Incompressible limit for the compressible viscoelastic fluids in critical space
-
Bin Han
Abstract
In this article, we consider the incompressible limit of global-in-time strong solutions with arbitrary large initial velocity for the compressible viscoelastic fluids in the sense of critical Besov framework. We decouple our compressible system into two coupling sub-systems by introducing a skew symmetric matrix, which is related to the deformation tensor. This work generalizes the similar result obtained by Hu et al. (Incompressible limit for compressible viscoelastic flows with large velocity, Advances in Nonlinear Analysis 12 (2023), 20220324) to the critical functional space with respective to the natural scaling of the system. The proof relies on the dispersive property of the linear system on the high-frequency regime and the parabolic property on the low-frequency regime. The dispersion tends to disappear when
1 Introduction
It is well known that viscoelasticity is a material property that exhibits both viscous and elastic characteristics when undergoing deformation. Viscoelastic fluids show intermediate behavior with some remarkable phenomena due to their elastic nature. They exhibit a combination of both fluid and solid characteristics and have received a great deal of interest. In this article, we focus on the Cauchy problem of the following compressible viscoelastic fluids in
The unknowns are the fluid density
with
It has been proved in [22] (also see [29]) that (1.2) is preserved by the flow, i.e.,
and
In this work, we consider a special case of the Hookean elasticity:
system (1.1) reads
where we denote
Before going any further, we first recall some known results on the viscoelastic fluids, including incompressible and compressible systems. For the incompressible system, Lin et al. [25] first proved the local and global existence (with small initial data) of classical solutions to the two-dimensional (2D) case (1.8). Zhang et al. [4,26] generalized this result to the three-dimensional (3D) case via the curl free quantity
For compressible viscoelastic fluids, motivated by the seminal works on the compressible Navier-Stokes equations [7], Qian and Zhang [29] established the global existence of system (1.1) near equilibrium in the critical space. Hu and Wu [16] proved the global existence of the strong solutions of (1.1) by the standard energy method under the condition that the initial data are close to the constant equilibrium state in
Motivated by [14], in this article, we are going to study the incompressible limit of system (1.1) in a larger functional framework. Before stating our main result, we use scaling transformation for (1.1) to guess which space may be critical. It is not difficult to verify that (1.1) is invariant by the transformation
provided that
Definition 1.1
A function space
We can directly verify that
Our main goal here is to prove the global existence of strong solution to (1.5) with the initial data belonging to the critical space defined in (1.7). More precisely, for any initial velocity field
As mentioned earlier, the global existence issue of strong solution to (1.8) supplemented with general data is open in
Suppose that
and
In addition, under the assumptions that if
(1.9)holds true for some small positive
(1.10)and
(1.11)where
Based on the aforementioned statement, we want to justify the incompressible limit for the compressible viscoelastic system (1.5) around the incompressible state
Theorem 1.2
Let
holds with
where
where
provided that
We should make the following remarks here.
Remark 1.3
The initial values of equations (1.5) and (1.8) are assumed so that the estimate of the perturbation quantity
Remark 1.4
As justified in [29], when
Remark 1.5
Our future work will focus on the stability problems of the compressible viscoelastic fluids in the largest functional space. Indeed, for
for some
This space described earlier was used by the first author and Zi in [10] for research of the global well posedness of the compressible viscoelastic fluids with small data.
1.1 Difficulties and key ideas
Formally, to study the limit of solutions of (1.5) by passing
where
From (1.4), we infer that
Applying the operator
1.1.1 Step 1. Reduction to two coupling hyperbolic systems
Motivated by interesting work on the isentropic compressible Navier-Stokes equations [7],
with
Then, applying
where on the right-hand side of (1.16), we have used
On the other hand, one cannot directly expect the coupling between
It follows immediately that
In the same way, we can obtain easily that
Applying
where
To simplify the presentation, we assume from now on that shear viscosity
which is the same as one linearized from the isentropic compressible Navier-Stokes equations. In order to bound
1.1.2 Step 2. Recovering the estimates of
E
We would like to emphasize that in (1.19), the incompressible part
Multiplying (1.4) by
i.e.,
Here, we use
Using (1.3) again, we infer that
Therefore,
Applying the div-curl decomposition to
Finally, noting that
It can be observed from (1.22) that all the derivatives in nonlinear terms are outside products. These key observations make full use of the compatibility conditions and allow us to accomplish the estimates of
The rest of this article unfolds as follows. In the next section, we introduce Besov space and recall basic facts about them. Section 3 is devoted to proving Theorem 1.2.
