Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter

This paper is dedicated to the global well-posedness issue of the compressible Oldroyd-B model in the whole space $\R^d$ with $d\ge2$. It is shown that this set of equations admits a unique global solution in a certain critical Besov space provided the initial data, but not necessarily the coupling parameter, is small enough. This result extends the work by Fang and the author [{J. Differential Equations}, {256}(2014), 2559--2602] to the non-small coupling parameter case.

In this paper, we focus on the case ǫ = 1. Setting a := b Re , then (a, u, τ) takes the form  [17]. They proved that (i) system (1.7) admits a unique local strong solution in suitable Sobolev spaces H s (Ω) for bounded domains Ω ⊂ R 3 ; and (ii) this solution is global provided the data as well as the coupling constant ω between the velocity u and the symmetric tensor of the constrains τ are sufficiently small. For extensions to this results to the L p -setting, see the work of Fernandéz-Cara, Guillén and Ortega [15]. Later on, Molinet and Talhouk [33] removed the smallness restriction on the coupling constant ω in [17]. The situation of exterior domains was considered first in [19], where the existence of a unique global strong solution defined in certain function spaces was proved provided the initial data and the coupling parameter ω are small enough. Recently, Fang, Hieber and the author [12] improved the main result given in [19] to the situation of non-small coupling constant.
For the scaling invariant approach, and Ω = R d , d ≥ 2, Chemin and Masmodi in [7] proved the existence and uniqueness of the global solution to the Oldroyd-B model (1.7) with initial data (u 0 , τ 0 ) belonging to the critical space Ḃ  [7,23,28]. Besides, we would like to mention that Constantin and Kliegl [9] proved the global regularity of solutions in two dimensional case for the Oldroyd-B fluids with diffusive stress. An approach based on the deformation tensor can be found in [25,26,27,29,30,31,35,39].
On the other hand, the studies on compressible Oldroyd-B model have thrown up some interesting results. Lei [24] and Gullopé, Salloum and Talhouk [18] investigated the incompressible limit problem of the compressible Oldroyd-B model in a torus and bounded domain Ω ⊂ R 3 , respectively. They showed that the compressible flows with well-prepared initial data converge to incompressible ones when the Mach number ǫ converges to zero. The case of ill prepared initial data was studied by Fang and the author [14] in the whole space R d , d ≥ 2. In particular, if ǫ = 1, we also obtained in [14] the existence and uniqueness of the global solution in critical spaces to system (1.5) with small coupling constant ω. The unique local strong solution to (1.4) with initial density ρ 0 vanishing from below and a blow-up criterion for this soltion were established in [13]. For the compressible Oldroyd type model based on the deformation tensor, see the results [11,20,21,36] and references therein.
The aim of this paper is to study the compressible Oldroyd-B model (1.8) in the critical framework. This approach goes back to the pioneering work by Fujita and Kato [16] for the classical incompressible Navier-Stokes equations. We refer to [3,4,22,38] for a recent panorama. Strictly speaking, the compressible Oldroyd-B model does not have any scaling invariance. However, if we neglect the coupling term divτ and the damping term τ, it is found that (1.4) is invariant under the transformation (ρ 0 , u 0 , τ 0 ) → (ρ 0 (ℓx), ℓu 0 (ℓx), τ 0 (ℓx)), (ρ(t, x), u(t, x), τ(t, x), p(t, x)) → (ρ(ℓ 2 t, ℓx), ℓu(ℓ 2 t, ℓx), τ(ℓ 2 t, ℓx), ℓ 2 p(ℓ 2 t, ℓx)), for any ℓ > 0. This motivates us to consider system (1.8) with initial data (a 0 , u 0 , τ 0 ) ∈Ḃ . Different from our previous results in [14], the coupling constant ω we investigate here is not small any more and thus the problem is much more complicated. As a matter of fact, in [14] (a, u) and τ are treated separately. To be more precise, we bound (a, u) by using the estimates obtained by Danchin [10] for the linearized system of barotropic compressible Naviter-Stokes equations, namely where Λ := (−∆) 1 2 . The linear coupling term divτ in the momentum equation is regarded as a source term, and the symmetric tensor of constrains τ is bounded with the aid of the well known estimates for transport equation. This is an easy way to get the global estimates of (a, u, τ) since the coupling between a and τ is neglected, nevertheless, in order to close the estimates, the price we have to pay is to impose some smallness restriction on the coupling constant ω. For general ω ∈ (0, 1), we must consider fully the coupling between a, u and τ, and deal with (a, u, τ) as a whole.
Let us now explain the main ingredients of the proof. Motivated by our previous result for incompressible Oldroyd-B model [40], we first consider the system of (a, u, divτ). Indeed, in view of the scaling above, u possesses the same regularity with divτ instead of τ, that is why it is more convenient to treat (a, u, divτ) as a whole in the process of energy estimates in Besov spaces. To do so, our proof relies heavily on the following decomposition on u and divτ: u = Pu + P ⊥ u, and divτ = Pdivτ + P ⊥ divτ, where P := Id + ∇(−∆) −1 div is the Leray operator, and P ⊥ := −∇(−∆) −1 div. Applying P ⊥ and P ⊥ div to the second and third equation of (1.8) respectively, we obtain and (1.11) Obviously, the linear part of (1.11) is the same with that of the auxiliary system of (u, Pdivτ) for the incompressible Oldroyd-B model (see [40] (1.12)), so the key point of this paper is to deal with the so called compressible part, i. e., system (1.10). For the same reason as the case of barotropic Navier-Stokes equations [10], we have to study the high frequency and low frequency part of system (1.10) in different ways. Roughly speaking, it is necessary to bound for some q 0 ∈ Z. In order to get the decay of a and divτ, we make full use of the linear coupling terms ∇a and divτ, and the estimates are very delicate both in low and high frequency cases. Furthermore, different from the barotropic Navier-Stokes equations, it is shown that there is a gap between the high frequency and low frequency estimates of k q . In other words, we can estimate (1.12) for q > q 0 and q ≤ q 1 with q 1 < q 0 . To overcome this difficulty, we introduce a new quantitỹ According to Bernstein's inequality,k q is equivalent to k q if q 1 < q ≤ q 0 . Therefor, it suffices to bound (1.12) with k q replaced byk q for q 1 < q ≤ q 0 . This is the main novel part of this paper. Once the estimates of the compressible part (a, P ⊥ u, P ⊥ divτ) of (a, u, divτ) is obtained, we can bound the incompressible part (Pu, Pdivτ) in a similar and easier way. Putting them together, we get the estimates of (a, u, divτ). On this basis, we bound τ directly via the third equation of (1.8), and hence obtain the global estimates of (a, u, τ) in the end. More details can be found in Section 3.
We shall obtain the existence and uniqueness of a solution (a, u, τ) to (1.8) in the following space. For T > 0 and s ∈ R, let us denote We use the notation E s if T = ∞, changing [0, T ] into [0, ∞) in the definition above.
Our main result reads as follows: Remark 1.1. In a forthcoming paper, we will deal with the general case p 2. Oldroyd-B model, using the estimates of the incompressible part in Section 3, we can give a new proof of the result in [40] for p = 2 without resorting to the Green matrix of the corresponding linearized system.

