2 + 1 − α and s 0, and Z s,s (t) := Z ℓ s,s (t) + Z h s (t). (5.10) and fors + sα  , ifs + sα > N 2 + 1 − α. (5.14) Proof of Lemma 5.2. By multiplying the first inequality of (5.6) with eμ 2 jα t and integrating over the time interval [0, t], we infer that Using the fact that sup t 0 j∈Z 2 αjs t s e −ct2 αj < +∞, we have that for every s ∈ [0, 1), Multiplying both sides of the above equation with 2 j(s+sα) and taking the ℓ 1 -norm with respect to j j 0 lead to where we have abbreviated ψ j (t, τ ) as ψ j .
For the terms g 1 j + g 4 j and g 8 j , taking advantage of inequalities (2.23), (5.10) and Lemma 2.3, we obtain For the terms g 2 j and g 5 j , using (2.12), (5.10) and Lemma 2.3, we deduce that For the terms g 3 j and g 6 j , thanks to inequality (2.14), (5.10) and Lemma 2.3 again, we infer that where δ is given by (5.14) so thats − 1 + α + δsα N 2 (to fit the norm Z s,s (τ )). For the term g 7 j , with the help of Lemma 2.7, we find For the last term g 9 j , thanks to the spectral support property of dyadic operators and (5.11), we see that where in the above we have used the fact that 2 j(1−δ)1 {j ′ j 0 } C for every j j 0 . Inserting (5.17)-(5.20) into (5.16), we thus conclude the inequality (5.12).

Lemma 5.3. Under the assumption of Proposition 5.1, ifs
Proof of Lemma 5.3. From the inequality (5.6), we have that for every j j 0 , where R j is given by (5.15). Multiplying the above inequality with eμ 2 jα t and integrating over [0, t], we obtain Multiplying the above equation with 2 j(s+sα) and taking the ℓ 1 -norm with respect to j j 0 , we find Inserting (5.24) into (5.23), we obtain the desired inequality (5.21).

5.2.
High-frequency estimates. Since X j exhibits a damping effect in the second inequality of (5.6), one can generally expect to derive an exponential decay for X j . But in order to be coincident with the low-frequency case, we instead will prove a polynomial decay estimate.
Proof of Lemma 5.4. Starting from (5.6) and letting t 0 = sµ −1 h , we have that for every t t 0 , d dt Thus, we deduce that Recalling notations (5.5) and (5.9), we multiply the above inequality by 2 j( N 2 +1−α) and take the ℓ 1 -norm over j > j 0 to get that for every t t 0 , (5.27) The above inequality also holds when t t 0 : Note that in view of (5.1), the above right-hand term is under control.
Next, we deal with the term in (5.27) containing R j (given by (5.26)). The last term in R j can be estimated as .
Hence, after an iteration of finite times and using the above 0 s < 1 result, we conclude the proof of (5.28) for every (s,s) satisfying (5.29).
Then there exists a positive real number T 0 such that for every t T 0 , Hence for any ε > 0, we have that for every t T 0 , In other words, the inequality (5.3) holds. We thus finish the proof of Proposition 5.1.