Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows System with Damping

In this paper, we consider the large time behavior of the strong solutions to the three dimensional compressible viscoelastic flows with damping. Based on the energy method and spectral analysis, we analyze the influences of the damping on the global existence and decay rates of compressible viscoelastic flows under some small assumptions in H-framework. Compared with the time decay rates of solutions to the compressible viscoelastic flows in [1], our results imply that the friction of the damping is stronger than the dissipation effect of the viscosities.


Introduction
In this paper, we are interested in three-dimensional compressible viscoelastic flows with damping in the following form: for ( ) [ ) 3   , 0, t x ∈ +∞ ×  .Here The corresponding elastic energy is chosen to be the special form of the Hookean linear elastic [2], ( ) ( ) In this paper, we investigate the Cauchy problem of system (1.1) with the initial data ( ) ( ) ( ) ( ) ( ) T div 0, 0 0 0 0 .
It is standard that the condition (1.3) is preserved by the flow, which has been proved in [3] [4].
The system (1.1) is a coupling system between Navier-Stokes equations and the deformation gradient with a damping term.When the damping term is absence in the system (1), there are many results about the global existence of solution to the compressible viscoelastic flows, refer to [5] [6] [7].Hu and Wang in [1] [8] have proved the global existence and the decay rates of multi-dimensional compressible viscoelastic fluids in the large time behavior.The global existence of the Cauchy problem with initial data close to an equilibrium state in Besov spaces was obtained in [3].For the Navier-Stokes equations, there are many mathematical results on the global existence of the solutions in [9] [10].Recently, for the Navier-Stokes equations with the electric potential, Wang in [11] proved the global existence of strong solution.They also obtained the decay rates ( ) ( ) in [1].For the classical solutions to the incompressible viscoelastic flows, refer to [12]- [18].And the global existence of weak solutions to the incompressible viscoelastic flows with large initial data is established in [3] [16] [19].
When we consider the damping term, there is very little progress on the global existence of solutions.And for the Navier-Stokes equations with the damping term ( ) − > , we refer to [20] [21] [22] [23] [24].In this paper, we consider the global existence and L 2 -norm decay rates of the compressible viscoelastic flows with the term for 1 β = in H 3 framework.We use the standard energy method to prove the global existence under the condition that the initial data are close to the constant equilibrium state.In order to prove the decay rates, we take the Hodge decomposition of the linear system, and then it becomes two similar systems which only involve two variables.It is different from the compressible viscoelastic fluids [1] [8], we divide the solution

( )( )
We use the energy method for the high frequency part and combine with the decay estimates of l U to obtain the decay estimates of the solutions.Here, when the damping term is taken into account, we obtain the improved global existence of the solutions for system (1.1).
Our main results are stated in the following theorem.

⋅
, respectively.We also denote ( ) , , We as- sume C to be a positive constant throughout this paper that may vary at different places and the integration domain 3  will be always omitted without any am- biguity.Finally , ⋅ ⋅ denotes the inner product in ( ) The rest of this paper is devoted to prove Theorem 1.1.In Section 2, we reformulate the system and establish a priori estimate for the strong solutions.
Then we consider the global existence of strong solutions.In Section 3, we give the decay estimates of the low frequency part and complete the proof of Theorem 1.1.Finally, Section 4 shows some useful inequalities.
then we have from ( 6) that , And here ( ) ( ) ( ) ( ) . We will assume that 1 α = for the rest of this paper.
then there exists a unique global solution ( ) Moreover, if ( ) ( ) , , n v E enjoys the following decay-in-time estimates: , , 1 .
We can get the global existence of solution by combining the local existence result with a priori estimates.In the following lemma we give the energy estimates of the lower and higher for ( ) To estimate the left-hand side of (2.7), we can use the first equation in (2.2).
The three terms on the right-hand side of the above equation can be estimated as follows.
First, we can use Hölder's inequality and Sobolev's inequality to get Similar to the proof of (2.8), we get ( ) For the second term of (2.7), we can get

