Dispersive effects of the incompressible viscoelastic fluids

We consider the Cauchy problem of the \begin{document}$N$ \end{document} -dimensional incompressible viscoelastic fluids with \begin{document}$N≥2$ \end{document} . It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group \begin{document}$e^{± it\Lambda}$ \end{document} . Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations.


1.
Introduction. At first, we introduce the incompressible Navier-Stokes system, namely,    u t + u · ∇u − ∆u + ∇Π = 0, x ∈ R N , t > 0, N ≥ 2, divu = 0, u| t=0 = u 0 , where u(t, x) and Π(t, x) denote the fluid velocity and the pressure, respectively. The existence of at least one global weak solution of (1.1) is well-known since the pioneering work of Leray [31] and Hopf [22], when u 0 ∈ L 2 . To prove the uniqueness of the solution in some time interval [0, T ), we need besides the condition u 0 ∈ L 2 a further regularity property of the initial value u 0 . The first sufficient condition on the initial value for a bounded domain seems to have been described in [25]. Then many results on sufficient conditions on u 0 to guarantee the existence and uniqueness of the solution were proved, see, e.g., [2,5,17,18,24,26] etc. For example, when u 0 ∈ H N 2 −1 , the global weak solution for (1.1) is unique on some local time interval [0, T ]. When u 0 Ḣ N 2 −1 is sufficiently small, the incompressible Navier-Stokes system has a unique global strong solution. Due to the energy conservation law u(t) 2 L 2 + 2 ∇u 2 L 2 ≤ u 0 2 L 2 , one sees that the key of the global well-posedness result is to control the high frequency part of the velocity u. We hope to prove that when the high frequency part of u 0 is small and the low frequency part of u 0 could be very large, then the incompressible Navier-Stokes system (1.1) has a unique global strong solution.
As an attempt, in this paper, we consider such problem to another incompressible system with elastic effect.
More precisely, we consider the so called incompressible viscoelastic fluids: (1.2) Here u, F, Π are the velocity, deformation tensor, and pressure of the fluids, respectively. Moreover, W (F ) is the elastic energy functional, ∂W (F ) ∂F is the Piola-Kirchhoff stress tensor and ∂W (F ) ∂F F is the Cauchy-Green tensor. Throughout this paper, we adopt the following notations: and summation over repeated indices will always be understood.
For the sake of simplicity, we only focus on the case of Hookean elastic materials: W (F ) := 1 2 |F | 2 . However, this simplify does not reduce the essential difficulties for analysis. Indeed, all the results we describe here can be generalized to more general cases. Obviously, the equilibrium (I, 0) solves (1.2). In this paper, we investigate the system (1.2) near (I, 0), and thus impose on (1.2) the following initial conditions: We further assume that E 0 (x) and u 0 (x) satisfy the following constraints (1.4) The first three of (1.4) are nothing but the consequences of the incompressibility condition and the last one can be understood as the consistency condition for changing variables between the Lagrangian and Eulerian coordinates [29]. We would like to remark that (1.4) is preserved for all time for sufficiently smooth solutions to (1.2), c.f. [29]. On this basis, denoting by E := F − I, then the system satisfied by (E, u) takes the form with constraints (1.6)

