Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows System with Damping ()
1. Introduction
In this paper, we are interested in three-dimensional compressible viscoelastic flows with damping in the following form:
(1.1)
for
. Here
denote the density, the velocity and the deformation gradient, respectively. The constants
and
are the shear viscosity and the bulk viscosity coefficients of the fluid, respectively, which satisfy
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
(1.2)
with
(1.3)
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
, which are faster than 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 L2-norm decay rates of the compressible viscoelastic flows with the term for
in H3 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
into two parts, the low frequency part
and the high frequency part
. Here, the key point is the highest order derivative estimates of the high frequency part can be governed by the decay estimates of the low frequency part, see Section 3. We use the energy method for the high frequency part and combine with the decay estimates of
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.
Theorem 1.1 Assume that the initial condition
satisfies the constraints (1.3), there exists a constant
such that if
(1.4)
then there exists a unique global solution
of the Cauchy problem (1.1)-(1.2) such that for any
,
(5)
Moreover, if
, the solution
enjoys the following decay-in-time estimates:
for some positive constant C.
Notation. We use
,
to represent the usual Lebesgue and Sobolev spaces on
and
, with norms
,
and
, respectively. We also denote
, where
,
. We assume C to be a positive constant throughout this paper that may vary at different places and the integration domain
will be always omitted without any ambiguity. 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.
2. Global Existence
Reformulations
In this subsection, we first reformulate the system (1.1). For
, (1.1) can be rewritten as
(2.1)
where we use the condition (1.3),
, then get the vector
,
Let
then we have from (6) that
(2.2)
where
(2.3)
And here
. We will assume that
for the rest of this paper.
Proposition 2.1 (Local existence). Let
and
. Then there exists a positive constant
depending on
, such that the initial value problem (2.1) has a unique solution
satisfying
Moreover, the following estimates hold,
and
where
.
Proof. The proof can be done by using the standard iteration arguments. Refer, instance, to [8][25].
Proposition 2.2 (A priori estimate). Assume that
(2.4)
then there exists a unique global solution
of the Cauchy problem (1.1)-(1.2) such that for any
,
(2.5)
Moreover, if
, the solution
enjoys the following decay-in-time estimates:
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
.
Lemma 2.1 Under the assumption (2.4), it holds that
(2.6)
Proof. Multiplying (2.2)1, (2.2)2, (2.2)3, by
, respectively, and then integrating them over
, we obtain
(2.7)
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
(2.8)
Similar to the proof of (2.8), we get
(2.9)
For the second term of (2.7), we can get
(2.10)
We also use Hölder’s inequality and Sobolev’s inequality to get
(2.11)
(2.12)
(2.13)
Substituting (2.11)-(2.13) into (2.10) gives
(2.14)
Combining (2.8), (2.9) and (2.14) with (2.7) since
is sufficiently small. This completes the proof of Lemma 2.1. ,
Lemma 2.2 Under the assumption (2.4), it holds that
(2.15)
Proof. For each multi-index k with
, by applying
to (2.2) and multiplying them by
and
respectively, and integrating over
, we have
(2.16)
Now we estimate the right-hand side of (2.16). Here we know
By Hölder’s inequality and Sobolev’s inequality, we have
(2.17)
Next we estimate the terms
one by one.
(2.18)
Similar to the estimate of
, we obtain
(2.19)
(2.20)
(2.21)
So we can get
(2.22)
Next, we deal with the term
and
.
(2.23)
And we have
(2.24)
Similarly, by using (2.2)3, we have
(2.25)
Similar to the proof of
, we have
(2.26)
So we get
(2.27)
Similarly, we can estimate the other terms.
(2.28)
For term
, we have
(2.29)
Then we have
And same as
, we have
Thus we can give the estimates of the
,
(2.30)
Finally we can estimate the term of
same as
,
(2.31)
Substituting (2.22), (2.30) and (2.31) into (2.16), we get (2.15). ,
In the following lemmas we give the dissipations on
and
.
