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Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Exterior Domains

We are concerned with the global existence of classical solutions to the barotropic compressible Navier-Stokes equations with slip boundary condition in a three-dimensional (3D) exterior domain. We demonstrate that the classical solutions exists globally in time provided that the initial total energy is suitably small. It is worth noting that the initial density is allowed to have large oscillations and contain vacuum states. For our purpose, some new techniques and methods are adopted to obtain necessary a priori estimates, especially the estimates on the boundary. Moveover, we also give the large-time behavior of the classical solutions what we have gotten.


Introduction
The viscous barotropic compressible Navier-Stokes for isentropic flows reveal the principles of conservation of mass and momentum in the absence of exterior forces: ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u) − µ∆u − (µ + λ)∇divu + ∇P (ρ) = 0, (1.1) where (x, t) ∈ Ω × (0, T ], Ω is a domain in R 3 , t ≥ 0 is time, x is the spatial coordinate. ρ ≥ 0, u = (u 1 , u 2 , u 3 ) and P (ρ) = aρ γ (a > 0, γ > 1) are the unknown fluid density, velocity and pressure, respectively. The constants µ and λ are the shear viscosity and bulk coefficients respectively satisfying the following physical restrictions: This paper can be regarded as a continuation of our previous work [6]. Unlike the previous article, we study the global existence of classical solutions of (1.1) in the exterior of a simply connected bounded domain in R 3 rather than in the inner bounded domain. More precisely, three-dimensional problem of the system (1.1) will be investigated under the assumptions: The domain Ω is the exterior of a simply connected bounded domain D in R 3 , i.e. Ω = R 3 −D and its boundary ∂Ω is smooth. In addition, the system (1.1) is discussed subject to the given initial data ρ(x, 0) = ρ 0 (x), ρu(x, 0) = ρ 0 u 0 (x), x ∈ Ω, (1.3) and slip boundary condition u · n = 0, curlu × n = 0 on ∂Ω, (1.4) with the far field behavior u(x, t) → 0, ρ(x, t) → ρ ∞ ≥ 0, as |x| → ∞, (1.5) where n = (n 1 , n 2 , n 3 ) is the unit normal vector to the boundary ∂Ω pointing outside Ω, ρ ∞ is a given constant. Our purpose is to show that such global classical solutions indeed exists provided that the initial energy is sufficiently small. It is worth noting that in this paper the initial density is allowed to have possibly large oscillations with constant state as far field which could be either vacuum or non-vacuum. Since Ω is no longer bounded, there are some distinguish differences from the previous work [6].
There are a considerable number of literatures on the large time existence and behavior of solutions to (1.1). The one-dimensional problem has been studied extensively, see [18,27,39,40] and the references therein. For the multi-dimensional case, the local existence and uniqueness of classical solutions are known in [35,41] in the absence of vacuum and recently, for strong solutions also, in [7-9, 28, 37] for the case that the initial density need not be positive and may vanish in open sets. The global classical solutions were first obtained by Matsumura-Nishida [33] for initial data close to a non-vacuum equilibrium in H 3 . In particular, the theory requires that the solution has small oscillations from a uniform non-vacuum state so that the density is strictly away from vacuum. Later, Hoff [19][20][21] studied the problem for discontinuous initial data. For the existence of solutions for arbitrary data, the major breakthrough is due to Lions [31] (see also Feireisl [14,16]), where the global existence of weak solutions when the exponent γ is suitably large are achieved. The main restriction on initial data is that the initial total energy is finite, so that the density vanishes at far fields, or even has compact support. However, little is known on the structure of such weak solutions, particularly, the regularity and the uniqueness of such weak solutions remain open. Here we refer the reader to the books by Lions [31], Feireisl [14], Novotný & Straškraba [36] and Giga & Novotný [47] for further details.
The choice of the boundary conditions is more complex when one discusses the system (1.1) in a region with boundaries. One of the frequent choices is the homogeneous Dirichlet boundary condition u = 0 on ∂Ω in the case when ∂Ω represents a fixed wall. This condition was formulated by G. Stokes in 1845, which is also called noslip boundary condition expressing the physical fact that a real fluid adheres to ∂Ω. However, In the case where the obstacles have a rough boundary, the no-slip boundary condition is no longer valid (see for instance [42]). An alternative was suggested by H. Navier even before (in 1824), who proposed the conditions as follows: u · n = 0, (2D(u) n + ϑu) tan = 0 on ∂Ω, (1.6) where D(u) = (∇u+(∇u) tr )/2 is the shear stress, ϑ is a scalar friction function, and the symbol v tan represents the projection of tangent plane of the vector v on ∂Ω. We also call it Navier-type