Global existence and optimal decay estimates of strong solutions to the compressible viscoelastic flows

This paper is dedicated to the global existence and optimal decay estimates of strong solutions to the compressible viscoelastic flows in the whole space $\mathbb{R}^n$ with any $n\geq2$. We aim at extending those works by Qian \&Zhang and Hu \&Wang to the critical $L^p$ Besov space, which is not related to the usual energy space. With aid of intrinsic properties of viscoelastic fluids as in \cite{QZ1}, we consider a more complicated hyperbolic-parabolic system than usual Navier-Stokes equations. We define"\emph{two effective velocities}", which allows us to cancel out the coupling among the density, the velocity and the deformation tensor. Consequently, the global existence of strong solutions is constructed by using elementary energy approaches only. Besides, the optimal time-decay estimates of strong solutions will be shown in the general $L^p$ critical framework, which improves those decay results due to Hu \&Wu such that initial velocity could be \textit{large highly oscillating}.


Introduction
We consider the following equations of multi-dimensional compressible viscoelastic flows: ∂ t (ρu) + div(ρu ⊗ u) − div(2µD(u) + λdivuId) + ∇P = αdiv(ρF F T ), ∂ t F + u · ∇F = ∇uF, (1.1) where ρ ∈ R + is the density, u ∈ R n stands for the velocity and F ∈ R n×n is the deformation gradient. F T means the transpose matrix of F . The pressure P depends only upon the density and the function will be taking suitably smooth. Notations div, ⊗ and ∇ denote the divergence operator, Kronecker tensor product and gradient operator, respectively. D(u) = 1 2 (∇u + ∇u T ) is the strain tensor. The density-dependent viscosity coefficients µ, λ are assumed to be smooth and to satisfy µ > 0, ν λ + 2µ > 0. For simplicity, the elastic energy W (F ) in system (1.1) has been taken to be the special form of the Hookean linear elasticity: which, however, does not reduce the essential difficulties for analysis. The methods and results of this paper could be applied to more general cases.
In this paper, we focus on the Cauchy problem of system (1.1), so the corresponding initial data are supplemented by (ρ, F ; u)| t=0 = (ρ 0 (x), F 0 (x); u 0 (x)), x ∈ R n . (1.2) It is well known that there are some fluids do not satisfy the classical Newtonian law. Also, there have been many attempts to capture different phenomena for non-Newtonian fluids, see for example [13,14,23,27]  Here let's first recall previous efforts related to viscoelastic flows. For the incompressible viscoelastic flows, there has been much important progress on classical solutions. Lin-Liu-Zhang [23], Chen-Zhang [7], Lei-Liu-Zhou [24] and Lin-Zhang [28] established the local and global well-posedness with small data in Sobolev space H s . Hu-Wu [21] proved the long-time behavior and weak-strong uniqueness of solutions. Chemin-Masmoudi [4] proved the existence of a local solution and a global small solution in critical Besov spaces, where the Cauchy-Green strain tensor is available in the evolution equation. Qian [29] proved the well-posedness of the incompressible viscoelastic system in critical spaces. Subsequently, Zhang-Fang [33] proved the global well-posedness in the critical L p Besov space. On the other hand, the global existence of weak solutions is still an open problem. Lions and Masmoudi [26] considered a special case that the contribution of the strain rate is neglected, and proved the global existence of a weak solution with general initial data.
For compressible viscoelastic flows, Lei-Zhou [28] proved the global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit. The local existence of strong solutions was obtained by Hu-Wang [19]. Shortly, Hu-Wang [18] and Qian-Zhang [30] independently proved the global existence in the critical L 2 Besov space with initial data near equilibrium. For convenience of reader, we would like to state their results as follows. .
Obviously, it is easy to see that Ḃ n/2 2,1 1+n 2 × Ḃ n/2−1 2,1 n is the critical space according to Definition 1.1. It should be emphasized that such basic idea is motivated by the seminal paper [8], where the author first proved the global well-posedness for the compressible Navier-Stokes equations near equilibrium. Compared to [8], there is an outstanding difficulty for the compressible viscoelastic system, that is, how to capture the damping effect of the deformation tensor among more complicated coupling between the velocity, the density and the deformation tensor. Hu-Wang [18] and Qian-Zhang [30] independently explored some intrinsic properties of the viscoelastic system and established uniform estimate for more complicated linearized hyperbolic-parabolic systems, which eventually leads to Theorem 1.1.