DISPERSIVE EFFECTS OF THE INCOMPRESSIBLE VISCOELASTIC FLUIDS 5263
Using the variant d ij := −Λ −1 ∂ j u i introduced in [34], where Λ s f = F −1 (|ξ| sf ), the system (1.5)-(1.6) can be reformulated as when the initial data satisfy the constraints (1.4). The theory of viscoelastic fluids recently gained quite some attention. According to the methods used to get the global estimates, the results on viscoelastic fluids fall into two categories. (I) Energy approach. The homogeneous linearized system of (1.5) is the following (1.8) From the standard energy estimate for (1.8), it is not difficult to find that the damping effect for E is partial. More precisely, the damping effect is only available for divE, but not for E itself. So the main difficulty to obtain the global solutions to system (1.5)-(1.6) is the lack of damping mechanism on E. To overcome this problem, in the two dimensional case, Lin, Liu and Zhang [32] write for some φ = (φ 1 , φ 2 ) due to the fact divE = 0, and then consider the new system for (u, φ). For the 3D case, (1.9) is unavailable anymore. Zhang et al [11,33] used the following quantities G := F −1 , U := G − I (1. 10) introduced by Sideris and Thomases [35] to replace the standard deformation tensor F . One very important property of this matrix G is that which is not true for F . Later, Liu, Lei and Zhou [29] proved that although the deformation tensor F is not curl free, curlF is actually of higher order nonlinear term, see (1.6) 3 . Combining the partial damping effect possessed by (1.8) with the fine property of curlG (see (1.11)) or curlF (see (1.6) 3 ), global solutions to (1.2) were established in certain Sobolev space H s , please refer to [11,33,29]. Some other results can be found in [27,28,30]. Besides, for the 2D case, Hu and Lin recently [23] proved the global-in-time existence of the Leray-Hopf type weak solutions to (1.2) in the physical energy space via the DiPerna-Lions theory.
(II) The scaling invariant approach. This approach goes back to the pioneering work by Fujita and Kato [17] for the classical incompressible Navier-Stokes equations. Danchin [12] first applied this approach to the compressible Navier-Stokes equations. In fact, Danchin studied carefully the following hyperbolic-parabolic 5264 DAOYUAN FANG, TING ZHANG AND RUIZHAO ZI system (with convection terms) (1.12) This system arises when linearizing the isentropic compressible Navier-Stokes equations on the density ρ and on the potential part of the velocity u. The smoothing properties for (1.12) were exploited in [12], and the different behaviors of the solution to (1.12) for low and high frequencies were revealed as well. The difficulty to understand (1.5)-(1.6) is similar to that of the isentropic compressible Navier-Stokes equations. As mentioned above, thanks to the new variant d ij := −Λ −1 ∂ j u i and the constraint (1.6) 3 , Qian [34] found that the system (1.5)-(1.6) is equivalent to (1.7), and the linearized system of (1.7) takes exactly the form (1.12). Accordingly, following the arguments in [12], Qian proved the global well-posedness of the system (1.5)-(1.6) if the initial data (E 0 , u 0 ) are small in (1.13) Later, the first two authors [36] extended the result in [34] to the L p setting. They showed that, for the high frequency part (E 0H , u 0H ) of the initial data (please refer to (1.32) for the definition of f L and f H with f ∈ S (R N )), the space in (1.13) can be replaced byḂ As a result, the large highly oscillating initial velocity as in [6] for the classical incompressible Navier-Stokes equations , for example, if N = 3, is also allowed in [36], see [9,10] for related results on the isentropic compressible Navier-Stokes equations. For the case that the anisotropy is taken into consideration, see the result of the second author [37]. Very recently, we [16] constructed a new class of solutions to the isentropic compressible Navier-Stokes equations with the aid of the dispersive properties of (1.12) in the low frequent part. In particular, the low frequency part of the initial velocity u 0L is not necessarily small inḂ N 2 −1 2,1 . Motivated by [16] and [36], in this paper, we study the global existence and uniqueness of the solutions to the incompressible viscoelastic fluids (1.5)-(1.6) with the initial data (E 0 , u 0 ) small in the following space with some α > 0 and p ≥ 2. In the following, we denote (1.17) If α = 0, E 0,0,p is chosen in [36] for the initial data. Furthermore, if α = 0 and p = 2, E 0,0,2 becomes the space in (1.13), chosen in [34] and [12]. Owing to Bernstein's inequality and the low frequency cut-off, it is easy to verify that In what follows, we denote E 0 := E 0,α,p , for simplicity. Due to the extra α-order regularity in the low frequency part, even the second component of the element in E 0 is not scaling invariant under the transformation u 0 (x) → lu 0 (lx), l > 0.
In that case, we will run into difficulties when estimating the nonlinear terms. However, the situation is different if we take the dispersive property of the system (1.12) into consideration. To shed some light on this property, we fall back on the eigenvalues associated with the linear system (1.12) in the low frequency case (|ξ| < 2): It is easy to see that there are two main parts − |ξ| 2 2 and ±i|ξ| in λ ± (ξ) if |ξ| 1. The two (semi)groups corresponding to them are e t∆/2 and e ±itΛ , respectively. Previous studies on the hyperbolic-parabolic system (1.12), such as [21] and [12], ignored the dispersive effect of the one-parameter group e ±itΛ . On the contrary, in this paper, e ±itΛ plays an important role. Indeed, the proof of the global well-posedness has two main components: • Energy estimates to control the high frequency part of (E, u). The main difficulty in this part is to find the damping effect of E H . • Energy estimates and Strichartz estimates to control the low frequency part of (E, u). The interplay between the dissipation and dispersive effect is present at this stage.
Different from our previous results in [16], we deal with the high frequency part of (1.12) in the criticle L p framework. The energy estimates for the high frequency part in [16] does not work here. We solve this problem by means of the so called effective velocity used by Haspot in [19], without resorting to the Green matrix of (1.12) used in [36].
We shall obtain the existence and uniqueness of a solution (E, u) to (1.5)-(1.6) in the following spaces.
p,1 ). We shall endow the space with the norm: .
In particular, if α = 0, E N p α (T ) coincides with the space used in [36]. We use the notation E p,1 ). At first, we can obtain the following local well-posedness results.
, then the solution (E, u) is unique. Then, we can obtain the main result as follows. (1.20) There exist positive constants c 0 and M 0 depending on N and p , such that for all (E 0 , u 0 ) ∈ E 0 satisfying the constraints (1.4) and Then we have φ l ⊂ B(0, l), and φ l Ḃ (1.23) Next, set where Q ∈ N will be determined below. Then it follows thaṫ ∆ q φ l = 0, for all q ≥ −Q. (1.25) Choosing Q ∈ N so large that and thus the data given in (1.28) apply to Theorem 1.2, even though Remark 1.5. In this paper, we take the Lebesgue index p = ∞ in the Strichartz estimates of the system (1.12) to minimize the restrictions on α. The same can not be achieved in our previous result [16] for the isentropic compressible Navier-Stokes equations due to the interactions between the potential part of the velocity (with dispersive property) and the divergence free part of the velocity (without dispersive property).
Remark 1.6. Compared with [16], the uniform estimates for the system (1.12) in the high frequency part is extended to the L p critical framework. This explains the restriction (1.19) for p and α. Clearly, if p = 2, the restriction (1.19) will disappear. Moreover, the estimates obtained in Proposition 4.1 can be used to the isentropic compressible Navier-Stokes equations.
2,1 , from Theorem 1.2, we immediately get the following corollary. The proof of which is similar to that of Theorem 1.2 in [16], we omit the details here. with (u 0 , E 0 ) satisfying the constrains (1.4). Then there exist positive constants c 1 , such that for Q ∈ N, if 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 spaces. The estimates for the system of acoustics, transport equation and heat equations are given in Section 3. Section 4 is devoted to a new property of the hyperbolic-parabolic system (1.12). In Section 5, we obtain the global a priori estimates of system (1.7), or equivalently, of system (1.5)-(1.6). In Sections 6-7, we prove the local well-posedness result Theorem 1.1 and Corollary 1.1. The proof of Theorem 1.2 is given in Section 8.

