A BKM’S CRITERION OF SMOOTH SOLUTION TO THE INCOMPRESSIBLE VISCOELASTIC FLOW

. In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in R n ( n = 2 , 3). We obtain a Beale-Kato-Majda-type criterion in terms of vorticity in two and three space dimensions, namely, the solution ( u ( t,x ) ,F ( t,x )) does not develop singularity until t = T provided that ∇ × u ∈ L 1 (0 ,T ; ˙ B 0 ∞ , ∞ ( R n )) in the case n = 2 , 3.

1. Introduction. In this paper, our study is concerned with the following incompressible Oldroyd model in R n (n = 2, 3) describing an incompressible non-Newtonian fluid: for any t > 0, x ∈ R n , where u(t, x) denotes the fluid velocity vector field, P = P (t, x) is the scalar pressure, F = F (t, x) ∈ R n × R n the deformation tensor, µ > 0 is the constant kinematic viscosity, while u 0 is the given initial velocity with ∇ · u 0 = 0. The above system (1) is one of the basic macroscopic model for viscoelastic flows (see [5] and references therein). This system corresponds to the so-called Hookean linear elasticity at a microscopic level. For more physical background to this system, see [7,13,23]. Significant progress has been made as regards viscoelastic flow by many physicists and mathematicians in the last few decades. The authors of [21] proved global existence in the two-dimensional case by introducing an auxiliary vector field to replace the transport variable F , while Lei and Zhou [19] proved the same results via the incompressible limit working directly on the deformation tensor F rather than exploiting a subtle damping effect of the deformation tensor. Then, the authors of [16] proved the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus 824 HUA QIU AND SHAOMEI FANG (n = 2, 3) in the case of near equilibrium initial data, while Lin and Zhang [22] established the global well-posedness of the initial-boundary value problem of the viscoelastic fluid system of the Oldroyd model with Dirichlet conditions. Further discussions of these topics can be referred to [3,4,6,8,10,11,14,15,16,17,18,19,21,22,25,26,27,28,29,31,34,35].
Before stating our main result, let us start with the standard description of general mechanical evolution to introduce some notations and definitions. In the context of hydrodynamics, the basic variable is the particle trajectory x(t, X), where X is the original labelling (Lagrangian coordinate) of the particle and referred to as the material coordinate, while x is the current (Eulerian) coordinate and referred to as the reference coordinate. For a given velocity field u(t, x), the flow map x(t, X) is defined by the the ordinary differential equation: ∂x(t,X) ∂t = u(t, x(t, X)), x(0, X) = X. The deformation tensor is then defined byF (t, X) = ∂x(t,X) ∂X . In the Eulerian coordinate, the corresponding deformation tensor F (t, x) is defined as F (t, x(t, X)) = F (t, X). Using the chain rule, one can see that F (t, x) satisfies the following transport equation, i.e. the second equation of (1): If ∇ · F T (0, x) = 0, then we have from the second equations of (1): Therefore, if ∇·F T (0, x) = 0, it will remain so for later times, namely, ∇·F T = 0 for any time t > 0. In what follows, we will make this assumption. Denote F k = F e k as the columns of F , then ∇ · (F F T ) = n k=1 F k · ∇F k by the fact ∇ · F k = 0. Hence the system (1) takes the following form equivalently: with k = 1, · · · , n, and u(0, x) = u 0 , F k (0, x) = F k,0 .
Yet, although much progress has been made on multi-dimensional Oldroyd model as mentioned above, the global existence of smooth solutions to (1) with general large initial data is still an outstanding open problem. More precisely, since the Oldroyd system (1) can be regarded as a combination between Navier-Stokes equations and with the source term ∇ · (F F T ) and the second equation of (1), we expect the local smooth solutions to blow up in a finite time. In three space dimension case, the authors [9] established a Beale-Kato-Majda [1] type criterion for smooth solutions as µ = 0 in (3), which reads that the smooth solution (u(t, x), In the case µ > 0, Yuan [32] obtained the blowup criterion of smooth solution to the Oldroyd model (3) in two and three space dimensions by means of only ∇u L ∞ , namely, the smooth solution (u(t, x), F (t, x)) does not appear breakdown until t = T provided that ∇u ∈ L 1 (0, T ; L ∞ (R n )).
In [30], the author obtained that the smooth solution (u(t, x), F (t, x)) is smooth at [33], Yuan and Li give the similar regularity criteria at the same time.
In this paper, we consider the blowup criterion of smooth solution to the Oldroyd model (3) and establish a Beale-Kato-Majda-type criterion of smooth solutions to the system (3) in the spaceḂ 0 ∞,∞ . This is motivated by Lei and Zhou's work [20], which they established a Beale-Kato-Majda-type criterion to MHD system under the norm BM O in the case of zero viscosity. Now, our main result is stated as follows.
We have the following corollary immediately. (6) is an extension to (4) in some sense, although they showed (4) for the Oldroyd system describing the motion of inviscid fluids. The result (6) improves the result (7) of Theorem 1.2 in [30]. This paper is organized as follows. Section 2 gives the preliminaries. Section 3 details the regularity criterion (6).