The goal of this paper is twofold: firstly, we aim at extending the above statement (Theorem 1.1) to the critical L p Besov space, which allows highly large oscillating initial velocity. Secondly, we shall exhibit the long time behavior of the constructed solution. Denote and In comparison with those results in critical L p framework for compressible Navier-Stokes equations (see for example [3] and [5]), Theorem 1.2 is not so surprising. Let's point out some new ingredients in the current proofs. To the best of our knowledge, there is a technical difficulty arising from a loss of one derivative for compressible N-S fluids, since there is no smoothing effect for the density in high frequency. To eliminate it, their proofs heavily rely on a paralinearized version combined with a Lagrangian change of variables, see [3,5] for details. To the compressible viscoelastic system, the situation becomes more complicated. As shown by [30], the damping effect of F can be produced by some intrinsic conditions (see Proposition 3.1), however, similar to the density, there is not any smoothing effect at high frequencies. Here, in order to solve (1.1) globally, we follow an elementary energy approach in terms of effective velocity rather than the elaborate Lagrangian change. The argument has been developed by Haspot [15,16] for compressible Navier-Stokes equations, which is based on the use of Hoff's viscous effective flux in [17]. Here, we introduce the following "two effective velocities", Indeed, the definition of w is almost the same as that in [15,16]. A slight difference lies on the coefficient of a, which comes from contribution of the deformation gradient F . Another effective velocity with respect to Ω ij is new, which allows to cancel the coupling between e ij and O ij at high frequencies (see Sections 4 and 5 for more details). In physical dimensions n = 2, 3, the value of p enable us to consider the case p > n for which the velocity regularity exponent n/p − 1 becomes negative. Consequently, Theorem 1.2 applies to large highly oscillating initial velocities (see [3,5] for more explanation).
An interesting question follows after gaining Theorem 1.2. One may wonder how the global strong solutions constructed above look like for large time. Although providing an accurate long-time asymptotic description is still out of reach, a number of results concerning the time decay rates of global solutions, sometimes referred to as L q − L r decay rates are available. For example, Hu-Wu [20] proved the global existence of strong solutions to (1.1) as initial data are the small perturbation (1, I; 0) in H 2 (R 3 ). Furthermore, with the extra L 1 (R 3 ) assumption, it was shown that those solutions converged to equilibrium state at the following way The decay rate in (1.4) turns out the same one for the heat kernel, which is sometime referred as the optimal time-decay rate.
Here f ℓ • and f h • represent the low and high frequency part of some norm f • to a tempered distribution f whose exact definition will be given in Section 2.
Some comments are in order.
3. Likewise, "two effective velocities" play a key role in establishing the nonlinear timeweighted inequality (1.7). Furthermore, the optimal decay estimates of L q -L r type can be derived from the definition of G p (t) by using interpolation tricks. The interested reader is referred to [6] for similar details.
The rest of this paper is arranged as follows: In Section 2, we first review the Littlewood-Paley theory and give definitions and estimates for the hybrid-Besov space. In Section 3, we reformulate our system into a hyperbolic-parabolic system coupled by the density, the velocity and the deformation gradient. Section 4 is devoted to presenting the proof of Theorem1.2. In Section 5, we prove the decay estimate in Theorem 1.3. Some analysis properties in the hybrid Besov space are also given in the Appendix.

Littlewood-Paley Theory and the Hybrid Besov Space
Throughout the paper, we denote by C a generic constant which may be different from line to line. The notation A B means A ≤ CB and A ≈ B indicates A ≤ CB and B ≤ CA.