Notation.
1. For f ∈ S (R N ), Q ∈ Z, denote f q :=∆ q f , and (1.31) In particular, Throughout the paper, C denotes various "harmless" positive constants, which may change line by line.

2.
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 N , and the readers may see more details in [3,7]. Let (χ, ϕ) be a couple of C ∞ functions satisfying and 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 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.
We then define the spacesḂ s p,r : Remark 2.1. The inhomogeneous Besov spaces can be defined in a similar way. Indeed, for u ∈ S (R d ), we set Then for all u ∈ S (R d ), we have the inhomogeneous Littlewood-Paley decomposition u = q∈Z ∆ q u, and for (p, r) ∈ [1, +∞] 2 , s ∈ R, we define the inhomogeneous Besov space B s p,r as Homogeneous Besov spaces fail to have nice inclusion properties, however, if there is a cut-off in the frequency space, the situation is different. The following low and high frequency embeddings will be used frequently in this paper. The proof of which follows from the definition of Besov norm immediately. and 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 [4], the paraproduct between f and g is defined bẏ 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.
). 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 froṁ In view of (2.5), Proposition 2.1 and Bernstein's inequality, one easily deduces the following product estimates: (2.6) The following Proposition will be used to prove the uniqueness of solutions obtained in Theorem 1.1.
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 [8].
3. Preliminary. We begin this section by recalling the Strichartz estimate for the acoustic system, which is of great importance in this paper.
be a solution of the following system of acoustics: Then, for any s ∈ R and T ∈ (0, ∞], the following estimate holds: , and 1 r + 1 r = 1. Next, we give the classical estimates in Besov space for the transport and heat equations. Please refer to, for example, [3] and [7] for the proofs.
we have the following a priori estimate ), and f solves

4.
A new property of the linear system (1.12). In this section, we focus on the following linear hyperbolic-parabolic system It is worth pointing out that the convection term (u · ∇)a can not be incorporated into f since there is no smoothing effect in the high frequency part of a. The purpose of this section is to establish the following Proposition: .

