Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations

Abstract:Westudy the initial-boundary value problems of the three-dimensional compressible elastic NavierStokes-Poisson equations under the Dirichlet or Neumann boundary condition for the electrostatic potential. The unique global solution near a constant equilibrium state inH2 space is obtained.Moreover, we prove that the solution decays to the equilibrium state at an exponential rate as time tends to in nity. This is the rst result for the three-dimensional elastic Navier-Stokes-Poisson equations under various boundary conditions for the electrostatic potential.


Introduction
As is known to all, solids have elastic behaviors so that the deformations happened in solids recover once the stress eld is removed, however, uids possess the viscous property which plays a role of internal frictions to dissipate the kinetic energy of uids. Viscoelastic uids we concern here lie between elastic solids and viscous uids, which ow like uids and demonstrate elastic behaviors. This kind of uids often have more complex microstructures than usual uids (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 ow, an elastic stress will appear to hinder the stretched deformations. In general, there are three kinds of stresses in viscoelastic uids: the hydrostatic pressure P, the viscous stress Tv and the elastic stress Te. So the total stress tensor T total can be expressed as where I is the identity matrix. The viscous stress Tv depends on the rate of strain, however, the elastic stress Te can be only determined by the deformation gradient. The combined stress governs the motion of viscoelastic uids. Many viscoelastic uids, from a microscopic point of view, are composed of a great many of charged particles, and at the macro level, they behave as the electrical conducting uid motion. About more backgrounds, the readers can refer to [23,32,42]. In this paper, we focus on the dynamics of the compressible viscoelastic electrical conducting uids. The readers will notice that there are many kinds of viscoelastic uid models, in which the elastic stress obeys a constitutive law of di erential or integral type, cf. [18,21,42].
However, we shall be devoted to studying the third type, i.e., the elastic stress depends only on the current deformation gradient.
To derive a PDE's model to describe the dynamics of the compressible viscoelastic electrical conducting uids, we start from the following energy dissipation law: (x, t) ∈ Ω × R + (Ω ⊂ R ), where ρ, u, F, ϕ denote the density, the velocity, the deformation gradient and the electrostatic potential, respectively. Here the total energy E total per unit volume includes the kinetic energy ρ|u| , the internal energy ω(ρ) depending on the uid density ρ, the elastic potential energy c ρ|F| and the electric energy |∇ϕ| . The above constant c > represents the speed of elastic wave propagation. For simplicity, we have assumed: (1) The electric energy is only generated by the electrostatic eld Es := −∇ϕ; (2) The dissipation is only caused by the uid viscosity; (3) The viscosity is chosen to follow Newton's law of viscosity.
Here the assumptions (2) and (3) imply that the dissipation (entropy production) per unit volume is equal to µ|∇u| + (µ + λ)|∇ · u| , where µ and λ represent the shear and bulk viscosity coe cients, respectively. The key point of the modeling is to derive the motion equation. The total force is the combination of the pressure, the viscous stress, the elastic stress and the Coulomb force due to the electrostatic eld. It is natural to use the momentum conservation law to write down a PDE: where ⊗ denotes the tensor product. Next one should gure out what is the total stress T total and how does it depend on other unknowns. The problem is tough since the total stress is relatively complicated for viscoelastic uids. However, using an energetic variational approach based on the energy dissipation law (1.1), the authors can derive the motion equation [46]: (ρu) t + ∇ · (ρu ⊗ u) = ρ∇ϕ − ∇P + µ∆u + (µ + λ)∇∇ · u + c ∇ · (ρFF T ), (1.2) from where one can know that Tv = λ div uI + µD(u), Te = c ρFF T , D(u) = [∇u + (∇u) T ] (1.3) 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 F and the Poisson's equation for ϕ, one can obtain the following closed system [46]: 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 indi erence. To illustrate those two points, we simply recall the derivation of the system (1.4). Based on the rst and second laws of thermodynamics, by assuming that the ow is isothermal and not a ected by external forces, we can deduce the formal energy dissipation law as follows. It holds that where the symbols K, I, W, Q, S and are the kinetic energy, the internal energy, the work of external forces, the heat, the entropy and the entropy production (or dissipation), respectively. Assume that the absolute temperature θ values a positive constantθ. So the total energy E total = K+I−θS. There is no external force which means that W = . By setting the total energy E total and the dissipation , like (1.1), and applying the energetic variational approach as done in [46], one can derive the closed system (1.4 [1,42,49]. This principle requires that the stress results only from deformations and is una ected (excepted for orientation) by rigid rotations of the body. For example, this restriction requires that the elastic stress T satis es where Q : R → R × and SO( ) is the × special orthogonal group (means that QQ T = I and det Q = ). It is easy to verify that Te = c ρFF T in (1.3) satis es the relation (1.5). In [46], Tan and the authors of this paper studied the global well-posedness of the three-dimensional Cauchy problem of (1.4) (Ω = R ). As a work of continuity, this paper will deal with the corresponding initial-boundary value problems in dimension three, in which the electrostatic potential shall be assigned to Dirichlet or Neumann boundary conditions. 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 ⊂ R is a bounded region and ν is the unit outward normal to ∂Ω. The unknown variables ρ = ρ(x, t) > , u = u(x, t) ∈ R , F = F(x, t) ∈ M × (the set of × matrices with positive determinants) denote the density, the velocity and the deformation gradient of viscoelastic electrical conducting uids, respectively. The electrostatic potential ϕ = ϕ(x, t) is coupled with the density through the Poisson equation. The pressure P = P(ρ) is a smooth function satisfying P (ρ) > for ρ > . Two constant viscosity coe cients µ and λ satisfy the usual physical constraints µ > and λ + µ . In the motion of uids, we useρ > to model a constant background charge distribution. Without loss of generality, we assumē 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 uid 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 L (R ). Based on the same L 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 L (R ) with the homogeneous Besov spaceḂ − / ,∞ (R ) due to L ⊂Ḃ − / ,∞ , Wu et al. [53] obtained the optimal time-decay rates of arbitrary spatial derivatives, where a pure energy method introduced in [7] was used. Besides, some related results of the system (1.9) can be seen in [11,14,15,41] and the references therein. As for the initial-boundary value problem of the system (1.9), Qian [40] proved that the global-in-time solution exists uniquely near the equilibrium state in H (R ) space and then Chen and Wu [3] showed the exponential decay rates. Hu and Wang [16] also obtained the unique global solution in a lower regularity space, say W ,q (R ) (q > ). When the density is constant, the system (1.9) will become the incompressible viscoelastic uid equations. For the incompressible problems, the readers can refer to [4, 12, 19, 20, 24-27, 30, 31, 55] and the references cited therein.
Without considering the elasticity, the system (1.6) becomes the compressible Navier-Stokes-Poisson equations: (1.10) 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 di cult 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 e ective 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 di culties 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 eld −∇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 eld enhances the decay of the density if adding some additional restrictions to the initial electric eld, cf. [51]. However, we found a very interesting phenomenon. Under the in uence of the elasticity, we can obtain the e ective 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 e ects of elasticity variables (not the deformation gradient F but the deformation φ = X(x, t) − x introduced in Section 2). Given two important relations (2.4) and (2.6), it su ces to solve the equations (2.3) about φ and then one can immediately obtain (ρ, F). Moreover, both the relation (2.6) and the Poisson equation (1.6) provide an e ective connection between the electrostatic potential ϕ and the deformation φ, say, ∆ϕ = ∇ · φ + O(|∇φ |), which plays an important role in deriving the estimates for ∇ϕ as stated in Lemma 3.3. The other point lies in that we can uniformly deal with two kinds of important boundary-value problems: Dirichlet type and Neumann type. Our results are relatively non-trivial since di erent di culties will appear under di erent boundary conditions. Here we make a series of delicate energy estimates (see Lemmas 3.1-3.10), which are all applicable for these boundary conditions. Our treatment is clean and e ective, which shall shed light on similar boundary value problems of other models.
Notation. Throughout this paper, we use a b if a Cb for a universal constant C > . The relation a ∼ b means that a b and b a. We denote the gradient operator ∇ = ∂x = (∂x , ∂x , ∂x ) T and ∇ j := ∂x j (j = , , ). We denote the Frobenius inner product of two matrices A, B ∈ R × by A : Particularly, |A| = A : A. The usual Sobolev spaces are denoted by H k = W k, (k = , , ...) equipped with the norm · H k . The usual Lebesgue spaces are denoted by L p ( p ∞) equipped with the norm · L p . We always write · = · L for brevity. The spaces involving time 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.