Littlewood-Paley decomposition
Let's begin with the Littlewood-Paley decomposition. There exists two radial smooth functions ϕ(x), χ(x) supported in the annulus C = {ξ ∈ R n : 3/4 ≤ |ξ| ≤ 8/3} and the ball B = {ξ ∈ R n : |ξ| ≤ 4/3}, respectively such that The homogeneous dyadic blocks∆ j and the homegeneous low-frequency cut-off operatorṡ S j are defined for all j ∈ Z bẏ The following Bernstein inequality will be repeatedly used throughout the paper.
Lemma 2.1 ( [2]) Let C be an annulus and B a ball. A constant C exists such that for any nonnegative integer k, any couple (p, q) in [1, ∞] 2 with q ≥ p ≥ 1, and any function u of L p , we have

The hybrid Besov space
We denote by Z ′ (R n ) the dual space of Firstly, we give the definition of the the homogeneous Besov space.
Secondly, we introduce the hybrid Besov space that will be used in this paper.
where R 0 is a fixed and sufficiently large constant which may depending on λ(1), µ(1), p and n.
Since we are concerned with time-dependent functions valued in Besov spaces, the spacetime mixed norm is usually given by u L q TḂ s,σ 2,p := u(t, ·) Ḃ s,σ 2,p L q (0,T ) . Here, we introduce another space-time mixed Besov norm, which is referred to Chemin-Lerner's spaces. The definition is as follows.
The index T will be omitted if T = +∞ and we shall denote byC b (Ḃ s,σ 2,p ) the subset of functions L ∞ (Ḃ s,σ 2,p ) which are continuous from R + toḂ s,σ 2,p . It is easy to check thatL 1 Also, for a tempered distribution f , we denote for s ∈ R.
Next, we collect nonlinear estimates in Besov spaces.  for p ≥ 2.

Remark 2.1 Lemma2.3 still remain true in the usual homogenous Besov spaces. For example the estimate in Lemma2.3(1) becomes
Let σ 1 , σ 2 , p 1 , p 2 satisfy Then we have There exists a universal interger N 0 such that for any 2 ≤ p ≤ 4, and σ > 0, There exists a constant C > 0, depending only on σ such that for all j ∈ Z, we have where (c j ) j∈Z denotes a sequence such that (c j ) ℓ 1 ≤ 1.

Reformulation of System (1.1)
Here, we present intrinsic properties of compressible viscoelastic flows, which have been explored in [30].
The density ρ and the deformation gradient F of (1.1) satisfy the following relations: By Proposition 3.1, the i-th component of the vector div(ρF F T ) can be written as where we used the first equality in (3.1).
For s ∈ R, we denote and introduce two variables as in [30]: Using the second equality in (3.1), we have Hence, with aid of (3.6), the system (3.4) can be reformulated as follows Additionally, we need the auxiliary equation in subsequent estimates which is deduced from the first equality in (3.1).

Proof of Theorem 1.2
Inspired by [5], we may extend those results in [30] to the L p critical framework. First of all, it is convenient to give the following interpolation inequalities The proof Theorem 1.2 is divided into several parts. The first one is to establish two a priori estimates.

