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A blow-up criterion for 3-D compressible visco-elasticity

In this paper, we shall prove a blow-up criteria in terms of the upper bound of the density, deformation tensor and the gradient of deformation tensor for 3D compressible visco-elasticity. Based on the structure of deformation tensor, we find a cancelation in higher order derivatives, which brings us the desired results.


Introduction
Visco-elastic fluids exhibit the characteristics for both fluid and solid. The elastic behavior of these materials is attributed to the underlying microstructure or configurations, the deformation of the structures will cause the exchange of kinetic energy and elastic internal energy. The energy exchange is realized through a coupling of the transport of the internal elastic variables and the induced elastic stress.
To illustrate this coupled interaction, we denote x(t, X) as the Eulerian coordinate, where X is the Lagrangian coordinate of the particles. The velocity field defined on the Eulerian coordiante be (1.1) u(t, x) = x t (t, X(t, x)), and the deformation tensor is (1.2) F (t, x) = H(t, X(t, x)) = ∂x ∂X (t, X), which is determined by a transport equation (see [3,8,16]), This equation is regarded as the compatibility condition between velocity u and deformation tensor F . For homogenous, hyper-elastic and isotropic material, the action function is (see [11,16,17]) where ρ 0 (X) is the density in the undeformed configuration, ρ is the density in the deformed configuration. Ω 0 is the original domain occupied by the material. E(H) is the elastic energy function. P (ρ) is the hydrostatic pressure, satisfying P (ρ) = C 0 ρ γ , with C 0 > 0 and the adiabatic index γ > 1. P (ρ) det H represents the internal energy. Taking a variation of A(x) with respect to the flow map x(t, X), we get the inviscid momentum equation in Eulerian coordinates ( [10]) ∂F F T is the Cauchy-Green tensor. When we take the dissipation energy into consideration, see [1,2], we can get the corresponding viscosity momentum equation Combining (1.3), (1.6) and mass conservation equation, with Hookean elasticity, we get the following viscoelasticity system The viscosity coefficient µ, λ are constants satisfying µ ≥ 0, 3λ + 2µ ≥ 0.
The system (1.7), when F do not exist, it is a compressible Navier-Stokes equations. The global strong solutions existence for large initial data is still open. Therefore, there are many efforts to study the blow-up criteria. See [7,19] and references therein.
When we take the deformation tensor F into consideration, the system (1.7) will be much more complicate, although the deformation tensor admits a transport equation which is similar to the continuity equation. The incompressible visco-elasticity has been studied by [9,10,11,12,14,16,17] etc. Precisely, in 2D, the local strong solutions to large data and global strong solutions to small data have been studied in [10,11,12,14,16]. The global strong solutions in 3D with small data have been presented in [12]. Subsequently, the blow-up criteria for 2D Oldroyd model have been presented in [13], a blow-up criterion in 3D with partial viscosity was proved in [15].
For the compressible case, The local strong solutions with large data and global strong solutions with small data to (1.7) were proved in [4] and [5,18] respectively. Subsequently, Hu-wang [6], presented a blow-up criterion for local strong solutions as ∇u L 1 (0,T,L ∞ (R 3 )) < ∞. To proceed, we firstly introduce some notations as following In Hu-Wang [6], they have proved the following results, Theorem 1.1. (Hu-Wang [6]). Assume that the initial data satisfy 0 . There exist classical solutions (ρ, u, F ) to (1.7) satisfying where T ⋆ is the maximal existence time. If T ⋆ < ∞ and 7µ > λ, then, Enlightened by the work of Sun-Wang-Zhang [19], in which the authors presented the blow-up criterion for a compressible Navier-Stokes in the terms of ρ L ∞ (0,T,L ∞ ) , we will establish a blow-up criterion in terms of the upper bounds of the density and deformation tensor for the local strong solution to the 3D compressible visco-elasticity. Our main result states as follows.
Theorem 1.2. Assume that (ρ, u, F ) is the local strong solution mentioned in (1.8), and µ, λ be as in Theorem 1.1. The initial data ρ 0 > ε 0 > 0, u 0 and F 0 as in Theorem 1.1, T ⋆ is the maximal existence time of the solution. If T ⋆ < ∞, then we have ∞, or equivalently detF = 0, the mass accumulate at one point or the material volume compress to zero. If F L ∞ (0,T,L ∞ (Ω)) = ∞,then the deformation leads to the blow-up phenomenon.
If ρ(t) L ∞ (0,T ;L ∞ (Ω)) and F L ∞ (0,T ;L ∞ (Ω)) are bounded, we can obtain a high integrability of velocity, which can be used to control the nonlinear term (See Lemma 3.2). The difficulty is to control the density and deformation, which satisfy transport equations. To do this, it requires the velocity is bounded in L 1 (0, T ; W 1,∞ (Ω)). On the other hand, we have to obtain some priori bounds for ∇ρ and ∇F to prove u ∈ L 1 (0, T ; W 1,∞ (Ω)), furthermore, the elasticity term ∇ · ( 1 det F F F T ) in the momentum equation will bring extra difficulty to us. Thanks to the structure of the equations, we get the cancelation to the derivatives of ρ, F during our computation, which brings us the desired result. Moreover, we get the result without the restriction on ρ −1 , it is an advantage compared with Sun-Wang-Zhang [20].
Generally speaking, the fluid is driven by three kinds of forces which present different mechanisms. They are gradient of pressure, elastic deformation, and inertial force. According to these mechanisms, we can decompose the velocity field to several parts. Here, since the pressure term and the elasticity term have similar form in the equation, we put them together and decompose velocity field into two parts, that is and (1.14) Lw = ρu, By the decomposition, in order to obtain the regularity of u, it is sufficient to consider the regularity of v and w, which admit the above elliptic equations.
Our paper is organized as following. In Section 2, we shall present some preliminaries. Section 3 is to present some priori estimates. The proof of Theorem 1.2 should be presented in Section 4.

