Local well-posedness and blow-up criteria of magneto-viscoelastic flows

In this paper, we investigate a hydrodynamic system that models the dynamics of incompressible magneto-viscoelastic flows. First, we prove the local well-posedness of the initial boundary value problem in the periodic domain. Then we establish a blow-up criterion in terms of the temporal integral of the maximum norm of the velocity gradient. Finally, an analog of the Beale-Kato-Majda criterion is derived.


1.
Introduction. In the past years, the study of magnetoelastic materials has attracted attention of scientists not only from the point of view of mathematical modeling but also for applications. A general magneto-viscoelastic model describing magnetoelastic materials was derived in Forster [8] based on an energetic variational approach (see, e.g., [13]). The resulting system of partial differential equations consists of the incompressible Navier-Stokes equations coupled with balance equations for the deformation gradient and the magnetization field. In the general magnetoviscoelastic model, the dissipative dynamical behavior of the magnetization M satisfies a Landau-Lifshitz-Gilbert (LLG) system with convection, which is mathematically involved due to its highly nonlinear structure (see [2,3,8] and the references therein). In this paper, we consider the following simplified magneto-viscoelastic system: in Ω × (0, T ), where T > 0 and Ω = T d for d = 2 or 3. In the system (1) − (4), v(x, t) : Ω × (0, T ) → R d is the velocity field, p(x, t) : Ω × (0, T ) → R is the pressure, F : Ω × (0, T ) → R d×d denotes the deformation gradient and M : Ω × (0, T ) → R 3 describes the magnetization vector. The magneto-viscoelastic fluid is sometimes exposed to an external effective magnetic field H ext (x, t) : Ω × (0, T ) → R 3 . The fluid viscosity µ is assumed to be a positive constant and α > 0 stands for the parameter that controls the strength of penalization on the deviation of |M | from 1.
The magneto-viscoelastic system (1) − (4) describes the motion of an incompressible fluid that responds mechanically to applied magnetic field and changes its magnetic property in response to mechanical stress. It can be viewed as a simplification of the general model derived in [8] such that the magnetization M now satisfies a gradient flow type equation with convection (see [27, Appendix A] for a sketch of the derivation for system (1) − (4)). In [8], the author proved the existence of global weak solutions to the magneto-viscoelastic system (1) − (4) with an additional regularizing term κ∆F (κ > 0) in the deformation equation (3). Later, Schlömerkemper andŽabensky [27] investigated the uniqueness of global weak solutions for the same problem with slightly different boundary conditions. It is worth mentioning that the artificial regularizing term κ∆F plays an essential role in their analysis. The deformation tensor F naturally satisfies the transport equation (3) that follows from the chain rule (see [20]). Introducing the term κ∆F to (3) is only to avoid the mathematical difficulty from the hyperbolic nature of (3). We note that if M ≡ 0, the system (1) − (4) reduces to a well-known model for incompressible viscoelastic flows [20]. Even for the incompressible viscoelastic fluids without magnetic effect, existence of global weak solutions with large initial data is still an outstanding open problem. Recently, Hu and Lin [11] proved the existence of a global-in-time weak solution to the Cauchy problem of the incompressible viscoelastic system in R 2 , provided that the initial deformation tensor is close to the identity matrix and the initial velocity is small. We also refer to Hu and Wu [12] for a weak-strong uniqueness result for the same system.
In this paper, we aim to establish the local well-posedness and some blow-up criteria of the magneto-viscoelastic system (1) − (4), in particular, without the regularizing term κ∆F as in [8,27]. Following the notation in [22], we define the usual strain tensor in the form of where I is the d × d identity matrix (d = 2, 3). Beside, in the remaining part of the paper, we always take H ext = 0 in (1) − (4) for the sake of simplicity. Then the magneto-viscoelastic system (1) − (4) can be rewritten into the following form: Here, we use the following notations The system (5)-(8) is subject to the following initial conditions: Now we are in a position to state the main results of this paper.
Theorem 1.1. Let d = 2, 3. Suppose that the initial data satisfy v 0 , E 0 ∈ H 2 (Ω), M 0 ∈ H 3 (Ω) and the following constraints Then there exists a positive time T depending on v 0 H 2 , E 0 H 2 and M 0 H 3 such that the periodic initial-boundary value problem (5) − (9) has a unique classical Remark 1.
(1) As indicated in [15], the first three relations in (10) are consequences of the incompressibility condition, while the last one can be understood as a consistency condition for changing of variables between the Lagrangian and Eulerian coordinates.
where E k = Ee k stands for the k-th column of the matrix E.
The magneto-viscoelastic system (1) − (4) has a highly nonlinear coupling structure due to the interconnection of viscoelasticity with magnetism. Nevertheless, it consists of two subsystems that have been extensively studied in the literature: one is the incompressible viscoelastic system (i.e., taking M ≡ 0), while the other one is the simplified Ericksen-Leslie (E-L) system for incompressible nematic liquid crystal flows with Ginzburg-Landau approximation (i.e., taking F ≡ 0). We recall that existence of local classical solutions as well as global classical solutions near-equilibrium of the incompressible viscoelastic system in the two-dimensional case was first proved in Lin et al. [20] (see also Lei and Zhou [16] for the same result via incompressible limit). Corresponding well-posedness results in the three dimensional case and a blow-up criterion can be found in Lei et al. [15] (see also Chen and Zhang [6]). We refer to Lin and Zhang [22] for the case of bounded domain, and to [5,7,10,29,30] for various types of blow-up criteria. On the other hand, concerning the simplified E-L system for nematic liquid crystal flow, Lin and Liu [19] first proved the existence of global weak solutions as well as local classical solutions and they also studied its long-time behavior (see Wu [28], Grasselli and Wu [9] for improved results on the uniqueness of asymptotic limit as t → +∞). Besides, some regularity criteria were derived in Liu and Cui [24] and Liu et al. [25] for the simplified E-L system in 3D (see Cavaterra et al. [4] for the generalized E-L system). For more detailed information on the mathematical analysis of these two subsystems, we refer to the recent review papers [18,21].
The rest of this paper is organized as follows. In Section 2, we provide some lemmas that will be used in the subsequent proofs. The local well-posedness of problem (5) − (9) (i.e., Theorem 1.1) is proved in Section 3. The last Section 4 is devoted to the proof of two blow-up criteria (i.e., Theorem 1.2 and 1.3).