Two a priori estimates
Let T > 0. We denote the following functional space E Then we have We introduce another functional space E n/2 T defined by The proof of Propositions 4.1-4.2 lie in the pure energy methods in terms of low-frequency and high-frequency decompositions.
Step1: Low-frequency estimates ( (4.4) Taking L 2 inner product of (4.4) 2 with e ij k , and then summing up the resulting equation with respect to indices i, j, we arrive at Taking L 2 inner product of (4.4) 1 and (4.4) 3 with a k and O k , respectively, and then adding the resulting equations to (4.5) together, we obtain To capture the dissipation arising from (a, O), we next apply the operator Λ to (4.4) 1 and take the L 2 inner product of the resulting equation with −d k . Also, we take the L 2 inner product of (4.4) 2 with Λ −1 ∂ i ∂ j a k . Therefore, we add those resulting equations and get On the other hand, we apply Λ to (4.4) 3 and then take the L 2 inner product of the resulting equation with e ij k . We also take the L 2 inner product of (4.4) 2 with ΛO ij k . By summing up those resulting equations, we obtain Now, we multiply a small constant ν 1 > 0 (to be determined) to (4.7) and (4.8), respectively, and then add the resulting equations with (4.6) together. Consequently, we are led to the following inequality It follows from (3.8) that Inserting (4.10) into (4.9), we can get For any fixed R 0 , we choose ν 1 ∼ ν 1 (λ 0 , µ 0 , R 0 ) sufficiently small such that (4.12) By using Cauchy-Schwarz inequality in (4.11), we can get the following equality owing to . (4.14) Next we begin to bound those nonlinear terms arising in G i (i = 0, 1, 3, 4). Since the quadratic terms containing a and v have already been done in [5], it suffices to deal with different terms involving in O as well as those cubic terms due to density-dependent viscosities. More precisely, we need to estimate the following terms according to the definitions of G i , and We write G j 0 = ∂ i aO ij + a∂ i O ij . Regarding ∂ i aO ij , by taking γ = −1, r 1 = ∞, r 2 = 1, r 3 = r 4 = 2, s 1 = s 2 = n/2 − 1, t 1 = t 2 = n/2 in (A.2) and using (4.1), we arrive at The terms a∂ i O ij , v · ∇O ij in G ij 3 and (4.15) may be treated along the same lines as ∂ i aO ij , so we omit the details for brevity. In order to bound ∂ k v i O kj in G ij 3 , by taking γ = 0, r 1 = ∞, r 2 = 1, r 3 = r 4 = 2, s 1 = s 2 = n/2 − 1, t 1 = t 2 = n/2 in (A.2) and using (4.1), we have (4.18) Next we bound the cubic term (4.16) in G ij 4 . Denote ∇μ(a)∇v To bound I 1 , we have (4.20) Inserting (4.20) and (4.21) into (4.19) and applying PropositionA.2, we can get To bound I 2 , we have .
Since bound of the cubic term 1 1+a div λ (a)divvId is the same as I, we omit the details. Summing up all the estimates and remembering (4.14), we get (4.27) Step 2: High-frequency estimates (2 k > R 0 ). Inspired by [15,16], we perform basic energy approaches in terms of effective velocities rather than the Lagrangian change as in [3,5]. (4.28) Note that (4.28), we get the following equation for the compressible part of v where The motivation using the system (4.29) is to make a comparison with the usual compressible Navier-Stokes equations. Here, we consider more complicated hyperbolic-parabolic coupled system Introduce two effective velocities as follows Noticing that the definition of w is almost the same as that in [15,16]. The subtle difference lies on the coefficient of unknown a, which comes from the contribution of deformation gradient F , see (4.28). The new effective velocity Ω ij is used to cancel the coupling between e ij and O ij in the high-frequency estimate. Firstly, we present those estimates for effective velocities. It follows from (4.31) that ) .

(4.35)
Owing to the high frequency cut-off 2 k > R 0 , we have ) .

(4.37)
Secondly, we see that (a, O ij ) satisfies the following damped equations in terms of effective velocities Applying∆ k to the above equations, we obtain .
Keep in mind that w =d + 2∇(−∆) −1 a, Ω ij = e ij + 1 µ 0 (−∆) − In addition, remembering (4.32), we have Likely, we need to bound those different terms inG i (i = 1, 3) and G i (i = 2, 4) compared to [5], for example,G  In order to bound ∂ k v i O kj , from (A.1) of PropositionA.1 with , r 1 = 1, r 2 = ∞, σ = τ = n/p, we have For O∇O, from (A.1) of PropositionA.1 with r 1 = r 2 = 2, σ = n/p, τ = n/p − 1 and by applying interpolation (4.1), we have (4.48) The estimate for div(aO) = a∇ · O + ∇aO may be handled with at the same away as O∇O. Next, we bound the cubic term (4.47) in G ij 4 . Following from the the same notation, we know that ∇μ(a)∇v For I 1 , we have On the other hand, regarding I 2 , we deduce that The computation for 1 1+a div λ (a)divvId totally follows from the same procedure as I, so we omit details. By putting above estimates together, remembering (4.46), we achieve that Step 3: Combination of two-step analysis. (4.51) The inequality (4.3) is followed by (4.50) and (4.51). Therefore, the proof of Proposition 4.2 is complete.

