Note On the blowup criterion of smooth solution to the incompressible viscoelastic flow

We study the blowup criterion of smooth solution to the Oldroyd models. Let $(u(t,x), F(t,x)$ be a smooth solution in $[0,T)$, it is shown that the solution $(u(t,x), F(t,x)$ does not appear breakdown until $t=T$ provided $\nabla u(t,x)\in L^1([0,T]; L^\infty(\mr^n))$, $n=2,3$.


Introduction
In this paper, we consider the blowup criterion of smooth solution to the incompressible Oldroy model in the two and three dimensional space: for any t > 0, x ∈ R n , n = 2, 3, where u(t, x) is the velocity field, p is the pressure, µ is the viscosity and F the deformation tensor. We denote (∇ · F ) i = ∂ xj F ij for a matrix F . The Oldroy model (1.1) describes an incompressible non-Newtonian fluid, which bears the elastic property. For the details on this model see [7].
The local existence and uniqueness of the Oldroy model on entire space R n or a periodic domain was established by Lin etc. in [7], where the global existence and uniqueness of smooth solution with small initial data was also established see also [5]. The wellposedness on a bounded smooth domain with Dirichlet conditions was established by Lin and Zhang in [8].
We remark some properties of the deformation tensor. Let x be the Euler coordinate and X the Lagrangian coordinate. For a given velocity field u(t, x) the flow map x(t, X) is defined by the following ordinary differential equation d dt x(t, X) = u(t, x(t, X)), x(0, X) = X.
The deformation tensor isF (t, X) = ∂x ∂X (t, X). In the Eulerian coordinate, the corresponding deformation tensor is define as F (t, x(t, X)) =F (t, X). Differentiating its both sides with respect to t by chain rule one obtain the second equation of (1.1), which says that ∂ t F ij + u k · ∂ x k F ij = ∂ x k u i F kj for i, j = 1, 2, · · · , n, in the (i, j)−th entries, where we use the Einstein summation convention that the repetition index denotes sum over 1 to n.
Denote the ith column of F as F ·i , then ∇ · (F F t ) = F ·i · ∇F ·i by the fact ∇ · F t = 0. So the system (1.1) can be rewritten in an equivalent form In reference [7], Lin, Liu and Zhang obtained the local existence and uniqueness of smooth solution for smooth initial data, and had a blowup criterion.
Theorem (Lin, Liu and Zhang) For smooth initial data (u 0 , F 0 ) ∈ H 2 (R n ), there exists a positive time T = T ( u 0 H 2 , F 0 H 2 ) such that the system (1.1) possesses a unique smooth solution on [0, T ] with Moreover, if T * is the maximal time of existence, then In reference [3], Hu and Hynd study the blowup criterion for the ideal viscoelastic flow, which is the Oldroy system (1.1) in the case of µ = 0. They showed an Beale-Kato-Majda [1] type blowup criterion that the smooth solution to the Oldroy flow do not develop singularity for t ≤ T provided that From the modeling of Oldroy system we know that the deformation tensor can be determined by the velocity u of the flow. Therefore we consider the blowup criterion of smooth solution by means of only ∇u ∞ . In fact, Zhao, Guo and Huang [12] constructed a set of finite time blowup solution in two dimension case: If α+β α−β f 0 > 0, α+β = 0 and α−β = 0, then the above solution will blow up at time T * = α−β (α+β)f0 . We see that There are other types of blowup criteria of smooth solutions to the Oldroy models, for example [6,2]. To this end, we state our main results.
is a smooth solution to the Oldroy system (1.3). Then the smooth solution do not appear breakdown until T * > T provided that In the second section we will prove the Theorem 1.1 for the case n = 2, which can be done by energy estimates. The L 2 and H 1 energy estimates are the same for the case n = 2 and n = 3. In the H 2 energy estimate, we use the Sobolev interpolation inequality ∇F 2 4 ≤ C ∇F 2 ∆F 2 . In case n = 3, however, the inequality is ∇F 2 2 which does not match the H 2 energy estimate, because it will result in the appearance of the term ∆F 3 2 that the power is higher that the left hand side. We obtain the H 2 energy estimate of u by virtue of the momentum equation, combining the H 2 estimate of u and F again with the estimate of ∇F L 6 we grasp the H 2 energy estimate of u and F finally. The section three will devote to the proof of the case n = 3.
In this paper C denote a harmless constant which may be dependent on dimension n, the norm of initial data, the viscosity µ, but not dependent on the estimated quantity. We denote the L p norm of a function f by f p or f L p . We denote the derivative with respect to x i by ∂ i or ∂ xi . We also use f t to denote the derivative of f with respect to t.