slip condition in which there is a stagnant layer of fluid close to the wall allowing a fluid to slip and the slip velocity is proportional to the shear stress. Such boundary conditions can be induced by effects of free capillary boundaries (see [3]), or a rough boundary as in [1,25], or a perforated boundary, which is then called Beavers-Joseph's law, see [5,38], or an exterior electric field as in [10]. The Navier-type slip condition has been employed in a wide range of problem, including free surface problems (see e.g., [43]), turbulence modeling (see e.g., [17]) and inviscid limits (see e.g., [13,46]). It is a remarkable fact that when ∂Ω is of constant curvature, (1.4) is equivalent to (1.6) with ϑ = κ (see [6]), where κ is the corresponding principal curvature of ∂Ω. One can see [13,21,34,43,44,46] for the studies of Navier-Stokes equations with Navier-type slip boundary condition. However, as far as we know, there are few research works on the compressible Navier-Stokes equations with slip boundary condition in an unbounded domain, except for the results of [21] and [15], which give the global existence of weak solutions in a half-space and the weak-strong uniqueness property in the class of finite energy weak solutions in an unbounded domain respectively. The outline of this paper is as follows. First, we will give our main results at the end of this section. In Section 2, some notations, known facts and elementary inequalities needed in later analysis are prepared for our discussion. Section 3 and Section 4 are devoted to deriving the necessary a priori estimates on classical solutions which can guarantee the local classical solution to be a global classical one. Finally, we will prove the main results, Theorems 1.1 and 1.2 in Section 5.
Before stating the main results, we explain some functional spaces, notations and conventions used throughout this paper. For a positive integer k and 1 ≤ q < +∞, the standard homogeneous Sobolev spaces are denoted as follows: . For simplicity, we denote L q (Ω), D k,q (Ω), D k (Ω), W k,q (Ω) and H k (Ω) by L q , D k,q , D k , W k,q and H k respectively, and set For two 3 × 3 matrices A = {a ij }, B = {b ij }, the trace of AB is presented by A : B, that is, Let (ρ 0 , u 0 ) and ρ ∞ be given by (1.3) and (1.5) respectively. The initial total energy of (1.1) is defined by where Now we give our first result, which indicates that the existence and large-time behavior of global classical solutions to the problem (1.1)-(1.5).
The following theorem shows the large-time behavior of the gradient of the density when vacuum states appear initially.
Moreover, it also follows from (1.12) 2 and (1.12) 3 that (1.16) due to the following simple fact that Finally, by (1.1) 1 , we have which together with (1.15) and (1.16) shows that the solution obtained by Theorem 1.1 is a classical one.
Remark 1.2 One also can prove the same conclusions of Theorems 1.1 and 1.2 under the following more wide boundary condition: 17) where A = A(x) is 3 × 3 symmetric matrix which has compact support in R 3 and A ∈ W 2,6 satisfying the conditions of Remark 1.3 in [6]. Since the treatments of some additional priori estimates are parallel to those of [6], we omit detailed proofs.
Due to the equivalence of norms ∇v L 2 ≃ D(v) L 2 for any v ∈ D 1 with v·n = 0 and v → 0 as |x| → ∞ (see [12]), similar to the proof of [6, Theorem 1.4], we immediately have the following conclusion. 6 (Ω) is a positive semidefinite 3 × 3 symmetric matrix. Under the same assumptions of Theorem 1.1 expect that the first condition of (1.8) is replaced by the conclusions of Theorems 1.1-1.2 with respect to the slip boundary condition (1.17) (instead of (1.4)) remain valid provided that 2µ + 3λ > 0.
As a direct result of Theorem 1.3, for compressible Navier-Stokes equations (1.1) with Navier-type slip boundary condition (1.6), we give the following conclusion on the global existence and large-time behavior of classical solutions. Corollary 1.4 Assume that ϑ ∈ W 2,6 , the shear viscosity coefficient µ and the bulk one λ satisfy one of the following two conditions: (1) A = ϑI − 2D(n) satisfies the assumption given by Remark 1.2.
We now comment on the analysis of this paper. Indeed, compared with the previous results ( [6]) where we treated the problem in a simply connected bounded domain, since the domain is unbounded, there are two difficulties that we have to overcome: one of them is how to estimate ∇u by means of divu and curlu, the other one is how to control the boundary integrals, especially (see (3.34)), For the former, we establish some necessary inequalities in Section 2 related to ∇u, divu and curlu, see Lemmas 2.5-2.9. For the latter, we introduce a smooth 'cut-off' function defined on a ball containing R 3 − Ω, which not only enables us to eliminate the first derivative in the boundary integral above by using the method in [6] based on the divergence theorem and the fact u = u ⊥ × n (see (3.27)), but also reduces the boundary integral to the integral on the ball. The reader can see (3.34) for details. All these treatments are the key to achieve our purposes in this paper.