(4.2)
Proof. The proof will be divided into three steps.
Step (I). Energy estimates in the high frequency part. Following the ideas of Haspot [19,20], we introduce the so called "effective velocity": Then it is easy to verify that Applying the operator P ≥Q , Q ∈ N to be determined below, to (4.3) , and then using Proposition 3.3, we are led to .
The convection term (u · ∇)a can be estimated as follows: Using the high frequency embedding (2.4), Bernstein's inequality and Proposition 2.1, we obtain and . (4.8) Substituting (4.5)-(4.8) into (4.4), and taking Q sufficiently large, we arrive at where the term 2 −Q P ≥Q w appearing in (4.5) has been adsorbed by the left hand side. To close the estimate of w, we turn to bound the high frequency part of a. In fact, by the definition of the effective velocity w, the equation of a in (4.1) can be rewritten as Applying the dyadic blocks∆ q , q ≥ Q to the above equation yields with R q :=Ṡ q−1 u · ∇a q −∆ q Ṫ u · ∇a . Arguing as in Theorem 3.14 in [3], we obtain for all t ∈ [0, T ], Next, following the computations in [16], we have Multiplying (4.11) by 2 q N p , summing up with respect to q over {Q, Q+1, Q+2, · · · }, and using (4.12)-(4.13), we find that . (4.14) Clearly, and T 0 q≥Q . (4.16) can be bounded in the same way as I 2 . Then, substituting (4.8), (4.15)-(4.16) into (4.14), one deduces that . (4.17) Combining (4.9) with (4.17), we find that there is a q 0 ∈ N, such that if Q ≥ q 0 , then . (4.18) From now on, we take Q = q 0 , and exploit the notation in (1.32). Thanks to the decomposition u = u L + u H and the interpolation, we have .

(4.19)
Similarly, using the low frequency embedding (2.3), we find that . (4.20) and . (4.22) Noticing that d = w + Λ −1 a, using the high frequency embedding again, we infer from (4.18)-(4.22) that . (4.23) Step (II). Energy estimates in the low frequency part. According to the energy estimates for the hyperbolic-parabolic system (4.1) obtained by Danchin [12], we get In view of Bony's decomposition, there holds (4.25) Using the low frequency embedding (2.3), Bernstein's inequality and Proposition 2.1, we find that and .

(4.27)
Owing to the decomposition a = a L + a H and the high frequency embedding, we have a .

(5.2)
Proof. The condition divE = 0 implies that ∂ k u i E kj = ∂ k (u i E kj ). Then by the definition of paraproduct, Bernstein's inequalities and Proposition 2.1, we have .

(5.3)
Moreover, using the low frequency embedding (2.3), we are led to , we take some precautions. Indeed, according to the low frequency embedding (2.3) and the interpolations, we find that . (5.7) Then using the high frequency embedding (2.4) and Corollary 2.1, we arrive at .

(5.8)
Collecting the estimates above, the proof of Lemma 5.1 is completed.

(5.14)
This completes the proof of Lemma 5.2.
The proof of Lemma 5.3 is completed.
6. Local well-posedness. In this section, we prove Theorem 1.1. We will proceed in two steps. First we prove the existence of the solution. The second part is devoted to the proof of uniqueness of the solution.
6.1. Existence of a local solution. First of all, we give a local version of Proposition 5.1. .
Then there exists a positive constant C 2 depending on p and N , such that and

DAOYUAN FANG, TING ZHANG AND RUIZHAO ZI
Proof. First of all, from (1.5) 1 , similar to (4.14), we have In view of Proposition 2.1 and Corollary 2.1, one easily deduces that . .
Next, we recall the following local well-posedness theorem. T (H s (R N )) for any T < T * . Moreover, if T * is the maximal existence time of the solution and T * < ∞, it is necessary that (6.14) Step (I). Approximate solutions.
In order to apply Theorem 6.1, we first construct approximate initial data. As a matter of fact, thanks to Lemma 4.2 in [1], we can choose (E n 0 , u n 0 ) ∈ H ∞ (R N ) such that div(E n 0 ) = divu n 0 = 0, for all n ∈ N.
(6.15) Then Theorem 6.1 ensures that system (1.5) with the initial data (E n 0 , u n 0 ) admits a solution (E n , u n ) ∈ L ∞ Tn (H s0 ) for some T n > and sufficiently large s 0 . Moreover, by virtue of Propositions 3.2 and 3.3, it is not difficult to verify that p,1 ). Furthermore, the properties of continuity with respect to time can be proved as those of Theorem 10.2 in [3]. Consequently, we actually have (E n , u n ) ∈ E N p (T n ), here and in what follows we denote by E N p (T n ) the solution space given by (1.18). Next, we will show that there exists a uniform time T 0 > 0, such that (E n , u n ) is uniformly bounded in E N p (T 0 ). (6.16) To this end, let us denote by T * n the maximal existence time of (E n , u n ), and define T 1 n be the supremum of all time T ∈ [0, T * n ) such that A n (T ) ≤ 4C 2 A 0 , and B n (T ) ≤ η, (6.17) with η to be determined below. Then from Proposition 6.1 and (6.15), for all T ∈ [0, T 1 n ], we infer that and

