Abstract
We study the initial-boundary value problems of the three-dimensional compressible elastic Navier-Stokes-Poisson equations under the Dirichlet or Neumann boundary condition for the electrostatic potential. The unique global solution near a constant equilibrium state in H2 space is obtained. Moreover, we prove that the solution decays to the equilibrium state at an exponential rate as time tends to infinity. This is the first result for the three-dimensional elastic Navier-Stokes-Poisson equations under various boundary conditions for the electrostatic potential.
1 Introduction
As is known to all, solids have elastic behaviors so that the deformations happened in solids recover once the stress field is removed, however, fluids possess the viscous property which plays a role of internal frictions to dissipate the kinetic energy of fluids. Viscoelastic fluids we concern here lie between elastic solids and viscous fluids, which flow like fluids and demonstrate elastic behaviors. This kind of fluids often have more complex microstructures than usual fluids (eg. water and air). A typical example is a polymer containing a large number of long-chain molecules. When these long-chain molecules are stretched out due to flow, an elastic stress will appear to hinder the stretched deformations. In general, there are three kinds of stresses in viscoelastic fluids: the hydrostatic pressure P, the viscous stress
where
To derive a PDE’s model to describe the dynamics of the compressible viscoelastic electrical conducting fluids, we start from the following energy dissipation law: (x, t) ∈ Ω × ℝ+ (Ω⊂ℝ3),
where
The electric energy is only generated by the electrostatic field Es≔ −∇ϕ;
The dissipation is only caused by the fluid viscosity;
The viscosity is chosen to follow Newton’s law of viscosity.
where ⊗ denotes the tensor product. Next one should figure out what is the total stress
from where one can know that
and the pressure P is determined by an ODE:
In the above, the superscript T denotes the transpose of the matrix. Coupled (1.2) with the continuity equation for ρ, the transport equations for
We say that such a system (1.4) is thermodynamically consistent and the constitutive relations for stresses in (1.4) satisfy the principle of material frame indifference. To illustrate those two points, we simply recall the derivation of the system (1.4). Based on the first and second laws of thermodynamics, by assuming that the flow is isothermal and not affected by external forces, we can deduce the formal energy dissipation law as follows. It holds that
where the symbols
where
To be precise, we will study the initial-boundary value problems of the compressible elastic Navier-Stokes-Poisson equations:
with the initial and boundary conditions
and
Here, Ω⊂ℝ3 is a bounded region and ν is the unit outward normal to ∂Ω. The unknown variables ρ=ρ(x, t) > 0, u=u(x, t) ∈ ℝ3,
Here we make an emphasis on physical meanings of the above boundary conditions. The vanishing velocity u on the boundary can be well understood as the non-slip boundary condition due to the fluid viscosity. The homogeneous Dirichlet-type boundary condition for the electrostatic potential ϕ implies that the boundary is grounded. In addition, the homogeneous Neumann-type boundary condition means that the boundary is well-insulated.
Now we review the history on the non-conducting viscoelastic system corresponding to the equations (1.6):
About the Cauchy problem of the system (1.9), Hu and Wang [13] proved the local existence and uniqueness of the strong solution with large initial data. Later, Hu and Wu [17] generalized the local unique strong solution to the global one in the framework of Matsumura and Nishida [34, 35] and got the optimal time-decay rates of lower-order spatial derivatives via semigroup methods developed in [9,39] under the condition that the initial data belong to L1(ℝ3). Based on the same L1 assumption for the initial data, the optimal time-decay rates of higher-order spatial derivatives were obtained by Li et al. [29] who used the Fourier splitting method (cf. [43,44]). Under the weaker assumption in the sense that one replaces L1(ℝ3) with the homogeneous Besov space
Without considering the elasticity, the system (1.6) becomes the compressible Navier-Stokes-Poisson equations:
For the Cauchy problem of the system (1.10), there are a lot of results, cf. [2,8,10,28,47,50,51,52,54] and the references therein. In a sense, the initial-boundary value problem of the Navier-Stokes-Poisson system is more difficult than its Cauchy problem. For initial-boundary value problems, it is necessary to estimate the boundary integral terms involving higher-order derivatives, however, these terms are often out of control due to the loss of the boundary information of derivatives. This means that one cannot obtain higher-order energy estimates by the usual energy methods. An effective method was introduced by Matsumura and Nishida [36,37] to deal with the initial-boundary value problems of the Navier-Stokes equations. However, the Poisson term ρ∇ϕ brings essential difficulties when considering the initial-boundary value problem of the system (1.10). The reason is that one cannot obtain the dissipation estimates of the electric field −∇ϕ whenever the boundary condition for the electrostatic potential is Dirichlet-type or Neumann-type or other else. Given this point, it is not like the Cauchy problem, where the electric field enhances the decay of the density if adding some additional restrictions to the initial electric field, cf. [51]. However, we found a very interesting phenomenon. Under the influence of the elasticity, we can obtain the effective dissipation estimates of ∇ϕ so that the Dirichlet or Neumann boundary value problems can be solved. Very recently, we learn that the Neumann problem for the system (1.10) has been solved by Liu and Zhong [33], while, the Dirichlet problem is still open.