Preliminaries.
We denote by L p (Ω), W m,p (Ω) the usual Lebesgue and Sobolev spaces on Ω, with norms · p , · W m,p respectively. For p = 2, we simply denote H m (Ω) = W m,2 (Ω) with norm · H m . The norm and inner product on L 2 (Ω) will be denoted by · and (·, ·), respectively. For simplicity, we do not distinguish functional spaces when scalar-valued, vector-valued or matrix-valued functions are involved. We denote by C a generic positive constant throughout this paper, which may vary at different places. Its special dependence will be indicated explicitly if necessary.
First, we recall the following important properties of the strain tensor E (see [15,20,23]).
is the solution of problem (5) − (9), with the initial data (9) satisfying the constraint (10). Then the following identities hold for all time t ≥ 0.
The following interpolation inequalities are consequences of the well-known Gagliado-Nirenberg inequality (see, e.g., [31]). They will be frequently used in the higher-order energy estimates.
, Ω ⊂ R d is bounded, and 0 ≤ j < s ≤ k, the following interpolation inequalities hold: for some constant C that is independent of v.
Besides, the following inequalities can be found in [14].
3. Assume that f, g ∈ H k (Ω), k ≥ 0 being an integer. Then for any multi-index β, |β| ≤ k, we have Finally, we derive the following basic energy law that reflects the energy dissipation property of magneto-viscoelastic flows.
Remark 2. Proposition 1 implies an important feature of problem (5) − (9) such that it only has a partially dissipative structure. Finding the hidden dissipative mechanism for E (or F ) will play an essential role in the study of global wellposedness of problem (5) − (9).
3. Local Well-posedness: Proof of Theorem 1.1. As in [15,19], one can use the Galerkin method to construct approximating solutions to the momentum equation (5), then use this approximating solution v and the equations for the deformation gradient (7) and the magnetization (8) to obtain approximating solutions for E and M . Thus, to prove the convergence of the approximating solutions, we need only a priori estimates. For the sake of simplicity, below we derive a priori estimates for smooth solutions of problem (5) Second estimate. Taking L 2 inner product of the momentum equation (5) with v t , using (6), (15) and integration by parts, we obtain . This together with the interpolation inequality (18) and Young's inequality implies that µ 2 where g(·, ·, ·, ·) stands for a generic nonnegative and increasing function of its variables.
Taking time derivative of the momentum equation (5) and taking L 2 inner product of the resulting equation with v t , using (6), (15), (17) and integration by parts, we obtain 1 2 where in the last step of the above derivation, we used the L 2 estimate (23). Third estimate. Taking the L 2 inner product of the momentum equation (5) with ∆v, using (17), (18) and integration by parts, we have and hence ∆v 2 ≤ g( ∇v , v t , ∆E , ∇∆M ).
From the transport equation (7), We can get the following estimate with the help of (17), Substituting (27) into (28), we find Similarly, taking ∇ to the magnetization equation (8), we have (17) and (18), we arrive at