Main results
In this section, we rst 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.

. Reformulation
We denote x as the current spatial (Eulerian) coordinate and X as the material (Lagrangian) coordinate for uid particles. These two coordinates are connected by the ow map x(X, t) de ned by the following system of ordinary di erential equations: where u(x(X, t), t) is a given velocity eld. Then the deformation gradient F is de ned as When considering it in the Eulerian coordinate, the deformation gradient F(x, t) will be de ned as By the chain rule, we easily prove that F(x, t) satis es the following transport equations: Next, we will reformulate the system (1.6). We introduce the inverse of F by is the inverse mapping of x(X, t). We de ne the quantity which was rst introduced by Sideris and Thomases [45]. Note that the matrix K = (K ij ) is curl free (cf. [31]), so there exists a vector valued function Due to u | ∂Ω = , it holds that φ | ∂Ω = . By (2.1)-(2.2) and the Taylor's expansion, we have where the absolute convergence of the matrix series is insured due to ∇φ H in later discussions. From the fact that ∇ · (ρF T )(t) = ∇ j (ρF jk ) = for all t stated in Lemma A.3, we shall deduce that the i−th component of the vector ρ ∇ · (ρFF T ) as follows: Through the fact ρ det F = for all t in Lemma A.3 and the determinant expansion theorem, we have Next, we de ne the material derivative Applying the divergence ∇· to both sides of (2.3), we have For simplicity, we take P ( ) = . 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 de ne Since ∇φ H in the following a priori estimates, by (2.6), we have ρ .
By the Taylor's expansion and (2.6), we get .