Lemma 2.3 Under the assumption (2.4), there exists a positive constant C such that
(2.32)
(2.33)
(2.34)
Proof. Notice that the condition
for all
gives
Thus we get
(2.35)
Then applying
to (2.2)2 and summing over i, we can get
(2.36)
where
Multiplying (2.36) by n, and then integrating over
, we get
where we use the Hölder’s, Sobolev’s and Cauchy’s inequalities to get
Multiplying (2.36) by
, and then integrating over
, we get
By Hölder’s, Sobolev’s and Cauchy’s inequalities we get
Then applying
to (2.2)2, summing over i and multiplying them by
, and then integrating them, we have
which gives (2.34) if
is small enough. This completes the proof of Lemma 2.3. ,
Lemma 2.4 Under the assumption (2.4), there exists a positive constant C such that
(2.37)
(2.38)
(2.39)
Proof. Taking
, then we have
(2.40)
where
. Note the condition
for all
, which means that
(2.41)
Then we get
(2.42)
Apply
to (2.2)2, we get
(2.43)
where the antisymmetric matrix
is defined as
Multiplying (2.43) by
, and then integrating over
, we get
By Hölder’s and Sobolev’s inequalities we have
where
. Now let us estimate the right-hand side term by term. We use Hölder’s and Sobolev’s inequalities have
Thus we get
(2.44)
Similarly, multiplying (2.43) by
, and then integrating over
, we get
then
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
, we get
The proof of Lemma 2.4 is completed. ,
Lemma 2.5 Under the assumption (2.4), it holds that
(2.45)
(2.46)
(2.47)
Proof. Combining (2.35) and (2.42), we get
(2.48)
Thus using the property of the Riesz potential, from (2.4), we have
and
and
From the above estimate, we may deduce from (2.41) that
and
Thus, the proof of Lemma 2.5 is completed. ,
Now we are in a position to verify (2.4). Since
is sufficiently small, from Lemma (2.1)-(2.5), we can choose a constant
suitably large such that
for any
, which implies
since
3. Decay Estimates
In this section, we show the decay estimates of the solution to the problem (2.2) in low-frequency regime.
3.1. Decay Estimates for the Linearized Problem
We note that the linearized system (2.2)1 and (2.36) depend only on
while the linearized system (2.40)-(2.43) depend only on
. Denote by
the pseudodifferential operator is defined by
and let
be the “compressible part” and “incompressible part” respectively. We finally obtain
(3.1)
and
(3.2)
Indeed, as the definition of d,
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
where
(3.3)
And there
is the solution semigroup defined by
,
with B being a matrix-valued differential operator given by
and
, we get
Now we need to consider the following linearized system (3.1),
,
(3.4)
Applying the Fourier transform to system (3.3), we have
By using the Fourier transform, we can get linearized system (3.1)
(3.5)
and
(3.6)
From (3.5), we have
(3.7)
Multiplying (3.5)2 by
,
(3.8)
Multiplying (3.8) by
and (3.7) to get
(3.9)
Similarly, from (3.10), we get
(3.10)
and
(3.11)
yields
(3.12)
Let
and
be Lyapunov-type functionals defined by
(3.13)
and
(3.14)
Then it is clear that there exists a small enough constant
, such that for any
,
(3.15)
and
(3.16)
From (3.9) and (3.16), we have
(3.17)
and
(3.18)
Using the Young inequality, we deduce that there exists a positive C and a sufficient small constant
such that for any
,
(3.19)
Therefore
(3.20)
where
and
.
We define a low frequency and high frequency decomposition
for a function
(3.21)
Let
be a function in
and some chosen constant R such that,
(3.22)
Thus, we get
(3.23)
Then
We show the decay estimates of
as follows.
Lemma 3.1 Let
, then for
, the following decay
(3.24)
(3.25)
(3.26)
(3.27)
Proof. By using the Plancherel theorem and the Hölder inequality, we get
Similarly, we can get (3.24)-(3.27). ,
Next, we consider decay rates for the nonlinear system.
3.2. Decay Estimates for the Nonlinear Problem
Lemma 3.2 It holds that for any integer
,
(3.28)
and
(3.29)
Proof. By using the Duhamel principle
(3.30)
By using (3.3) and the Hausdorff-Young inequality, the nonlinear source terms can be estimates as follows:
(3.31)
(3.32)
(3.33)
Put these estimates into (3.30), this completes the proof of Lemma 3.2. ,
3.3. Optimal Decay Estimates
Now we will show the proof of Proposition 2.2. Define the temporal energy functional
(3.34)
for any
, where it should be mentioned that
is equivalent to
, we get
(3.35)
From (3.23), we have
(3.36)
Adding
to both of the inequality and using the smallness of
, we have
(3.37)
Here
can be large enough. If we define
(3.38)
Then, by using (3.38) and Lemma 3.2 that for any
,
(3.39)
From (3.37) and (3.39), we get
(3.40)
and also can get
(3.41)
Therefore we obtain that for any
,
(3.42)
Noticing
, with the smallness of
, we have
(3.43)
Considering (2.35) and Lemma 2.5, we get
(3.44)
Similarly, by Lemma 3.2 we can obtain
(3.45)
From (2.2) and use the estimates above, we get
(3.46)
Appendix
In this appendix, we state some useful inequalities in the Sobolev space.
Lemma 4.1 Let
. Then
Lemma 4.2 Let
be an integer, then we have
(4.1)
and
(4.2)
where
and
(4.3)
Proof. Please refer for instance to [26] . ,
Lemma 4.3 Let
, then
(4.4)
Lemma 4.4 If
and
, then it holds that
(4.5)