WENJING ZHAO
So we find the following estimate with the help of (27) such that Plugging (27), (29) and (31) into (26), one has Fourth estimate. Noting (24) and (32), it is clear that we need to gain the estimates for ∆E and ∇∆M . Applying ∆ to the transport equation (7), multiplying ∆E and integrating over Ω, we get Again using (17) and (18), we obtain On the other hand, we take ∇ to the momentum equation (5), and then take the L 2 inner product of the resulting equation with ∇∆v, we can get Applying (17) and (18), we find that  Combining (27), we arrive at Inserting (34) into (33), we get Fifth estimate. Now we take triple derivatives to the magnetization equation (8), multiply ∇∆M both sides, integrate over Ω, we get Using again the interpolation inequalities (17) and (18), we have Noticing the orders of ∇∆v are all smaller than two, using the Young inequality and substituting (21), and (34) into (36), we find Sixth estimate. Combining (24), (32), (35) and (37), we finally arrive at It follows from the momentum equation (5) that This last estimate together with (38) yields that there exist positive constants T , Besides, we deduce from (27), (29), (31) and (34) that Keeping the above estimates in mind, by a standard argument, we can obtain the existence of local classical solution to problem (5) − (9). Uniqueness of local classical solutions can be easily derived by using the energy method and Gronwall's lemma.
The proof of Theorem 1.1 is complete.
Keeping in mind that we have the global L 2 -estimate (23), thanks to the basic energy law, then we proceed to get the H 1 -estimate. Taking ∇ to the equations (5) and (7), multiplying with ∇v and ∇E respectively, and then integrating over Ω, we get 8 . (43) Applying ∆ to the magnetization equation (8) (44) Using the incompressibility of v, we have Next, using integration by parts, we get Similarly, we obtain , ∆∇ i v i ) = 0. Moveover, using Hölder's inequality, the Cauchy-Schwarz inequality and the Sobolev imbedding theorem, we get Summing up (42) − (44), we infer from the above estimates that Then by Gronwall's inequality, we deduce that for all 0 ≤ t ≤ T * , and some universal constant C. We proceed to obtain H 2 -estimate. Taking ∆ to (5) and (7), ∇∆ to (8), multiplying the resultants with ∆v, ∆E and ∇∆M respectively, and integrating over Ω, we get 1 2 d dt ∆v 2 + µ ∇∆v 2 = −(∆(v · ∇v), ∆v) − (∆∇p, ∆v) On account of (23) and integration by parts, we deduce 3 + H Similarly, we obtain Collecting the above estimates and using the bound (46), we obtain the H 2 -estimate Using the equation for E by integration along the characteristic passing [17] one can get With the help of Gronwall's lemma, we obtain On the other hand, applying the Sobolev imbedding theorem and (46), we have Then going back to (47), by Gronwall's lemma we infer that where C is a constant depending on C 1 , v 0 H 2 , E 0 H 2 and M 0 H 3 . Hence, the local solution (v, E, M ) can be extended beyond t = T * . This leads to a contradiction of the definition of maximal existence time T * . The proof of Theorem 1.2 is complete.

4.2.
Proof of Theorem 1.3. The proof is again based on a contradiction argument. Assume now From the assumption (51), we see that for any small constant 0 < ε 1, there exists a corresponding T = T (ε) ∈ (0, T * ) such that Recalling the well-known Kato's inequality (see [1]) valid on T d and the lower-order estimate (23), we infer that where C 2 depends on T d and C 3 > e depends on v 0 , E 0 , M 0 H 1 , α and T d . For all t ∈ (T, T * ), we define Recalling (45), by Gronwall's lemma and (53), we find that where C(ε) is a positive constant depending on ε, T * , and Ω, which may change from line to line. Next, applying ∇ 3 to the equations (5) and (7), ∇ 4 to (8), taking L 2 inner product with ∇ 3 v, ∇ 3 E and ∇ 2 ∆M respectively, and adding the resultants together, we obtain We estimate the right-hand side of (55) term by term. Using the incompressibility of v and Lemma 2.3, we have and H Similarly, using the property (15) and applying the estimate (20), we deduce that Concerning the terms involving M , we see that Using Hölder's inequality, Young's inequality, the Gagliardo-Nirenberg inequalities (17), (18) and the following inequalities we can estimate J 1 , J 2 and J 3 as follows  Inserting the above estimates into (55), and choosing ε > 0 small enough such that 12C 2 ε ≤ 1, we obtain ≤ C(ε)(1 + ∇v L ∞ + ∇E L ∞ )(C 3 + H(t)), ∀ t ∈ (T, T * ). (56) Then integrating the above inequality with respect to time from T to t ∈ (T, T * ), by the inequality (53), we have for all T ≤ t < T * . From Gronwall's lemma, we deduce that Applying Gronwall's lemma again, we have for all t ∈ [T, T * ) (recalling (52)) ln(C 3 + H(t)) and thus Hence, we can extend the local classical solution (v, E, M ) beyond T * , which leads to a contradiction with the definition of T * .
The proof of Theorem 1.3 is complete.