DISPERSIVE EFFECTS OF THE INCOMPRESSIBLE VISCOELASTIC FLUIDS 5285
We take T 0 and η > 0 so small that Then (6.18) and (6.19) reduce to and respectively. Therefore, T * n ≥ T 1 n ≥ T 0 for all n ∈ N, and hence (6.16) holds.
Using the classical compactness method, we will prove that the approximate sequence (E n , u n ) tends to some function (E, u) which satisfies the system (1.5). In fact, for any fixed T ∈ [0, T 0 ], from (6.16), we see that u n L is uniformly bounded in Therefore, u n is uniformly bounded in L 2 T (Ḃ N p p,1 ). (6.22) Moreover, (6.16) also implies that E n is uniformly bounded in L ∞ T (Ḃ N p p,1 ). Then using Corollary 2.1, we obtain , (6.23) where we have used the fact divu n = div(E n ) = 0. It follows from (6.22) and (6.23) that ∂ t E n is uniformly bounded in L 2 T (Ḃ N p −1 p,1 ). Let us denoteĒ n := E n −E n 0 . Then from above estimates we get that E n is uniformly bounded in C T (Ḃ To estimate {∂ tū n } n≥1 , we write the equation forū n as ∂ tū n = ∆ū n + divE n + Pdiv −u n ⊗ u n + E n (E n ) .
Clearly, owing to Bernstein's inequality, there holds ∆ū n is uniformly bounded in L ∞ T (Ḃ . (6.28) Thanks to the decomposition u n = u n L + u n H , there holds div(u n ⊗ u n ) = div(u n L ⊗ u n L ) + div(u n L ⊗ u n H + u n H ⊗ u n L ) + div(u n H ⊗ u n H ). Similar to (6.28), we get div(u n L ⊗ u n L ) , (6.29) and div(u n L ⊗ u n H + u n H ⊗ u n L ) .

6.2.
Uniqueness. For any fixed T ∈ [0, T 0 ], assume that (E i , u i ) ∈ E N p (T ), i = 1, 2, are two solutions of the system (1.5) with the same initial data (E 0 , u 0 ). Set δu = u 1 − u 2 , δE = E 1 − E 2 . Then (δE, δu) satisfies the following system, To simplify the presentation, let us denote By using the low frequency embedding (2.3) and the uniform estimate (6.38), it is easy to verify that, for i = 1, 2, there holds 2N ). Applying Proposition 3.2 to (6.39) 1 , and using Corollary 2.1, (6.40), and (6.38), we have for all t ∈ [0, T ]. Next, applying first the operator P to (6.39) 2 , and then using Proposition 3.3 and Corollary 2.1 again, we are led to for all t ∈ [0, T ]. Similar to (6.22), we easily have Therefore, we can chooseT so small that

DAOYUAN FANG, TING ZHANG AND RUIZHAO ZI
Then for all t ∈ [0,T ], there holds ds. (6.42) Substituting (6.42) into (6.41) yields .ds Gronwall's inequality and (6.40) imply then that does not work for p = 2N , next we will work inḂ N p −1 p,∞ as in [15]. However, our case is more complex than that of [15] due to the fact that the higher regularity of δu L precludes the possibility of using a logarithmic interpolation inequality on δu . We have to deal with δu in a new way. Firstly, similar to (6.41), thanks to Proposition 2.2, we get In view of (6.38), using Gronwall's inequality, we have . (6.44) Next, to bound δu, we estimate δu L and δu H separately with different norms. In fact, thanks to Proposition 3.3, we are led to .