The novelty of this paper mainly includes two points. One is to develop the effects of elasticity variables (not the deformation gradient
Notation. Throughout this paper, we use a ≲ b if a ⩽ Cb for a universal constant C > 0. The relation a ∼ b means that a ≲ b and b ≲ a. We denote the gradient operator
This paper is organized as follows. We make a reformulation for the original problem (1.6)–(1.8) and list the main results in Section 2. In Section 3, we establish the delicate energy estimates of solutions for the linearized system. In Section 4, we complete the proof of Theorem 2.1 by deducing the a priori estimates from the energy estimates in Section 3. In the appendix, we list some auxiliary lemmas needed in the previous sections.
2 Main results
In this section, we first make a reformulation of the original problem and then state the main results on the existence, uniqueness and large-time behaviors (exponential decay rates) of solutions.
2.1 Reformulation
We denote x as the current spatial (Eulerian) coordinate and X as the material (Lagrangian) coordinate for fluid particles. These two coordinates are connected by the flow map x(X, t) defined by the following system of ordinary differential equations:
where u(x (X, t), t) is a given velocity field. Then the deformation gradient
When considering it in the Eulerian coordinate, the deformation gradient
By the chain rule, we easily prove that
Next, we will reformulate the system (1.6). We introduce the inverse of
where X=X(x, t) is the inverse mapping of x(X, t). We define the quantity
which was first introduced by Sideris and Thomases [45]. Note that the matrix
Due to u∣∂Ω=0, it holds that φ∣∂Ω=0. By (2.1)–(2.2) and the Taylor’s expansion, we have
where the absolute convergence of the matrix series is insured due to
Through the fact
which implies
Next, we define the material derivative
Applying the divergence ∇· to both sides of (2.3), we have
For simplicity, we take P′(1)=1. Thus, using (2.3)–(2.6), we can rewrite (1.6) into the linearized form as
which is subject to the initial and boundary conditions
In the above, we define
Since
By the Taylor’s expansion and (2.6), we get
which infers
2.2 Main results
Our main results are stated in the following: the global existence, uniqueness and exponential decay of solutions.
Theorem 2.1
Let Ω⊂ℝ3 be a bounded domain with
Then there exists a suitably small constant δ0 > 0 such that if
then the initial-boundary value problem (1.6)–(1.8) admits a unique global solution
Furthermore, there exists a constant α>0 such that for all t=0,
where C0 > 0 depends only on the initial data.
Finally, we give some remarks.
Remark 2.1
Since it is hard to impose the boundary conditions for the deformation gradient
Remark 2.2
Given the relations (2.4) and (2.6), we can drop the equations for ρ and
Remark 2.3
In this paper, we try to pursue global-in-time solutions with minimal regularity. In fact, the global H2-regularity solution obtained in Theorem 2.1 can become more regular if improving the smoothness of the boundary ∂Ω and the initial data.
Remark 2.4
We easily deduce the regularity of the electrostatic potential ϕ from Theorem 2.1:
by Lemma A.1 (Poincaré's inequality).
3 Energy estimates
For completeness, we first give the local existence and uniqueness of the strong solution of the problem (1.6)–(1.8) and omit its proof, cf. [22].
Proposition 3.1
Let Ω⊂ℝ3 be a bounded domain with
and
where C1 > 1 is some fixed constant.
To obtain the global-in-time solution of the problem (1.6)–(1.8), we shall make many efforts to derive the a priori estimates. Note that the relations (2.4) and (2.6)
It suffices to derive the energy estimates of the solution (u, φ, ▽ϕ) to the linearized system (2.8).
