STRONG SOLUTIONS TO COMPRESSIBLE BAROTROPIC VISCOELASTIC FLOW WITH VACUUM

. We consider strong solutions to compressible barotropic viscoelastic ﬂow in a domain Ω ⊂ R 3 and prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Inspired by the work of Kato and Lax, we use the contraction mapping principle to get the result.

1. Introduction. Materials in which the stress at a point depends on the entire history of the strain are called viscoelastic material, which can be viewed as the intermediate state between the fluid and solid. In general, viscoelastic material exhibits a combination of the microscopic elastic behavior, such as memory effects, and the macroscopic fluid properties. Many complicated hydrodynamic and rheological behaviors of complex fluids can be regarded as consequences of internal elastic properties. For example, Stokes fluids can be considered as special cases of viscoelastic materials. Viscoelastic flows play an important part in studying viscoelastic materials and have a wide range of applications, which give not only the rich and complicated rheological phenomena, but also formidable challenges in analysis and numerical simulations. This paper is concerned with the initial boundary value problem to the following compressible viscoelastic fluid system of Oldroyd model ∂ t ρ + div(ρu) = 0, (1.1) ∂ t (ρu) + div(ρu ⊗ u) + ∇P (ρ) = µ∆u + λ∇divu + div(ρF F T ), (1.2) in (0, T ) × Ω, where the unknown functions ρ = ρ(x, t) denotes the density, u(x, t) ∈ R 3 the velocity of fluid and F ∈ M 3×3 the deformation gradient. The symbol ⊗ denotes the Kronecker tensor product, F T means the transpose matrix of F .
Generally speaking, the pressure P depends on the density and temperature. In that case, the system (1.1)-(1.3) is not closed and should be complemented by the energy equation. However, there are physically relevant situations in which we assume that the fluid flow is barotropic, i.e., the pressure depends only on the density. This is the case when either the temperature or the entropy is supposed to be constant. The typical expression is P (ρ) = ρ γ . Here, we assume that the pressure P satisfies a more general law: The viscosity coefficients µ and λ satisfy the physical conditions: is a bounded domain with smooth boundary. Supplement the system with the boundary and initial conditions  [4,5,6,12,13,14,15,20,21,22,23], due to its physical importance, complex, rich phenomena and mathematical challenges. Without the deformation gradient F , the compressible viscoelastic system reduces to the compressible Naiver-Stokes equations. There is a huge literatures on the compressible Naiver-Stokes equations, for instance, [1,8,16,18,19].
As to the compressible viscoelastic case, the mathematical analysis is much more complicated. Comparing with Navier-Stokes equations, we will encounter extra difficulties in studying the compressible viscoelastic system. From the viewpoint of mathematical structure, the mathematical analysis is difficult due to the loss of dissipation of the deformation gradient in (1.3). From the viewpoint of partial differential equation, the system (1.1) − (1.3) is a highly nonlinear system coupling between hyperbolic equations and parabolic equations. Thus, the extension of known results for the Navier-Stokes equations to the compressible viscoelastic flows is not simple. In spite of these, there is much recent important progress for the compressible viscoelastic flows. More precisely, Qian [23] proved the existence and uniqueness of global strong solution for the initial boundary value problem near the equilibrium state in H 2 . Hu and Wang [5], proved the global well-posedness with small data in Besov space. In [4], Hu and Wang used the Schauder-Tychonoff fixed point theorem obtained the local existence of strong solutions in R 3 , with initial density has strictly lowed bound and upper bound. It should be pointed out that our viscoelastic model is called Oldroyd-A model. A related model is Oldroyd-B model, which have been attracted attention in the past decade. There is some differences between the two model. Readers who are interested in the Oldroyd-B model can refer to [3,7,10].
The aim of this paper is to prove the existence of unique local strong solutions to (1.1)-(1.3) with inf ρ 0 = 0. Here we use the contraction mapping to obtain the result, in some sense simply and extend the result of [1] and [4]. Before stating the local existence result, we need to specify the definition of strong solutions which we will address.
Our main result can be summarized as follows: If, in addition, the following compatibility condition holds for some g ∈ L 2 , then there exist a positive time T 0 and a unique strong solution This paper is written as follows. In Section 2, we give the notations and working function space which will be needed in later analysis. In Sections 3, we consider a linearized problem and derive some local estimates for the solutions independent of the lower bound of initial density. In Section 4, we use the contraction mapping principle to get the existence and uniqueness of local strong solutions.

