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Global classical solutions to the elastodynamic equations with damping
Journal of Inequalities and Applications volume 2021, Article number: 88 (2021)
Abstract
In this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.
1 Introduction
The Oldroyd model for an incompressible viscoelastic fluid is governed by the following system of equations in \(\mathbb{R}^{3}\):
Here u denotes the fluid velocity, \(F:=(F_{ij})_{3\times 3}\) stands for the deformation tensor, p represents the fluids pressure, and \(\mu >0\) is a viscosity constant. The system (1.1) is also called as the viscoelastic Navier–Stokes equations. It have been studied by many authors (see [1–7] and the references cited therein) since the pioneering work of Renardy [8] and Baranger et al. [9]. There have been several interesting works on the initial value problem of (1.1), for instance, the short time existence of a smooth solution and the global existence of a smooth solution that is initially small have been established in various settings [10–12]. For large (rough) initial data, the global existence of weak solutions to (1.1) has been achieved by [13, 14] in dimension two. Recently, Jiang and Jiang [15] proved the global well-posedness of strong solutions for (1.1) in some classes of large data in dimension three.
The main difficulty in proving the global existence result of the viscoelastic Navier–Stokes equations lies in the equation of the stress tensor F in (1.1), which does not show any dissipative mechanism. In pursuing global weak solutions of (1.1), the authors of [11] proposed the following system as a way of approximating solutions of (1.1):
where \(\nu >0\) is a damping constant. We call (1.2) the viscoelastic Navier–Stokes equations with damping. It is not hard to establish the existence of a global in time weak solution of (1.2) by following the scheme of [16] on the incompressible Navier–Stokes equations. There are many studies about behaviors of solutions to (1.2), for example, partial regularity of weak solutions and forward self-similar solutions of (1.2) have been obtained in [17] and [18], respectively. Chemin and Masmoudi [19] established the global existence of small solutions to the Cauchy problem. Guillopé and Saut [20] also studied the initial-boundary value problem of this modified system (1.2). Finally, we would like to mention that the classical inviscid case of (1.2), i.e., \(\mu =0\), is a challenging problem to show the existence of global classical solutions. In this article, we are interested in studying the Cauchy problem only with the initial deformation that is a small displacement from equilibrium and the initial data have some symmetry.
Motivated by [15, 21], we investigate the global existence of the classical solutions to the following Cauchy problem in \(\mathbb{R}^{3}\):
with periodic boundary conditions
In what follows, we will make a fundamental simplification and assume that
This means that the deformation F has divergence-free columns. It can be obtained by taking the divergence of the second equation in (1.3), which yields the following equation:
From the above transport equation, we can obtain that if \(\operatorname{div}F^{\mathrm{T}}(x,0)=0\), then \(\operatorname{div}F^{\mathrm{T}}=0\) for any \(t>0\).
And we assume that
moreover,
Before stating our main result, we shall introduce some simplified notations in this article:
(1) Sobolev’s spaces and norms:
where \(1< p\leq \infty \) and k are nonnegative integers;
(2) Estimates of the product of functions in Sobolev spaces (denoted as product estimates):
which can be easily verified by Hölder’s inequality and the embedding inequality (see [22, Theorem 4.12]).
Under the assumptions of (1.6) and (1.7), now we can state our main result in the following theorem.
Theorem 1.1
Consider the 3D elastodynamic system (1.3) and (1.4) with initial data satisfying the conditions (1.6) and (1.7). Assume that \((u_{0},F_{0})\in H^{3}(\mathbb{T}^{3})\) with \(\operatorname{div}u_{0}=\operatorname{div}F_{0}^{\mathrm{T}}=0\). Then there exists a small constant \(\epsilon >0\) depending on α such that system (1.3) admits a global classical solution provided that
Without loss of generality, we assume that \(\alpha =(2\pi )^{3}\), as our results do not change qualitatively as \(\nu >0\) is varied, so we set
Obviously, \((0,I)\) is an equilibrium-state solution of the system (1.3). Now, we denote the perturbation quantities by
where I denotes an identity matrix. By (1.7), we have
Then, \((u,U)\) satisfies the perturbation equations:
And the properties of initial data (1.6) and (1.7) persist. Indeed,
and
Setting \(U_{j}:=Ue_{j}\) for \(j=1,2,3\), from the assumption \(\operatorname{div}U^{\mathrm{T}}=0\), we have
For the system (1.11), now we define the following weighted energies which will enable us to achieve our desired estimates:
The energies above are defined on the domain \(\mathbb{R}^{+}\times \mathbb{T}^{3}\).