We assume that for some sufficiently small ϵ>0 and some T > 0,
which implies
We first establish the dissipation estimate for
Lemma 3.1
It holds that
Proof. Integrating the resulting identity u · (L1−R1)−Δφ·(L2−R2)=0 over Ω, integrating by parts and using u∣∂Ω=φ∣∂Ω=0 and (1.8), we obtain
Here we have used the facts
and if ϕ∣∂Ω=0, then
and if ▽ϕ·ν∣∂Ω=0 (implying ▽ϕt·ν∣∂Ω=0), then
Then, noting (2.9), we easily use Hölder’s inequality and Lemmas A.1–A.2 to bound the right-hand side of (3.3) by ϵ∣∣(∇ u, ▽φ, ∇2φ)∣∣2. □
Next, we construct the dissipation estimate for
Lemma 3.2
It holds that
Proof. From
we obtain
We integrate the resulting identity ut·(∂tL1−∂tR1)−Δφt·(∂tL2−∂tR2)=0 over Ω to obtain
Here we have used the facts
and if ϕt∣∂Ω=0, then
and if ∇ϕtt·ν∣∂Ω=0, then
Since
Note that
Then, by (3.6)–(3.7), we can use Hölder’s inequality and Lemmas A.1–A.2 to bound the right-hand side of (3.5) by
The following estimate is very important since it gives the estimate independent of ∇2u for the electric field −∇ϕ.
Lemma 3.3
It holds that
Proof. Multiplying (2.8)1 by −φ and integrating the resulting identity over Ω by parts, by Hölder’s and Cauchy’s inequalities and Lemmas A.1–A.2, we can obtain
Here we have estimated this term in light of different boundary conditions:
(1) If ϕ∣∂Ω=0, by Lemmas A.1–A.2, then
(2) If ∇ϕ·ν∣∂Ω=0, by Lemmas A.1–A.2, then
It is necessary to derive the following estimate so that the energy located under the time derivative contains
Lemma 3.4
It holds that
Proof.By integrating the resulting identity ut × (L1−R1)=0 over Ω, we obtain
By (2.9), Hölder’s inequality and Lemmas A.1–A.2, we easily obtain
Plugging (3.12) into (3.13), by Cauchy’s inequality, we deduce (3.10). □
So far, we have used the above four lemmas to establish the lower-order energy estimates for (u, φ, ϕ). To obtain the estimates of the higher-order derivatives of (u, φ, ϕ), we have to split the estimates into the interior estimates and the estimates near the boundary, cf. [36,37]. We first establish the interior estimates.
Lemma 3.5
Let
Proof.Integrating the identity
where we have computed
Then, by (3.6), Hölder’s and Cauchy’s inequalities and Lemmas A.1–A.2, we can bound the right-hand side of (3.17) by
Note that
By Cauchy’s inequality, we have
Thus we prove (3.13). Integrating the identity
Integrating the identity
where we have computed
Hence, since ϵ≪1, we deduce (3.15) from (3.18). Similarly, we can deduce (3.16) from
Next, we shall construct the estimates of higher-order derivatives of (u, φ, ϕ) near the boundary, where we use a method introduced in [36,37]. The main idea is to straighten the boundary by introducing a suitable coordinate transformation on the restricted domain containing the boundary, thus we can integrate by parts to obtain the desired higher-order energy estimates since the tangential derivatives under new coordinates are always equal to zero on that flat boundary.
We shall choose a finite number of bounded open sets
(1) The surface Θj∩∂Ω is the image of a smooth vector function
where δ is some positive constant independent of j, j=1, 2, ⋅, N.
(2) Any x=(x1, x2, x3) ∈ Θj is expressed as
where
We shall omit the superscript j in what follows for simplicity without causing any misunderstanding. And we define the unit vectors
with
By the Frenet-Serret’s formula (cf. [5]), there exist smooth functions (α1, β1, γ1, α2, β2, γ2) of (y1, y2) satisfying
An easy computation shows that the Jacobian J of the transform (3.19):
We observe that the transform (3.19) is regular through choosing y3 so small that J ⩾ δ/2 from (3.20). Hence, the function ϒ(y)≔ (ϒ1, ϒ2, ϒ3)(y) is invertible. Moreover, the derivatives
where
And we can deduce from (3.21) that
and
Thus, in each Θj, (2.7), (2.8)1 and ∇·φ can be rewritten in the local coordinates (y1, y2, y3) as follows:
where
We denote the tangential derivatives by
where 0 ⩽ k ⩽ 2 and
Next, we will use the following four lemmas (Lemmas 3.6–3.9) to give the energy estimates near the boundary. Then we can obtain the desired higher-order estimates for (u, φ) (Lemma 3.10).
Lemma 3.6
Let
Proof. It is similar to the proof of Lemma 3.5, so we omit it. □
Lemma 3.7
Let
Proof. Integrating the identity
We immediately deduce (3.27) from the above. □
Lemma 3.8
Let
and
where κ+ι=1.
Proof. Applying
In order to eliminate the terms
Multiplying (3.33) by
We easily estimate the right-hand side of (3.34) as follows:
and
Substituting (3.35)–(3.36) into (3.34), we obtain (3.28).