2.
Proof of the case n = 2 (1) L 2 -energy estimate and L p estimate of the deformation tensor F The L 2 -energy estimate can be easily obtained by the standard L 2 inner product process.
Multiplying both sides of the second equation of (1.3) by p|F ·k | p−2 F ·k for 2 ≤ p < ∞ and integrating both sides on R n it follows that Summing up the estimate (2.2) with respect to k one has Let p → ∞, we have (2)Ḣ 1 -energy estimate We differentiate the equations (1.3) with respect to x i , then multiply the resulting equations by ∂ i u and ∂ i F ·j for i = 1, 2, integrate with respect to x and sum them up. It follows that where use has been made of the facts (3)Ḣ 2 -energy estimate Applying operator ∆ on both sides of (1.3), we have Taking the L 2 inner of equation (2.6) with ∆u and ∆F ·k and summing them up, one can obtain that 1 2 Here use has been made of the the facts that (u · ∇∆u, ∆u) = 0, (u · ∇∆F ·k , ∆F ·k ) = 0, Noting that |(∆u · ∇u, ∆u)|, where we have used the Sobolev interpolation inequality ∇F 2 4 ≤ C ∇F 2 ∆F 2 . Arguing similarly as the above, one has Ct ∇u ∞ ds .
Next we derive the higher derivative estimate of u and F . For this purpose we need the following commutator estimate. Proposition 2.1. (Kato and Ponce [4], [9]) Let 1 < p < ∞ and 0 < s. Assume that f, g ∈ W s,p , then there exists a abstract constant C such that Applying Λ s on both sides of (1.3) and taking the inner product with Λ s u and Λ s F , it can be derived that where we have used the facts The commutator estimate (2.10) implies that where the Sobolev embedding H s−1 (R n ) ֒→ L ∞ (R n ) for s > 1 + n 2 is applied. Inserting the above estimates into estimate (2.11), it follows where we have used the fact So, for s ≥ 3, applying Gronwall's inequality to (2.12), by induction for u's estimate, we obtain the higher derivative estimate: Therefore, we complete the proof of the case n = 2.
3. Proof of the case n = 3 In the three dimensional case the L 2 and H 1 energy estimates are the same as the case of dimension two. To estimate the H 2 energy estimate we need the following estimates.
Multiplying the first equation of (1.3) by u t and integrating both sides over R 3 with respect to x, and noting div u = 0, it follows Integrating both sides with respect to t it yields where the Sobolev embedding H 2 (R 3 ) ֒→ L ∞ (R 3 ) has been used. Differentiating the first equation of (1.3) with respect to t, we arrive at Taking L 2 inner product of the equation (3.2) with respect to u t , it can be similarly derived that . Applying the Gronwall' inequality, it yields It need still to estimate F t 2 2 . From the second equation of (1.3) it can be derived that Inserting it to the estimate (3.3) we obtain the estimate of u t 2 : where C(t) is explicit increasing function of t dependent on t 0 ∇u ∞ ds. From the first equation of (1.3), ∇p can be solved by Riesz transformation being the jth Riesz transformation.
In virtue of the boundedness of Riesz operator R in L p space for 1 < p < ∞, we obtain that For details about Riesz transformation see [10,11].
Thus from the first equation of (1.3) we have µ ∆u 2 ≤ u t 2 + u · ∇u 2 + ∇p 2 + F ·k · ∇F ·k 2 where the interpolation inequality u ∞ ≤ C u 1 4 2 ∆u 3 4 2 has been used. So we derive (3.5) ∆u 2 ≤ C( u t 2 + u 2 ∇u 4 2 + F ∞ ∇F 2 ). Next we derive the estimate of ∆F 2 . Applying ∆ on the both sides of equation (1.3) and taking the L 2 inner product with ∆u and ∆F ·k respectively, we have where use has been of the facts (u · ∇∆u, ∆u) = (u · ∇∆F ·k , ∆F ·k ) = 0, (F ·k · ∇∆F ·k , ∆u) + (F ·k · ∇∆u, ∆F ·k ) = 0. Next we estimate the right hand sides. By the communicator estimate (2.10) one has For the second term on the right hand side of (3.6) we estimate as follows Summing up (3.6) and (3.7), and inserting the above estimates into the summation, we arrive at We still have to estimate ∇F 6 . Differentiating the second equation of (1.3) with respect to x i , one has ∂ t ∂ i F ·k + ∂ i u · ∇F ·k + u · ∇∂ i F ·k = ∂ i F ·k · ∇u + F ·k · ∇∂ i u. Multiplying both sides of the above equation by 6|∂ i F ·k | 4 ∂ i F ·k , and integrating both sides with respect to x over R 3 , it can be derived that (3.9) d dt ∇F 4 6 ≤ C ∇u ∞ ∇F 4 6 + C F ∞ ∆u 6 ∇F 3 6 .