Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating

In this paper, we consider the initial value problem for the compressible viscoelastic flows with self-gravitating in \begin{document}$\mathbb{R}^n(n≥ 3)$\end{document} . Global existence and decay rates of classical solutions are established. The corresponding linear equations becomes two similar equations by using Hodge decomposition and then the solutions operator is derived. The proof is mainly based on the decay properties of the solutions operator and energy method. The decay properties of the solutions operator may be derived from the pointwise estimate of the solution operator to two linear wave equations.


1.
Introduction. The compressible viscoelastic flows with self-gravitating in multidimensional space is governed by        ∂ t ρ + ∇ · (ρu) = 0, ∂ t (ρu) + ∇ · (ρu ⊗ u) + ∇P (ρ) = µ 1 ∆u + µ 2 ∇(∇ · u) + ∇ · (ρF F T ) + ρ∇Φ, ∂ t F + u · ∇F = ∇uF, ∆Φ = ρ −ρ, lim |x|→∞ Φ = 0 (1) The variables are the density ρ, the velocity u, the deformation tensor F and the electrostatic potential Φ. Furthermore, P = P (ρ) is the pressure function satisfying P (ρ) > 0 for ρ > 0. The viscosity coefficients satisfy µ 1 > 0, 2µ 1 + nµ 2 > 0. The compressible viscoelastic flows with self-gravitating have strong physical background, we may refer to [20]. For instance in semiconductor devices, it can be used to simulate the transport of charged particles under the electric field of electrostatic potential force. When µ 1 = µ 2 = 0, (1) reduce to self-gravitating Hookean elastodynamics. The three dimension Hookean elastodynamics has been studied and global classical solutions have been established by Hu [2]. The compressible viscoelastic flows with self-gravitating may be viewed as the compressible viscoelastic equations coupled with the self-consistent Poisson equation. Since the 80's of the last century, the local existence, global existence and asymptotic behavior 348 YINXIA WANG AND HENGJUN ZHAO of solutions to the initial value problem for the compressible viscoelastic equations have been investigated extensively, we refer to [3,4,5,6,7,8,19,26]. The global existence with initial data close to an equilibrium state has been established in Besov spaces By Hu and Wang [4]. Several important estimates are also achieved, including a smoothing effect on the velocity, and the L 1 -decay of the density and deformation gradient. The existence and uniqueness of the local strong solutions near the equilibrium have been established in [5]. We also refer to [3] for local large solutions. Local well-posedness of solutions in critical spaces has been obtained by Qian and Zhang [19], provided that the initial density is bounded away from zero. Moreover, global solutions for small data were also established. Recently, Hu and Guo [7] established global existence and optimal decay rates of solutions.
When the density is constant, the compressible viscoelastic equations is reduced to the incompressible viscoelastic equations. Many mathematican's investigated the incompressible viscoelastic equations and lots of interesting results have been established, we may refer to [1,9,10,11,12,13,14,15,17,16,18,27]. Local existence with large initial data and global existence with small initial data were established in [1,12,18]. To obtain global solutions, some important relations are also used to prove that some linear terms are in fact high order terms. Lei, Liu and Zhou [13] find E = F − I satisfies the relation which implies ∇ × E ia a high order term.
Let m = ρu be the momentum, the compressible elastodynamics equations (1) may be written as (2) In this paper, we investigate global existence and optimal decay of classical solutions to (2) with the following initial value Our main goals of this paper are to prove global existence and the optimal decay estimate of solutions to the initial value problem (2), (3). By introducing the Hodge decomposition, (2) may be written as the system (13) whose linear system are decoupled. By the decay estimate of solution operator to (13) and energy method, global existence and the optimal decay estimate of solutions are established. For the details, please refer to Theorem 4.2, 5.1 and Remark 1, 2.
There are also two-folds in our present paper: firstly, it is very difficult to obtain the solutions operator to (2) since (2) has n 2 + n + 1 equations and the unknown functions are coupled. To overcome this difficulty, we introduce the Hodge decomposition and use some special relations (8), (9), then (2) may be written as the system (13) whose linear system are decoupled; Secondly, we clarify the decay property of the solutions operator by investigating the solutions operator to two linear wave equation in (15) and (20). The advantage of this method in the paper is avoiding the complex calculation by using Taylor formula.
The paper is organized as follows. We make rearrangement for the problem by using the Hodge decomposition and some relations in Section 2. In Section 3, we discuss the decay properties of solution operator to (13). Section 4 is devoted to establishing global existence and asymptotic behavior of solutions in odd space dimensions, while Section 5 is devoted to global existence and asymptotic behavior of solutions in even space dimensions.
Thus we have 350 YINXIA WANG AND HENGJUN ZHAO By (6), we have Owing to (9) and (10), we get were L is antisymmetric matrix, which is defined by Let Then using (9) and (10), (5) may be transformed into where L is antisymmetric matrix, which is given by (12) and The following Lemma comes from [7].