Main results
Our main results are stated in the following: the global existence, uniqueness and exponential decay of solutions. (2.10) Then there exists a suitably small constant δ > such that if Furthermore, there exists a constant α > such that for all t ,

12)
where C > depends only on the initial data.
Finally, we give some remarks.

Energy estimates
For completeness, we rst give the local existence and uniqueness of the strong solution of the problem (1.6)-(1.8) and omit its proof, cf. [22].  and where C > is some xed constant.
To obtain the global-in-time solution of the problem (1.6)-(1.8), we shall make many e orts to derive the a priori estimates. Note that the relations (2.4) and (2.6) (3.1) It su ces to derive the energy estimates of the solution (u, φ, ∇ϕ) to the linearized system (2.8).
We assume that for some su ciently small ϵ > and some T > , We rst establish the dissipation estimate for ∇u.
Next, we construct the dissipation estimate for ∇u t .
The following estimate is very important since it gives the estimate independent of ∇ u for the electric eld −∇ϕ.
It is necessary to derive the following estimate so that the energy located under the time derivative contains ∇u.
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 rst establish the interior estimates.
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 at boundary.
We shall choose a nite number of bounded open sets {Θ j } N j= in R , such that ∂Ω ⊂ ∪ N j= Θ j . The local coordinates y = (y , y , y ) in each open set Θ j will satisfy the conditions as follows: (1) The surface Θ j ∩ ∂Ω is the image of a smooth vector function z j (y , y ) = (z j , z j , z j )(y , y )(eg. take the local geodesic polar coordinate), satisfying |z j y | = , z j y · z j y = and |z j y | δ > , where δ is some positive constant independent of j, j = , , . . . , N.
(2) Any x = (x , x , x ) ∈ Θ j is expressed as 19) where N j i (y , y ) denotes the unit outward normal vector at the boundary point with the coordinate (y , y , ). We shall omit the superscript j in what follows for simplicity without causing any misunderstanding. And we de ne the unit vectors e = zy and e = zy |zy | , with e = (e i ), e = (e i ), i = , , . So, we have N = e × e . By the Frenet-Serret's formula (cf. [5]), there exist smooth functions (α , β , γ , α , β , γ ) of (y , y ) satisfying  .
Proof. It is similar to the proof of Lemma 3.5, so we omit it.

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. . u, ∇φ, ∇ϕ ∇u -.
u t , ∇φ t , ∇ϕ t ∇u t -. - Note that the energy estimates obtained in Section 3 are all of form where B(t) contains some bad large terms. For clarity, we make a table to illustrate that the bad terms B(t) appeared in some row can be controlled by the dissipation D(t) located in other rows after multiplying them by a small constant. For simplicity, we omit the transformation of the domains of integration between x-domain and y-domain since their equivalence in norms due to (3.20)-(3.21). We keep in mind that the summation N j= will be taken once χj occurs in the following. Now, let us do the derivations in detail. Letε > be any small constant, which can be di erent from line to line. Applying (3.8) ×ε + (3.2) + (3.4), we have d dt (u, ∇ · φ, ∇φ, ∇ϕ, u t , ∇ · φ t , ∇φ t , ∇ϕ t ) + (∇u, ∇φ, ∇ϕ, ∇u t ) ϵ ∇ (u, φ) . (4.1) Adding (3.10) ×ε to (4.1), we have d dt (u, ∇u, ∇ · u, ∇ · φ, ∇φ, ∇ϕ, u t , ∇ · φ t , ∇φ t , ∇ϕ t ) + (∇u, ∇φ, ∇ϕ, u t , ∇u t ) ϵ ∇ (u, φ) .  From the above, we have proved the following a priori estimates: where C > is some xed 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.

A Appendix
In the appendix, we list some useful lemmas which are frequently used in previous sections. First, we recall the Poincaré's inequality: (2) If u ∈ W ,p (Ω), denoting the average of u over Ω by uΩ = |Ω| Ω u dx, then u − uΩ L p (Ω) C ∇u L p (Ω) .
The above constant C > 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. The above two positive constants C and C depend only on n, m, k, p, q, α and Ω. A special case: If u ∈ W m,p (Ω) ∩ L q (Ω), then we have ∇ k u L r (Ω) C u α W m,p (Ω) u −α L q (Ω) .
Next, we give some important time-invariant relations for the density ρ and the deformation gradient F. Lemma A.3. If the initial data (ρ , F ) satisfy then ρ and F in (1.6) shall satisfy Proof. The proof can be found in [41].
Then, we shall give the regularity estimates for the Stokes problem: for k = , .