2.
Preliminaries. This section is devoted to providing some lemmas and some basic facts on Littlewood-Paley theory, which will be used in the proofs of our main results.
Let h = F −1 φ. The frequency localization operator is defined by We now introduce the homogeneous Besov spaces.
and Z (R n ) can be identified by the quotient space of S /P with the polynomials space P, the set S (R n ) of temperate distributions is the dual set of S for the usual pairing.
Let us recall the well-known Gagliardo-Nirenberg inequality.
there exists a constant C such that The following lemma follows from Majda-Bertozzi [24].

Lemma 2.3. The following inequality holds
for m ≥ 1.
The following lemma is well known (for example, see [2]).

Lemma 2.4 (Bernstein inequality). The following inequalities hold
for any m ∈ N, 1 ≤ p ≤ q ≤ ∞ and f ∈ L p (R n ). Here c and C are positive constants independent of f and k.
At last, let us recall the following logarithmic Sobolev inequality which is proved by Kozono-Ogawa-Taniuchi [12]. For completeness, the proof will be also sketched here.
Lemma 2.5. There exists a uniform positive constant C such that holds for all vectors f ∈ H 3 (R n )(n = 2, 3) with ∇ · f = 0.

Proof. First of all, it follows from the Littlewood-Paley decomposition and Lemma 2.4 that
Consequently, it follows from the above estimates and Calderon-Zygmand theory that Lemma 2.5 holds. We have proved Lemma 2.5.
3. Proof of Theorem 1.1. In this section we give the proof of Theorem 1.1. For ease of notation, k is denoted as n k=1 (n = 2, 3). Multiplying (3) 1 by u and (3) 2 by F k respectively, and integrating on R n , then we have where the fact Summing over k for (9) and combining with (8), we have which follows that Then u L ∞ (0,T ;L 2 ) + u L 2 (0,T ; and Now, applying ∇ to (3), multiplying the resulting equations by (∇u, ∇F k ), and integrating on R n , then it follows that by summing them up 1 2 where the following fact is used in the second equality of (12) by ∇ · u = ∇ · F k = 0. From the Gronwall inequality, we get In terms of (6), we know that for any small constant > 0, there exists T * < T , such that Let From (13), (14), (15) and Lemma 2.5, we get Here C 0 is an absolute positive constant, C 1 depends on T * ≤ t < T and ∇u(T * ) 2 Now, applying ∇ m to the first equation of (3), taking L 2 -inner product of the resulting equation with ∇ m u, and integrating over R n , we have Likewise, By using ∇ · u = ∇ · F k = 0, integrating by parts the above two equalities infers that For simplicity, we shall set m = 3 in what follows. First of all, using the Hölder inequality and Lemma 2.3, we have Then by (15), we have For the term I 3 , by integrating by parts and the Hölder inequality, one has for any δ > 0 in 2D case, and in 3D case: for any δ > 0. Next, using Lemma 2.2, the Young inequality and (16), it follows that 13 10 for any δ > 0 in 2D case, and in 3D case: 19 12 for any δ > 0. Hence, if we take and for any δ > 0. Noting that the first term in right hand of (19) k 7 ∇u(t) L ∞ ∇ 3 F k (t) 2 L 2 ≤ C ∇u(t) L ∞ (e + Φ(t)) .