Lemma 2.2. Assume that (σ, m, E, ∇Φ) H s is suitably small, then we have 3. Decay properties of solution operator to (13). This section is devoted to study the decay properties of solution operator to (13). To do so, we first consider the linear equations of (13) Noting that the first equation and second equation in (14) is coupled. σ satisfies the following problem By Fourier transform, we havê whereĜ and Ω satisfies the following problem Similarly, we arrive at Γ satisfies the following problem By Fourier transform, we havê whereĜ E − E T satisfies the following problem By Fourier transform, we havê Owing to (16), 19, (21) and (24), we have Taking F −1 to (25), we arrive at where In order to study the decay properties of solution operator G , it is suffice to study the decay properties of G, H, G and H since the solution operator G is given in term of G, H, G and H. Therefore, we firstly study the decay properties of G and H. Taking Fourier transform of (15), we have Lemma 3.1. Let σ be the solution to the problem (15). Then its Fourier imageσ verifies the pointwise estimate for ξ ∈ R n and t ≥ 0, where ω(ξ) = |ξ| 2 1+|ξ| 2 . Proof. The proof may be found in [23]. Here we omit the details.
By the method in [21] and [22], the pointwise estimate (29) together with the solution formula ( (16)) to the problem (28) give the corresponding pointwise estimates forĜ andĤ. The result is stated as follows.
Lemma 3.2. LetĜ andĤ be the fundamental solutions of (15) in the Fourier space, which are given explicitly in (17). Then we have the pointwise estimates The proof may be found in [24]. Here we omit the details.
Similar to the proof of Lemma 3.2, by Lemma 3.3, it is not difficult to get Lemma 3.4. LetĜ andĤ be the fundamental solutions to (23) in the Fourier space, which are given explicitly in (22). Then we have the pointwise estimates of G andĤ for ξ ∈ R n and t ≥ 0, where ω(ξ) = |ξ| 2 1+|ξ| 2 . Lemma 3.5. Let 1 ≤ p ≤ 2, and let k, j and l be nonnegative integers. Assume that ϕ ∈Ẇ j,p H k+l+2 . Then we have Next we state the decay property of solution operator G to (13). By Lemma 3.5 and (26), it is not difficult to derive the following decay estimate for solution operator G . Lemma 3.6. Let 1 ≤ p ≤ 2, and let k, j and l be nonnegative integers. Assume that all norms appearing on the right-hand side of the following inequalities are bounded. Then we have 4. Asymptotic behavior of solutions in odd space dimensions. The purpose of this section is to prove global existence and asymptotic decay of solutions to the initial value problem (2), (3) in odd space dimensions. We need the following lemma for composite functions, which can be found in Lemma 5.2.6 on pp188 of [25].
We apply ∂ ι x (ι ≤ n−1 2 ) to the third equality in (46) and take the L 2 norm, it gives Making use of (41) with j = 0, l = 0 to K 1 , it yields We obtain from (42) with j = 0, l = 1 Finally, we note On one hand, noting that We arrive at from (8) It follows from (11) and (103)-(108) that Due to he above two inequalities and Lemma 2.2, we deduce that In what follows, we estimate the higher order derivatives. Multiplying the first, second equation in (5) by , m, respectively, adding the resulting equations and integrating with respect to x, using integration by parts, we have Thanks to integration by parts and (57), we arrive at Substituting (113) into (112) yields Similarly, we obtain for k = 0, . . . , s. We claim that for any t ∈ [0, T ], it holds 5. Asymptotic behavior of solutions in odd space dimensions. The purpose of this section is to prove global existence and asymptotic decay of solutions to the initial value problem (2), (3) in even space dimensions. We state our result as follows: Theorem 5.1. Let n ≥ 4 be an even integer and s ≥ n 2 + 3. Let p ∈ [1, 2n n+2 ) Suppose that (ρ 0 − x 1 , F 0 − I, m 0 ) ∈ L p , (∂ x1 (ρ 0 − x 1 ), F 0 − I, m 0 ) ∈ H s and put E 1 = (ρ 0 − x 1 , F 0 − I, m 0 ) L p + (∂ x1 (ρ 0 − x 1 ), F 0 − I, m 0 ) H s . Then there is a positive constant δ 1 such that if E 1 ≤ δ 1 , then the problem (2) Moreover, for n 2 ≤ k ≤ s − 2, we have Remark 2. Under the same assumptions of Theorem 5.1, for 2 ≤ q ≤ n, by Gagliardo-Nirenberg inequality, L q decay estimate hold.
Proof. We may extend local solutions to global solutions and prove decay estimate of solutions to the problem (2), (3) by establishing the uniform a priori estimates.
To do so, we define the following norm We omit the details. The proof is completed.