Approximate solutions and uniform estimates
The construction of approximate solutions is based on the following local-in-time existence.
In order to apply Theorem 4.1, we need the following lemma, which could be shown by the proof of Lemma 4.2 in [1]. when k → 0. we also have ρ 0,k ≥ c 0 2 for any k ∈ N.
Let (ρ 0,k , F 0,k ; u 0,k ) be the sequence for initial data stated in Lemma 4.1. Then Theorem 4.1 indicates that there exists a maximal existence time T k > 0 such that system (1.1) with initial data (ρ 0,k , F 0,k ; u 0,k ) has a unique solution (ρ k , F k ; u k ) with ρ k bounded away from 0, and satisfies Then using the definition of Hybird Besov spaces and Bernstein inequality in Lemma 2.1, we have First we claim that With aid of the continuity argument, it suffices to show for all k ∈ N, Then By applying Proposition 4.1, we obtain By choosing M = 3CC 0 and η sufficient small enough such that so (4.53) is followed by (4.54) directly. Therefore, we obtain a sequence of approximate solutions (ρ k , F k ; u k ) to the system (1.1) for any k ∈ N. From (4.3) and (4.55), we have where we chose η sufficiently small. Consequently, based on Proposition 4.2, the continuity argument ensures that T k = +∞ for any k ∈ N.

Passing to the limit and existence
Next, the existence of the solution will be proved by the compact argument. We show that, up to an extraction, the sequence (a k , O k ; v k ) converges in the distributional sense to some function (a, O; v) such that ). By interpolation, we also de- with ζ = min{ 2n p − 1, 1}, which is a direct consequence of (4.59) Recalling (3.4), we have By interpolation and Lemma 2.3, it follows from (4.55) that (∂ t a k , ∂ t O k ) ∈ L 2 locḂ n/p−1 p,1 . On the other hand, It's easy to see that (4.60) Thanks to Lemma 2.3 and Proposition A.
. As before, we write Then by applying Lemma 2.3 and Proposition A.2, we get and 1 1+a k div λ (a k )divv k Id may be treated along the same way. Consequently, combining (4.60) − (4.63), we conclude that ). Therefore the claim (4.58) is proved. Furthermore, we see that (a k , O k ; v k ) is equicontinuous on R + valued in Ḃ n/p−1 p,1 n . Let {φ j } j∈N be a sequence of smooth functions supported in the ball B(0, j + 1) and equal to 1 in B(0, j). It follows from (4.58 By applying Ascoli's theorem and Cantor's diagonal process, there exist a (a, O; v) such that for any smooth function φ ∈ C ∞ 0 (R n ), when k → +∞ (up to an extraction). Actually, by interpolation, we also have Then, using the so-called Fatou property in Besov spaces and the uniform bound in (4.55), we conclude that (4.57) is fulfilled. It is a routine process to verify that (a, O; v) satisfies the system (3.4) in the sense of distributions. Below is to check the desired regularity of solutions. Noticing that ).
On the other hand,