Remark 1.2. For incompressible
The rest part of this paper is organized as follows. In Section 2, we introduce the tools ( the Littlewood-Paley decomposition and paradifferertial calculus) and give some nonlinear estimates in Besov space. Section 3 is devoted to the global estimates of the paralinearized system (3.1) of (1.8). The proof of Theorem 1.1 is given in Section 4.

Notation.
(1) For a, b ∈ L 2 , (a|b) denotes the L 2 inner product of a and b.

The Functional Tool Box
The results of the present paper rely on the use of a dyadic partition of unity with respect to the Fourier variables, the so-called the Littlewood-Paley decomposition. Let us briefly explain how it may be built in the case x ∈ R d which the readers may see more details in [1,5]. Let (χ, ϕ) be a couple of C ∞ functions satisfying and Set ϕ q (ξ) = ϕ(2 −q ξ), h q = F −1 (ϕ q ), andh = F −1 (χ). The dyadic blocks and the low-frequency cutoff operators are defined for all q ∈ Z bẏ holds for tempered distributions modulo polynomials. As working modulo polynomials is not appropriate for nonlinear problems, we shall restrict our attention to the set S ′ h of tempered distributions u such that lim Note that (2.1) holds true whenever u is in S ′ h and that one may writė Besides, we would like to mention that the Littlewood-Paley decomposition has a nice property of quasi-orthogonality: One can now give the definition of homogeneous Besov spaces.
Since homogeneous Besov spaces fail to have nice inclusion properties, it is wise to define hybrid Besov spaces where the growth conditions satisfied by the dyadic blocks are different for low and high frequencies. In fact, hybrid Besov spaces played a crucial role for proving global well-posedness of compressible barotropic Navier-Stokes equations in critical spaces [10]. Let us now define the hybrid Besov spaces that we need. Here our notations are somehow different from those in [10].
We then define the spaceḂ s,t then it is easy to verify that u B s,t Notation. We will use the following notation: Obviously, The following lemma describes the way derivatives act on spectrally localized functions.
Next we recall a few nonlinear estimates in Besov spaces which may be obtained by means of paradifferential calculus. Firstly introduced by J. M. Bony in [2], the paraproduct between f and g is defined byṪ f g = q∈ZṠ q−1 f∆ q g, and the remainder is given byṘ We have the following so-called Bony's decomposition: The paraproductṪ and the remainderṘ operators satisfy the following continuous properties.