10) Journal of Applied Mathematics and Physics
We also use Hölder's inequality and Sobolev's inequality to get ( ) , , Proof.For each multi-index k with 1 2) and multiplying them by , and k E ∇ respectively, and integrating over .
Now we estimate the right-hand side of (2.16).Here we know By Hölder's inequality and Sobolev's inequality, we have .
Next we estimate the terms ( ) .
Similar to the estimate of 1   1 So we can get ( ) Next, we deal with the term 2 I and 3 I . .
And we have ( ) .
So we get ( ) Similarly, we can estimate the other terms.
Then we have .
And same as 2   1 .
Thus we can give the estimates of the 2 I , ( ) Finally we can estimate the term of 3 I same as 1 I , ( ) Proof.Notice that the condition ( ) Then applying i ∇ to (2.2) 2 and summing over i, we can get ( ) ( ) where ( ) Multiplying (2.36) by n, and then integrating over 3  , we get where we use the Hölder's, Sobolev's and Cauchy's inequalities to get Multiplying (2.36) by n ∆ , and then integrating over 3  , we get By Hölder's, Sobolev's and Cauchy's inequalities we get Proof.Taking 2) , then we have where ( ) Then we get Apply curlv to (2.2) 2 , we get where the antisymmetric matrix  is defined as
By Hölder's and Sobolev's inequalities we get Now let us estimate the right-hand side term by term.We use Hölder's and Sobolev's inequalities have , , , , .
Similarly, applying ∇ to (2.43) and multiplying by ( ) , and then integrating over 3  , we get The proof of Lemma 2.4 is completed. Lemma 2.5 Under the assumption (2.4), it holds that ( ) , , Thus using the property of the Riesz potential, from (2.4), we have From the above estimate, we may deduce from (2.41) that Thus, the proof of Lemma 2.5 is completed. Now we are in a position to verify (2.4).Since 0 δ ≥ is sufficiently small, from Lemma (2.1)-(2.5),we can choose a constant 1 0

Decay Estimates
In this section, we show the decay estimates of the solution to the problem (2.2) in low-frequency regime.

Decay Estimates for the Linearized Problem
We note that the linearized system (2.
be the "compressible part" and "incompressible part" respectively.We finally obtain Indeed, as the definition of d, ω and the relation involve pseudodifferential operators of degree zero, the estimates in space ( ) for the original function v will be the same as for ( ) Here, we just discuss the system (3.1),since the system (3.2) is the same as system (3.1).To use the estimates of the linear problem for the nonlinear system (3.1) and (3.2), we rewrite the solution of (3.1) as .
And there ( ) K t is the solution semigroup defined by ( ) tB K t e − = , 0 t > with B being a matrix-valued differential operator given by ( ) , we get ( ) Now we need to consider the following linearized system (3.1), Applying the Fourier transform to system (3.3),we have By using the Fourier transform, we can get linearized system (3.1) Similarly, from (3.10), we get ) ˆˆˆˆ. 2 2 the Young inequality, we deduce that there exists a positive C and a sufficient small constant 0 0 C > such that for any , , 0 , where ( ) ( ) ( ) ( ) We define a low frequency and high frequency decomposition and some chosen constant R such that, ( ) Thus, we get ( ) ( ) ( ).
We show the decay estimates of ( ) l U t as follows.

Decay Estimates for the Nonlinear Problem
Lemma 3.2 It holds that for any integer Proof.By using the Duhamel principle By using (3.3) and the Hausdorff-Young inequality, the nonlinear source terms can be estimates as follows: Put these estimates into (3.30),this completes the proof of Lemma 3.2.

Optimal Decay Estimates
Now we will show the proof of Proposition 2.2.Define the temporal energy functional for any 0 t T ≤ ≤ , where it should be mentioned that ( ) , , , , , , , , .
, , to both of the inequality and using the smallness of δ , we have Here 3 D can be large enough.If we define Then, by using (3.38) and Lemma 3.2 that for any 0 { } ) ( )       , , div div 1 .

Appendix
In this appendix, we state some useful inequalities in the Sobolev space.
 Lemma 4.3 Let , the velocity and the deformation gradient, respectively.The constants µ and λ are the shear viscosity and the bulk viscosity coefficients of the fluid, respectively, t

.
2Let 1 m ≥ be an integer, then we have