Finally, applying
Lemma 3.9
Let
Proof.. By (2.7) and (2.8)1, we have,
Applying Lemma A.4 to (3.40), we obtain
where we have estimated
From (3.40)1, we deduce,
and
Thus we can deduce (3.37) from (3.41)–(3.43). Similarly, we can obtain
To estimate the term
Then applying Lemma A.4 to (3.45), we obtain
where we have estimated
and
Thus we deduce (3.38)–(3.39) from (3.44) and (3.46). Hence, we complete the proof of Lemma 3.9. □
Now we can use the estimates (3.37)–(3.39) in Lemma 3.9 to derive the desired higher-order dissipation estimates for (u, φ).
Lemma 3.10
Let
Proof. First, we can establish the following estimates:
In fact, applying ∇2 to (2.8)2, multiplying it by ∇2φ and integrating over Ω, we have
Therefore, we obtain (3.50). Similarly we can prove (3.51) and (3.52).
Next, plugging (3.50) ×
We estimate
Plugging (3.54) into (3.53), we deduce (3.47). Similarly, we can deduce (3.48) from (3.51) and (3.38), as well as (3.49) from (3.52) and (3.39). □
4 Proof of Theorem 2.1
In this section, we will establish the a priori estimates based on the lemmas in Section 3. Once we have the a prior estimates, the proof of Theorem 2.1 is natural.
Note that the energy estimates obtained in Section 3 are all of form
where
Now, let us do the derivations in detail. Let
Adding (3.10) ×
Applying [(3.15)+(3.16)] ×
Applying [(3.25)+(3.26)]×
Applying (3.47) ×
Applying (3.29)k=1 ×
Applying (3.48) ×
Applying (3.27) ×
We collect all the terms under the time derivative in (4.8) and then denote all of them by Y(t). Then (4.8) becomes
By Poincaré's inequality (cf. Lemma A.1), we easily check that
Applying Gronwall’s inequality to (4.9), we obtain
By (2.8)1, we easily estimate
Combining (4.10)–(4.12) with (3.1), by (2.8), there exists a functional
such that
From the above, we have proved the following a priori estimates:
Proposition 4.1
(A priori estimates). Let T > 0. Assume that for sufficiently small ϵ>0,
Then we have for any t ∈ [0, T] and some α > 0,
where C2 > 1 is some fixed constant.
Then the local solution given in Proposition 3.1 can be extended to the global one by combining the a priori estimates given in Proposition 4.1 with a standard continuous argument, cf. [35]. The exponential decay rate (2.12) follows from (4.13). Hence, we complete the proof of Theorem 2.1.
In the appendix, we list some useful lemmas which are frequently used in previous sections. First, we recall the Poincaré's inequality:
Lemma A.1
Let Ω be a bounded, connected, open subset of ℝn, with a C1 boundary ∂Ω. Assume 1 ⩽ p ⩽ ∞.
(1) If
(2) If u ∈ W1,p(Ω), denoting the average of u over Ω by
The above constant C > 0 depends only on n, p and Ω.
Proof. The detailed proof can be found in [6]. □
Then we recall the classical Gagliardo-Nirenberg-Sobolev inequality on a bounded domain.
Lemma A.2
Let Ω be a bounded domain of ℝn with a Cm boundary ∂Ω. Assume u ∈ Lq(Ω) and ∇mu ∈ Lp(Ω) with 1 ⩽ p, q ⩽ ∞ and
where
with
The above two positive constants C1 and C2 depend only on n, m, k, p, q, α and Ω.
A special case: If u ∈ Wm, p(Ω)∩ Lq(Ω), then we have
Proof. See [38]. □
Next, we give some important time-invariant relations for the density ρ and the deformation gradient
Lemma A.3
If the initial data
then ρ and
Proof. The proof can be found in [41]. □
Then, we shall give the regularity estimates for the Stokes problem:
Lemma A.4
For the Stokes problem (A.1) on a bounded region Ω with ∂Ω ∈ C3, we have
for k=0, 1.
Proof. Refer to [48]. □
Acknowledgements
Wenpei Wu would like to thank Professor Guochun Wu for several helpful discussions on this topic.
Yong Wang was partially supported by Guangdong Provincial Pearl River Talents Program (No. 2017GC010407), Guangdong Province Basic and Applied Basic Research Fund (Nos. 2021A1515010235 and 2020B1515310002), Guangzhou City Basic and Applied Basic Research Fund (No. 202102020436), the NSF of China (No. 11701264) and Science and Technology Program of Guangzhou (No. 2019050001).
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Conflict of Interest: The authors declare that they have no conflict of interest.
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