The rest of this paper is organized as follows. In Sect. 2, we will derive a priori estimates of the higher order energy \(\mathcal{E}_{0}\) and lower order energy \(\mathcal{E}_{1}\), and we only need to consider the highest-order norms in each energy estimate due to the condition (1.13) and the Poincaré inequality. And in Sect. 3, we will prove Theorem 1.1.
2 Energy estimate
First, we will deal with the higher-order energy \(\mathcal{E}_{0}\). It shows that the highest-order norm \(H^{3}(\mathbb{T}^{3})\) of \(u(t,\cdot )\) and \(U(t,\cdot )\) can be bounded uniformly.
Lemma 2.1
Under the condition (1.10), it holds that
Proof
We divide the proof into two parts. Instead of deriving the estimate of \(\mathcal{E}_{0}(t)\) directly, we will first get the uniform bound of \(\mathcal{E}_{0,1}\) which is defined by
First, to get the estimate of \(\mathcal{E}_{0,1}\), we apply the \(\nabla ^{3}\) derivative on system (1.11). Then, we take the inner product with \(\nabla ^{3}U\) in the first equation of (1.11) and also the inner product with \(\nabla ^{3}u\) in the second equation of the same system. Adding them up, we get
where
First, for the term \(M_{1}\), using integration by parts and the divergence-free condition, we have
For the \(M_{2}\), by integration by parts, we get
By using the Hölder’s inequality, we have
Then applying div to \(\text{(1.11)}_{1}^{\mathrm{T}}\), where T represents the transpose of the matrix, we find that
where we have used the condition \(\operatorname{div}U^{\mathrm{T}}=0\). From the regularity theory of elliptic equations [23, 24], thus we get
Substituting the above inequality into (2.6), we obtain that
Thus, from (2.9), we obtain
For the term \(M_{3}\), by using the Hölder’s inequality and product estimates, we obtain
Hence,
For the estimate of \(M_{4}\), using the Hölder’s inequality and product estimates, we have
Hence,
For the last term \(M_{5}\), by integration by parts, we can obtain
By the Hölder’s inequality and product estimates, we have
thus, we can obtain that
Summing up the estimates for \(M_{1}\)–\(M_{5}\), i.e., (2.4), (2.10), (2.11), (2.13), and (2.16), then integrating (2.3) with respect to time, we now get the estimate of \(\mathcal{E}_{0,1}(t)\) which is defined in (2.2) as
Here, we have used the Poincaré inequality to consider the highest-order norm only.
Next, we work with the left term in \(\mathcal{E}_{0}(t)\). Applying the \(\nabla ^{2}\) derivative on the first equation of system (1.11), and taking the inner product with \(\nabla ^{2}\nabla u\), we get
where
As when getting the estimate of \(\mathcal{E}_{0,1}\), we shall derive the estimate of the each term on the right-hand side of (2.18).
For \(M_{6}\), by the Hölder’s inequality and product estimates, we get
where we have used the Poincaré inequality in the last inequality.
Thus, we conclude
The estimate for \(M_{7}\) is almost the same, by using the Hölder’s inequality, we can obtain that
For the last term \(M_{8}\), using integration by parts, we can write this term as
By (1.11)2, we can obtain
First, by using the product estimates, we get
On the other hand, by the Hölder’s inequality and product estimates, we get
where we have used the Poincaré inequality in the last inequality.