Some known inequalities and facts
In this subsection, some facts and elementary inequalities, which will be used frequently later, are collected.
First, similar to the proof of [22,Theorem 1.4], we have the local existence of strong and classical solutions.
The following Lemma can be found in [11].
Lemma 2.2 (Gagliardo-Nirenberg) Assume that Ω is the exterior of a simply connected domain D in R 3 . For p ∈ [2, 6], q ∈ (1, ∞) and r ∈ (3, ∞), there exists some generic constant C > 0 which may depend on p, q and r such that for any f ∈ H 1 (Ω) and g ∈ L q (Ω) ∩ D 1,r (Ω), Generally, (2.1) and (2.2) are called Gagliardo-Nirenberg's inequalities. Next, in order to arrive at the uniform (in time) upper bound of the density ρ, we need the following Zlotnik inequality [48]. Lemma 2.3 Assume that g ∈ C(R) and y, b ∈ W 1,1 (0, T ), and the function y satisfy where ζ is a constant such that Consider the Neumann boundary value problem ∂v ∂n = −f · n on ∂Ω, ∇v → 0, as |x| → ∞, Thanks to [36, Lemma 5.6], we have the following conclusion.
Lemma 2.4 For the system (2.5), one has (1) If f ∈ L q for some q ∈ (1, ∞), then there exists a unique (modulo constants) solution v ∈ D 1,q such that (2) If f ∈ W k,q for some q ∈ (1, ∞), k ≥ 1, then ∇v ∈ W k,q and ∇v W k,q ≤ C f W k,q .