Proposition 2.2.
For all s ∈ R, σ > 0, and 1 ≤ p, p 1 , p 2 , r, r 1 , r 2 ≤ ∞, the paraproductṪ is a bilinear, continuous operator from L ∞ ×Ḃ s p,r toḂ s p,r and fromḂ −σ ∞,r 1 ×Ḃ s p,r 2 toḂ s−σ p,r with 1 r = min{1, 1 In view of (2.3), Proposition 2.2 and Bernstein's inequalities, one easily deduces the following product estimates: In the following, we shall give a commutator estimate, which will be used to deal with the convection terms.
The study of non-stationary PDEs requires spaces of the type L ρ T (X) = L ρ (0, T ; X) for appropriate Banach spaces X. In our case, we expect X to be a Besov space, so that it is natural to localize the equations through Littlewood-Paley decomposition. We then get estimates for each dyadic block and perform integration in time. But, in doing so, we obtain the bounds in spaces which are not of the type L ρ (0, T ;Ḃ s p,r ). That naturally leads to the following definition introduced by Chemin and Lerner in [6].

Linearized system
We begin this section by giving the paralinearized version of system (1.8) where v, F, G and L are some known functions. The purpose of this section is to establish the following property of system (3.1).
Proof. Before proceeding any further, let us localize the system (3.1). Similar to (1.10) and (1.11), we find that (a q , P ⊥ u q , P ⊥ divτ q ) and (Pu q , Pdivτ q ) satisfy respectively and where v q :=Ṡ q−1 v,

The Compressible Part
Since the parabolic-hyperbolic system (3.3) behaves differently in high and low frequency, we have to deal with the high and low frequency part of (3.3) in different ways. To simplify the presentation, in the following we give all the estimates which are needed both for high and low frequency cases.
To begin with, taking the L 2 inner product of (3.3) 1 with a q and ∆a q respectively, integrating by parts, we obtain Next, taking the L 2 inner product of (3.3) 2 with P ⊥ u q and of (3.3) 3 with P ⊥ divτ q yields In order to deal with the low frequency part, we also need to give the L 2 estimate of Λ −1 P ⊥ divτ q . To do so, applying first the operator Λ −1 to third equation of (3.3), and then taking the L 2 inner product of the resulting equation with Λ −1 P ⊥ divτ q , we are let to Owing to the fact that the linear operator associated with (3.3) can not be diagonalized in a basis independent of ξ, coercive estimates can not be achieved by means of a linear combination of (3.5)-(3.9). We must make full use the linear coupling terms in (3.3). Accordingly, the following three equalities of cross terms are given. More precisely, first of all, applying ∇ to the first equation of (3.3), taking the L 2 inner product of the resulting equation with P ⊥ u q . Secondly, taking the L 2 inner product of the second equation of (3.3) with ∇a q . Summing up these two results, we find that d dt Similarly, we get the equalities involving the quantity (P ⊥ u q |P ⊥ divτ q ) and (P ⊥ u q |∆P ⊥ divτ q ), d dt Now let us define , for q > q 0 , where q 0 , q 1 , M, M ′ , α and β will be determined later. Next, we will estimate Y q for all q ∈ Z step by step.
Multiplying (3.6) and (3.10) by 2 1−ω Re 2 and 1−ω Re respectively, summing up the resulting equations yields Multiplying (3.7) and (3.13) by M and ReWe respectively, adding them to (3.14), we obtain By using Cauchy-Schwarz inequality, we have It follows that and hence The rest terms on the left hand side of (3.15) can be bounded as follows. 2Re(ReWe + 2) (1 − ω) 3 We , then it is easy to verify that (3.22) Next, we estimate the right hand side of (3.15). Indeed, using Cauchy-Schwarz and Bernstein's inequality, we have In view of commutator estimates, cf. [1], we infer that Now from (3.16), (3.20)-(3.24), we find that there exist two constants c 1 and C depending on d, Re, We and ω, such that if q > q 0 , then (3.15) implies that Step 2: low frequencies.
Part (i). q ≤ q 1 . Multiplying (3.10) and (3.11) by 1 Re and We Re respectively, and then adding them to (3.5) and (3.7), we get and We Re respectively, adding them to the above inequality yields

