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

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 Tv and the elastic stress Te . So the total stress tensor Ttotal 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 fluids. Many viscoelastic fluids, 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 fluid 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 fluids. The readers will notice that there are many kinds of viscoelastic fluid models, in which the elastic stress obeys a constitutive law of differential 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 fluids, we start from the following energy dissipation law: (x, t) ∈ Ω  ×  ℝ⁺ (Ω⊂ℝ³),

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 1/2ρ∣u∣², the internal energy ω(ρ) depending on the fluid density ρ, the elastic potential energy 12c2ρ|F|2 and the electric energy 1/2∣∇ϕ∣². The above constant c > 0 represents the speed of elastic wave propagation. For simplicity, we have assumed:

1. The electric energy is only generated by the electrostatic field E_s≔ −∇ϕ;

2. The dissipation is only caused by the fluid viscosity;

3. 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 Ttotal and how does it depend on other unknowns. The problem is tough since the total stress is relatively complicated for viscoelastic fluids. However, using an energetic variational approach based on the energy dissipation law (1.1), the authors can derive the motion equation [46]:

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 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 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 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 constant θˉ . So the total energy Etotal=K+I−θˉS . There is no external force which means that W=0 . By setting the total energy Etotal and the dissipation △, like (1.1), and applying the energetic variational approach as done in [46], one can derive the closed system (1.4). Reversely, by multiplying Eqs. (1.4)₁, (1.4)₂ and (1.4)₃ by ω′(ρ)−ϕ, u and c2ρF , respectively, summing them up and then integrating over Ω, where integrating by parts under the conditions of u∣_∂Ω=0 and ϕ∣_∂Ω=0 or ∇ϕ·ν∣_∂Ω=0 (ν-unit outward normal to ∂Ω), one can back to the energy dissipation law (1.1). In this sense, the system (1.4) is thermodynamically consistent. Moreover, all the constitutive relations for stresses satisfy the principle of material objectivity or frame indifference [1,42,49]. This principle requires that the stress results only from deformations and is unaffected (excepted for orientation) by rigid rotations of the body. For example, this restriction requires that the elastic stress T˜ satisfies

where Q:R→R3×3 and SO(3) is the 3 × 3 special orthogonal group (means that QQT=I and detQ=1) . It is easy to verify that Te=c2ρFFT in (1.3) satisfies 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=ℝ³). 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⊂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, F=F(x,t)∈M3×3 (the set of 3 × 3 matrices with positive determinants) denote the density, the velocity and the deformation gradient of viscoelastic electrical conducting fluids, 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)>0 for ρ>0. Two constant viscosity coefficients μ and λ satisfy the usual physical constraints μ>0 and 3λ+2μ⩾0. In the motion of fluids, we use ρˉ>0 to model a constant background charge distribution. Without loss of generality, we assume

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 B˙2,∞−3/2(R3) due to L1⊂B˙2,∞−3/2 , 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 H²(ℝ³) 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^1,q(ℝ³) (q > 3). When the density is constant, the system (1.9) will become the incompressible viscoelastic fluid equations. For the incompressible problems, the readers can refer to [4,12,19,20,24,25, 26,27,30,31,35] and the references cited therein.

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 F but the deformation φ=X(x, t) − x introduced in Section 2). Given two important relations (2.4) and (2.6), it suffices 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)₄ provide an effective connection between the electrostatic potential ϕ and the deformation φ, say, ∆ϕ=∇·φ+O (∣∇φ²∣), 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 different difficulties will appear under different 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 effective, 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 > 0. The relation a ∼ b means that a ≲ b and b ≲ a. We denote the gradient operator ∇=∂x=(∂x1,∂x2,∂x3)T and ∇j:=∂xj (j=1, 2, 3). We denote the Frobenius inner product of two matrices A, B∈R3×3 by A:B:=∑i,j=13AijBij . Particularly, |A|2=A:A . The usual Sobolev spaces are denoted by H^k=W^k,2 (k=1, 2, ...) equipped with the norm ∥⋅∥Hk . The usual Lebesgue spaces are denoted by L^p (1 ⩽ p ⩽ ∞) equipped with the norm ∥⋅∥Lp . We always write ∥⋅∥=∥⋅∥L2 for brevity. The spaces involving time L^p([0, T];Z) denote all the measurable functions f:[0, T] → Z with the norm ∥f∥Lp([0,T];Z):=(∫0T∥f(t)∥Zpdt)1/p<∞ for 1 ⩽ p<∞. The spaces involving time C([0,T];Z) denote all the continuous functions f:[0, T] → Z with the norm ∥f∥C([0,T];Z):=max0⩽t⩽T∥f(t)∥Z<∞ .

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.

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].

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 ∇u .

Next, we construct the dissipation estimate for ∇ut .

The following estimate is very important since it gives the estimate independent of ∇²u for the electric field −∇ϕ.

It is necessary to derive the following estimate so that the energy located under the time derivative contains ∇u .