Uniqueness
Due to techniques, allow us to only deal with the case 2 ≤ p ≤ n in the proof uniqueness of solutions. We will consider the remaining interval with respect to p in near future. The proof depends on a logarithmic inequality. For convenience of reader, we present it by a lemma. Assume that (ρ i , F i ; u i )(i = 1, 2) are two solution to the system (1.1) with the same initial data. Without loss of generality, we may assume that (4.66) Using embedding and (4.66), we have for η > 0 sufficiently small. Set Thanks to (3.4), we find that (δa, δv, δO) satisfies with δF = −δv · ∇a 1 − ∇ · δv − a 1 ∇ · δv − δa∇ · v 2 , δH = δv · ∇O 1 + ∇δv + ∇δvO 1 + ∇v 2 δO, In the following, we denote it is suffices to prove the uniqueness in E 1 . So we take p = n in the subsequent process. Applying Proposition A.3, we get Inserting the equality into (4.70), we arrive at by Gronwall's inequality According to our a prior estimates, by choosing η small, we have Consequently, inserting (4.73) into (4.72) to implies that Combining (4.71) and (4.74), we get Applying Lemma 4.2 with s = r = ε = 1 and f = δv, we obtain which together with (4.74) and (4.71) indicates that

The proof of time-decay estimates
In this section, we aim at proving the time-weighted energy inequality (1.6) taking for granted Theorem 1.2. We will proceed the proof into the three subsections, according to three terms of the definition of G p (t). Subsection 5.1 is devoted to the low-frequency estimates. In the spirit of [6], we need to perform nonlinear estimates in terms of deformation tensor in the Besov space with negative regularity. In Subsection 5.2, in order to overcome the technical difficulty that there is loss of one derivative for the density and deformation tensor at high frequencies, we develop "two effective velocities" to obtain the upper bound for the second term in G p (t).
To close the high-frequency estimates, in Subsection 5.3, a crucial observation enables us to establish gain of regularity and decay altogether for the velocity, which strongly depends on Proposition A.4. For simplicity, we denote In what follows, we will use the two key lemmas repeatedly. For convenience, we denote by · δ,L p := ( · p L p + δ p ) 1/p for 1 ≤ p < ∞.

Bounds for the low frequencies
From (4.11) and (4.12), we have It follows from Lemma 5.2 that d dt Then integrating in time and letting δ → 0, then there exists a c 0 such that Regarding the first term in (5.5), we multiply the factor t s+s 0 2 2 ks and sum up on 2 k ≤ R 0 to get where we have used the fact Furthermore, the corresponding nonlinear term in (5.5) can be estimated as We claim that if p fulfills the assumption as in Theorem 1.3, then we have for all t ≥ 0, where G p (t) and X p (t) are defined by (1.7) and (5.1). Since those quadratic terms containing a and v in G i (i = 0, 1, 3, 4) have already been done in [6], it suffices to give suitable decay estimates for some terms involving in O. Precisely, we need to hand the following integral As far as we know, the regularity level remains the same between the density and deformation tensor. Hence, these terms ∂ i aO ij , a∂ i O ij , O jk ∂ j O ik , O lj ∂ l O ik , O lk ∂ l O ij can be treated along the same line. In principle, the above integral can be reduced to In order to handle I ℓ in terms of with a ℓ , O ℓ and v ℓ , we use the following Lemma. where we have used the Berstein inequality (p * = 2p p−2 ≥ p) and the fact that only finite middle frequencies of g are involving inṠ k 0 +N 0 g h . 1 1 The limit case p = n follows from f g h ℓḂ (5.24) By embedding, the definition of G p (t) and the fact that α ≥ n 2p for sufficiently small ε > 0, we have  • If σ, τ ≤ n/p and σ + τ > 0, then TḂ n/2−n/p+σ,σ 2,p g L r 2 TḂ n/2−n/p+τ,τ 2,p . (A.1) • If s 1 , s 2 ≤ n/p and s 1 + t 1 > n − 2n p with s 1 + t 1 = s 2 + t 2 and γ ∈ R, then • If s 1 , s 2 ≤ n/2 and s 1 + t 1 > n 2 − n p with s 1 + t 1 = s 2 + t 2 , then k∈Z 2 k(s 1 +t 1 −n/2) ∆ k (f g) L r  For the heat equation, one has the following optimal regularity estimate. where λ and µ are constants such that µ > 0 and λ + µ > 0(up to the different dependence on the viscous coefficients). Indeed, both Pu and P ⊥ u satisfy the heat equation. We can apply P and P ⊥ to (A.7) to get the heat estimate (A.6).