1.2 Notations
For
(1.23)(1.24)Throughout this article,
In what follows, we denote by
(1.25)and initial quantities are given by
2 Littlewood-Paley decomposition results
The proof of most of the results presented in this article requires a dyadic decomposition of Fourier variables, which is called the Littlewood-Paley decomposition. Let us briefly explain how it may be built in the case
Let
and
Set
Then,
holds for tempered distributions modulo polynomials. As working modulo polynomial is not appropriate for nonlinear problems, we shall restrict our attention to the set
Note that (2.1) holds true whenever
The homogeneous decomposition has nice properties of quasi-orthogonality:
A series of functional space may be characterized in terms of Littlewood-Paley decomposition. Let us now give the definition of homogeneous Besov space:
Definition 2.1
[1] For
Then, the homogeneous Besov space is defined as
The following lemmas will be needed to estimate the nonlinear terms of (1.16) and (1.19). One of them is just a consequence of Bony decomposition and of continuity results for the paraproduct and remainder operators.
Lemma 2.2
(Bernstein inequality [1]) For all
and
The following properties of continuity for the product of two functions in Besov space will be constantly used in this article.
Lemma 2.3
Let
Then,
For the details of the proof on the aforementioned lemma, please refer to [1] (Chapter 2).
Lemma 2.4
(Interpolation inequality [1]) Let
The following lemma is used in (3.24).
Lemma 2.5
(Commutator estimate [21]) For
We end this section by introducing the Chemin-Lener space [1], which will be used for studying the evolution equations in the Besov framework.
Definition 2.6
For
when
3 Proof of the main results
In this section, we aim to construct the global a priori estimates on the perturbation equations in low- and high-frequency regimes. We first assume that
Then, by a standard continuation principle, we may extend the solution beyond
which indicates that
Now, we derive the estimates on
In what follows, we will show the desired norm of the right-hand side in the aforementioned inequality that
In order to obtain the key estimates of
Now, localizing (3.9) by the Littlewood-Paley operators
The commutator term of (3.10) may be bounded with the help of Lemma 2.100 in [1]. Testing (3.10) by
Using Gronwall’s inequality, we conclude that
which implies that
Thus, from (3.13), we have
3.1 Estimates on
(
b
,
d
)
To estimate the potential part of the velocity
where
We follow an energy method to bound
and
To obtain an estimate on the high-frequency part of
where
Performing the inner product with
To eliminate the highest order term, namely, the one with
By integration by parts, we have
Hence, adding
Multiplying the aforementioned equality by
with
According to the definition of
and second,
In addition, using Lemma 2.5, we have
Therefore, (3.21)–(3.24) lead to
whence, integrating over
According to (3.25), we note that there will be loss of the expected parabolic smoothing effect of
whence, integrating in time variable,
Together with (3.25), by using (3.22) and multiplying by
where notations
and using the Leibniz rule and, again, Lemma 2.3,
The commutator term may be bounded as follows (the first inequality stems from Lemma 2.100 in [1] and the second one may be deduced from that lemma and Leibniz rule):
Hence, putting (3.27)–(3.30) together, we obtain
Now, let us bound
and using the stability of the Besov space
As for
From (3.32) to (3.38) and (3.13), applying the Hölder inequality in time variable and recalling the setting of
Returning to equation (3.14)2, it is worth mentioning that
The first term of the right-hand side satisfies
To bound
Assuming that
Using Gronwall’s inequality, we conclude that
3.2 Estimate on
Ω
and
E
¯
We are required to consider the momentum and continuity equations altogether. Now, localizing (1.19) according to Littlewood-Paley operators, we discover that
where
By introducing
where notations
Therefore, according to (3.45)–(3.49), we can obtain
Now, let us bound
Combining the aforementioned inequalities (3.50)–(3.55) and (3.13) and using the Hölder inequality in time variable, we conclude that
Then, bounding
Hence, by the Gronwall lemma,
3.3 Global-in-time closure of the estimates
In this section, we will complete the proof of (3.1). Assuming that
and
These assumptions combined with (3.40) imply that
which gives that
Then, we choose
while (3.57), (3.59), and the fact that
yield
which implies that
Next, assuming that for a suitably large (universal) constant
then set
Thus, we can derive that
Then, we choose
Hence, we end up with
From (3.61)–(3.63), we see that assumption (3.2) is recovered, from which we see that assumptions (3.58) and (3.59) are recovered. This also ensures that
If
Acknowledgement
The authors would like to thank the anonymous referees for their valuable comments and suggestions that help to improve and clarify the article greatly.
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Funding information: The first author was partially supported by Zhejiang Province Science Fund (LY21A010009).
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Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.
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Conflict of interest: The authors state no conflict of interest.
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Data availability statement: No data were used for the research described in the article.
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