Now, applying div to (1.11)2, we get
thus, from the regularity theory of elliptic equations and product estimates, we get
where we have used Poincaré inequality in the last inequality. Thus, we can get
and so
Integrating (2.18) with respect to time, using the estimates of (2.20), (2.21), (2.30), and Young’s inequality, we obtain
Multiplying (2.17) by a suitably large number and adding (2.31), we then complete the proof of Lemma 2.1. □
Next, we want to give the estimate of the lower-order energy \(\mathcal{E}_{1}(t)\) defined in (1.15). The result is given in the following lemma.
Lemma 2.2
Under the condition (1.10), it holds that
Proof
Like the proof Lemma of 2.1, we divide this proof into two parts. Also, we will first get the estimate of \(\mathcal{E}_{1,1}\) which is defined by the following:
Now apply ∇ derivative on system (1.11), take the inner product with ∇U in the first equation of (1.11), and also take the inner product with ∇u in the second equation of (1.11). Adding them up and multiplying by the time weight \((1+t)^{2}\), we get
where
First, the term \(N_{1}\) is equivalent to the following form:
Thus,
hence, by using the Hölder’s inequality, we can get
For the term \(N_{2}\), by using integrating by parts and Hölder’s inequality, we can get
Firstly, by (2.7) and the product estimates, we can obtain that
then putting (2.36) into (2.35), we obtain
Hence, we get
Also for the term \(N_{3}\), by using the Hölder’s inequality and product estimates, we obtain
By (2.36), we have
Thus, we get
Next, we turn to estimating the last term \(N_{4}\). By using integration by parts, we have
thus, by using the product estimates, we get
Hence,
Now, summing up the estimates for \(N_{1}\)–\(N_{4}\), i.e., (2.34), (2.38), (2.40), and (2.43), and integrating (2.33) with respect to time, we get the estimate of \(\mathcal{E}_{1,1}\),
Here, we have used the Poincaré inequality to consider the highest-order norm only.
By an identical argument as in the proof of Lemma 2.1, multiplying the first equation of (1.11) by ∇u and taking the inner product, then multiplying by the time weight \((1+t)^{2}\), we get
where
Using the product estimates, we obtain
hence, we conclude that
Similarly, we have
thus,
For the last term \(N_{7}\), we first rewrite it using integration by parts and have
thus, by using the product estimates, we get
On the other hand, by using (1.11)2, we get
Thus, by the product estimates, we get
Now, by (2.26), we get
thus,
Hence, we obtain
thus, we get
Integrating (2.45) with respect to time, using (2.46), (2.47), (2.55), and Young’s inequality, we get
Now, multiplying (2.44) by a suitably large number and adding (2.56), using the Young’s inequality, we complete the proof of Lemma 2.2. □
3 Proof of Theorem 1.1
Now, we will combine the above a priori estimates of all the energies defined in (1.15) together, and give the proof of Theorem 1.1. First, we define the total energy as follows:
Multiplying (2.1) and (2.32) in the above two lemmas by a different suitable number and summing them up, we can get the following inequality:
for some positive constant \(C_{1}\).
Under the setting of initial data (1.9), there exists a positive constant \(C_{2}\) such that the initial total energy satisfies
According to the standard local well-posedness theory which can be obtained by classical arguments, there exists a positive time T such that for \(C_{3}=C_{1}C_{2}\),
Let \(T^{*}\) be the largest possible time of T satisfying (3.3), it is then left to show that \(T^{*}=\infty \). Noticing the estimate (3.1), we can use a standard continuation argument to show that \(T^{*}=\infty \) provided that ϵ is small enough. We omit the details here. Hence, we finish the proof of Theorem 1.1.
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The authors would like to thank the anonymous referee for invaluable suggestions, which improved the presentation of this paper.
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This work was carried out in collaboration between both authors. XL designed the study and guided the research. ML performed the analysis and wrote the first draft of the manuscript. XL and ML managed the analysis of the study. Both authors read and approved the final manuscript.
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Liu, M., Lin, X. Global classical solutions to the elastodynamic equations with damping. J Inequal Appl 2021, 88 (2021). https://doi.org/10.1186/s13660-021-02608-9
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DOI: https://doi.org/10.1186/s13660-021-02608-9