Preliminaries
Consider the following boundary value problem for the Lamé operator L where Ω is R 3 or a bounded domain in R 3 . We present the following Lemmas, see [19,20] for details. Lemma 2.1. Let q ∈ (1, ∞) and U be a solution of (2.1). There exists a constant C depending only on λ, µ, q and Ω such that the following estimates hold.

Lemma 2.3.
Let Ω = R 3 or be a bounded Lipschitz domain and f ∈ W 1,q (Ω) with q ∈ (3, ∞). There exists a constant C depending on q and the Lipschitz property of Ω such that The following lemma is well-known.
Lemma 2.4. Let ρ, F as defined before, and ρ 0 is the initial data of ρ, then In the following, we use the notation, for the matrix (A) 3×3 and (B) 3×3 where summation applied to terms with repeated index.
Proof. Multiplying the equation (1.7) 2 by r|u| r−2 u and integrating the resulting equation on Ω to obtain By using the fact |∇u| ≥ |∇|u||, the term in second integrand can be estimated by The pressure term where C is constant depends on ρ L ∞ (0,T,L ∞ (Ω)) , F L ∞ (0,T,L ∞ (Ω)) and the initial data.
From the estimates for Lamé operator and the energy estimates Lemma 3.1, we have Similarly, we have Then, the results (3.9) follows from (3.11)-(3.16) and energy estimates Lemma 3.1.   Proof. We first estimate ∇ 2 w L 2 (0,T ;L q (Ω)) . By Lemma 2.1 and (1.14), we have Noting, taking ∂ t to this equation, we get Applying u⊗ and divergence to (4.5), we have Adding (4.6) and (4.7) together, we get Multiplying above equation byu and integrating on Ω, we get The estimates of the second and the third terms on the left hand side, as well as the first and the second terms on the right hand side in (4.9), are the same as in [19,20]. For completeness, we give a brief proof as following − Ω (∇divu t + div(u ⊗ ∇divu)) ·udx (4.11) We continue to estimate the pressure term.