2.
Preliminaries. First, it is necessary for us to give some notations by reason of the convenience of discussions.
(1) Ω f dx = Ω f , and T 0 Now, we will introduce the working function space which plays an important role in the process of proof:

Remark 1.
Before the proof, we point out that the approach to proving Theorem 1.1 is to apply the contraction mapping principle. Since the system (1.1)-(1.3) is of mixed hyperbolic-parabolic type and the initial density may vanish, we encounter a well-known difficulty in the theory of symmetric quasilinear hyperbolic systems. For these systems, contraction cannot be proved in the usual setting, that is, to consider self-mapping and contraction in the same regularity class W. To resolve this problem, Kato [9] and Lax [11] offered an ingenious idea by studying contraction in a larger space. Taking up this idea, we are able to establish the contraction in the space L (see in Section 4). Chu et. al. [2] adopted the same idea to tackle the compressible liquid crystal system.
3. Existence for the linearized equations. In this section, we reformulate the nonlinear equation (1.1)-(1.3) such that the left-hand becomes linear and the starting problem can be transferred to a fixed point equation.
with the given v ∈ W, the boundary condition (1.6) and initial conditions We denote ρ δ 0 = ρ 0 + δ, where δ > 0 is a constant and ρ 0 ≥ 0. If the initial density vanishes from below, we cannot expect the density ρ is bounded away from zero. As a result, the lack of a positive lower bound of ρ causes (3.1) to become a degenerate linear parabolic equation. This prevents us from using the standard argument to construct the local solutions. For this reason, we consider the linearized problem (3.1)-(3.3) with initial density bounded away form zero and derive some uniform bounds, which are independent of the lower bounds of initial density. Firstly, we solve out the density, and obtain estimates for density.

6)
where C is a constant.
(3.6) can be obtained similarly by taking 1-th derivative with respect to t of (3.7).
Combining the hyperbolic structure of (3.3) and the corresponding previous results, we can solve the deformation gradient F similarly as Lemma 3.1.
Proof. The existence can be founded in [23], and estimates can be obtained similarly as Lemma 3.3 in [4].
The next lemma gives the estimates on the velocity.

TONG TANG AND YONGFU WANG
Proof. Since the transport equation (3.7), initial condition and Lemma 3.1, we get ρ(x, t) > 0. The existence of the solution to (3.2) can be obtained by a semidiscrete Galerkin method of parabolic equations, or by Lax-Milgram theorem as [4,24].
To ensure the higher regularity, specially as to the term u t , we need some compatibility condition. The same process for higher regularity has been obtained in the whole space R 3 in [4]. However, there are some differences in our proof. In the following, we will give the detail of proof and show the use of compatibility condition.
In order to derive estimate for ∇u t , we differentiate (3.2) with respect to t and get Multiplying the identity by u t and using integration by parts, we obtain Using the continuity equation and the Gagliardo-Nirenberg inequality, we get Substituting I 1 − I 6 into (3.12), choosing a sufficiently small , and integrating with respect to t, we have To estimate || √ ρu 0t || 2 L 2 , we observe from (3.2) and the compatibility condition (1.8) Adding (3.13) and (3.14), we obtain the following estimates by Growall's inequality: To obtain further estimates, we rewrite (3.2) as which is a strongly elliptic system. By the classical elliptic regularity theory, we deduce From the previous lemmas, we get and In a similar way, we can obtain ||u|| L 2 (0,T ;W 2,q ) ≤ C + T α (α > 0). (3.19) Gathering (3.15)- (3.19) and choosing T small enough, we obtain the estimates and complete the proof. Hence, the proof is finished.
Combining all the lemmas, we get the existence for the linearized equations where T ∈ (0, T ). Thus, there are essentially two main tasks to prove: the self-mapping and contraction. The former has been done due to Lemma 3.3, which guarantees the selfmapping. As mentioned in Remark 2.1, we need to prove a contraction estimate in the lager space L. The following lemma implies that the map J is contracted in the sense of weaker norm for v ∈ M. Lemma 4.1. There exists a constant 0 < θ < 1 such that for any v i ∈ M, i = 1, 2, Multiplying (4.1) by ρ and integrating over Ω, we get where . In a similar way, we have where . Multiplying (4.3) by u and integrating over Ω, we deduce where where E(t) = E 1 (t) + E 2 (t) + E 3 (t) satisfies, for small T , where K 2 is a constant dependent on initial data, thanks to Lemma 3.1-3.3. Let T small enough, we obtain the following by Gronwall's inequality Since u is zero on boundary, we finish the proof.
By the contractibility of J and utilizing the iteration methods used in [11] and [17], we can obtain a unique fixed point u. This proves the existence of a strong solution. Then adapting the arguments in [1], we can easily prove the time-continuity of the solution (ρ, u, F ).
Thus, we complete the proof of Theorem 1.1.