12). (3.27)
Now we estimate the cross terms on the left hand side of (3.27).
From (3.28)-(3.32), it is easy to see that Using Cauchy-Schwarz inequality and Lemma 2.1 over and over again, the rest cross terms on the left hand side of (3.27) can be bounded in a similar way. In fact, Moreover, (3.40) 2ω Re It follows from (3.34)-(3.40) that , and Next, we estimate the right hand side of (3.27). To this end, notice first that integrating by parts yields Substituting this equality to (3.27), using the fact q ≤ q 1 , it is not difficult to verify that Similar to (3.24), we have Part (ii). q 1 < q ≤ q 0 . Multiplying (3.9) by We 2ωRe , adding the resulting equation to (3.5) and (3.7), we arrive at 1 2 We Re respectively, adding them to (3.51), we are led to 1 2 Now we estimate the cross terms in the left hand side of (3.52) one by one. Indeed, all of them can be bounded by using Cauchy-Schwarz inequality and Lemma 2.1. More precisely, Consequently, if (3.54) and (3.56) hold, we have Collecting Then (3.57) and the following inequality hold, Now we are left to bound the right hand side of (3.52). To do so, noting first that then Lemma 2.2 implies As a result, by virtue of Cauchy-Schwarz inequality and the fact q ≤ q 0 , we arrive at Using the commutator estimates in [1] once more, one easily deduces that From (3.57), (3.70) and (3.72)-(3.73), we conclude that there exist two constants c 3 and C depending on d, Re, We and ω, such that if q 1 < q ≤ q 0 and α, β satisfy (3.68) and (3.69), then (3.52) implies

The Incompressible Part
We begin this part by giving the following five equalities in which the L 2 estimates and cross terms of the corresponding incompressible part (Pu q , Pdivτ q ) of (u q , divτ q ) are involved. Since they are obtained in the same way with those in the compressible part, we give a list of the results directly.
where q 0 , q 1 , M ′ , β are given as in the compressible part. Next, we shall boundỸ q for all q ∈ Z.
It is easy to see that We .
Part (i). q ≤ q 1 . Similar to (3.27), a linear combination of (3.75), (3.76), (3.78) and (3.79) yields where M ′ is the same as in (3.27). The cross terms in (3.88) and the right hand side can be estimated in a similar manner as those in (3.27), accordingly, we infer from (3.88) that there exist constantsc 2 and C depending on d, Re, We, and ω, such that if q ≤ q 1 , there holds Part (ii). q 1 < q ≤ q 0 . Similar to (3.52), by a linear combination of (3.75), (3.77) and (3.78), we get where β is the same as in (3.52). Arguing as in the corresponding compressible case, we find that there exist constantsc 3 and C depending on d, Re, We, and ω, such that if q 1 < q ≤ q 0 , (3.90) implies Global estimates of (a, u, divτ) Let X q := Y q +Ỹ q . Moreover, we set Recalling the definition of Y q andỸ q , using Bernstein's inequalities, we infer from (3.16), (3.33), (3.57) and the corresponding estimates for incompressible part that (3.92) X q ≈ s(q) a q L 2 + u q L 2 + divτ q L 2 . (3.93) Performing a time integration in (3.93), multiplying the resulting inequality by 2 q(s−1) , and taking sum w. r. t. q over Z yields Using the fact (3.92), this inequality is nothing but Gronwall's inequality implies then that The smoothing effect of u Applying∆ q to the equation of (3.1) 2 yields Taking the inner product of the above equation with u q , integrating by parts, we have By virtue of Bernstein's inequality and the commutator estimates, we easily get with positive constant κ depending on d. Multiplying this inequality by 2 q(s−1) and sum over q > q 0 , we arrive at Consequently, using Gronwall's inequality, thanks to (3.94), we are led to The damping effect of τ Applying∆ q to the equation of (3.1) 3 yields We D(u q ) = L q + (v q · ∇τ q −∆ qṪv · ∇τ).
Similar to the estimates of u h , one easily deduces that Multiplying the above equation by 2 qs , and summing over q ∈ Z, we find that τ(t) Ḃ s Recalling that H k := (divL) k − 1≤i, j≤dṪ∂ i v j ∂ j τ i,k , by virtue of Lemma 2.1 and Proposition 2.2, it is easy to see that Since the local existence and uniqueness of the solution (a, u, τ) to (1.8) have been proved in [14], it suffices to show T * = ∞, where T * is the maximal existence time of (a, u, τ). To this end, let us now denote V(t) := t 0 ∇u(t ′ ) L ∞ dt ′ , , , and X 0 := a 0 Ḃ Clearly, U(t) is continuous with respect to time t. Applying Proposition 3.1 with s = d 2 , v = u and F = −div(Ṫ ′ a u), and employing the product estimates in Besov spaces, it is not difficult to verity that 2,1 ) , and L L 1 .