Definition 2.1
Let Ω be a domain in R 3 . If the first Betti number of Ω vanishes, namely, any simple closed curve in Ω can be contracted to a point, we say that Ω is simply connected. If the second Betti number of Ω is zero, we say that Ω has no holes.
The following two lemmas are given in [2,45], more precisely, Theorem 3.2 in [45] and Propositions 2.6-2.9 in [2]. Lemma 2.5 Let k ≥ 0 be a integer, 1 < q < +∞, and assume that D is a simply connected bounded domain in R 3 with C k+1,1 boundary ∂D. (2.6) In particular, for k = 0, we have Lemma 2.6 Suppose that D is a bounded domain in R 3 and its C k+1,1 boundary ∂D only has a finite number of 2-dimensional connected components. Then for an integer In particular, if D has no holes, then The following conclusion is shown in [45, Theorem 3.2].
Lemma 2.7 Let D be a simply connected domain in R 3 with C 1,1 boundary, and Ω is the exterior of D.
Lemma 2.9 Let D be a simply connected domain in R 3 with smooth boundary, and Ω is the exterior of D. For any p ∈ [2, 6] and integer k ≥ 0, there exists some positive constant C depending only on p, k and D such that Proof. First, letting B R {x ∈ R 3 ||x| < R} be a ball whose center is at the origin such thatD ⊂ B R , it follows from (2.1) that there exists some C depending only on D such that every v ∈ {D 1,2 (Ω)|v(x, t) → 0 as |x| → ∞} satisfies for any p ∈ [2,6] which together with the standard Sobolev's inequality gives Then, it suffices to prove (2.9) in the case v · n = 0 since the case v × n = 0 can be handled similarly, where we utilize Lemma 2.6 instead of Lemma 2.5. We introduce a cut-off function η( On the other hand, the standard L p estimate implies that which together with (2.12) leads to (2.13) Hence, for k = 0 and p ∈ [2, 6], by Holder's and Gagliardo-Nirenberg's inequalities, where in the second inequality we have used (2.10). Finally, choosing k = 1 in (2.13) gives where in the last inequality we have utilized (2.14). Combining this, (2.13), Gagliardo-Nirenberg's inequality, and an inductive derivation leads to (2.9) and finishes the proof of Lemma 2.9. Finally, we state the Beale-Kato-Majda type inequality which was first proved in [4,26] when divu ≡ 0 (see also [23] for the case that divu ≡ 0). Lemma 2.10 For 3 < q < ∞, assume that u · n = 0 and curlu × n = 0, ∇u ∈ W 1,q , there is a constant C = C(q) such that the following estimate holds Proof. The proof is similar to that of [6, Lemma 2.7], and we omit it.

Estimates for F , curlu and ∇u
where F is called the effective viscous flux and plays an important role in our analysis. Now we give some a priori estimates for F , curlu and ∇u, which will be used frequently later.

A priori estimates(I): lower order estimates
In this section, Assume Ω is always the exterior of a simply connected bounded domain in R 3 . Choosing a positive real number R such thatD ⊂ B R , one can extend the unit outer normal n to Ω such that In what follows, we will use the convention that C may denote a generic positive constant depending on µ, λ, γ, a,ρ, ρ ∞ , Ω and M , and use C(δ) to emphasize that C depends on δ. We have the following standard energy estimate for (ρ, u) and preliminary L 2 bounds for ∇u and ρ Proof. Multiplying (1.1) 1 by G ′ (ρ) and taking advantage of slip boundary condition (1.4) show that Multiplying (3.4) by u, along with (3.3) and (1.4), we have which together with (2.8) gives (3.2) and finishes the proof of Lemma 3.1.
For σ(t) min{1, t}, we define and The rest of section is devoted to proving the following proposition, which guarantees the existence of a global classical solution of (1.1)-(1.5).
then sup Proof. Proposition 3.2 is a direct consequence of Lemmas 3.5, 3.7 and 3.8 below.
The following Lemmas 3.3-3.8 will be proven under the assumption (3.9).

11)
and Proof. The proof is motivated by [6,19,24]. First, the following two identities will be used more than once: (3.14) Next, for m ≥ 0, multiplying (1.1) 2 by σ mu yields (3.15) We will estimate the three terms in the last equality. It follows from the slip boundary condition u · n| ∂Ω = 0 that u · ∇u · n = −u · ∇n · u on ∂Ω. (3.16) For I 1 , we check that where we have used (3.16), (3.13) and the relation For the boundary term in the last of (3.17), one has by (2.11) ∂Ω σ m P u · ∇n · uds ≤ C(ρ)σ m u 2 As a result, together with (3.2), Similarly, by (3.13), On the other hand, it follows from Hölder's inequality, (2.11), and (2.16) that Hence, (3.20) For I 3 , a straightforward computation along with (3.14) gives that (3.21) Together with the estimates of I 1 , I 2 and I 3 above, it follows from (3.15) that Imposing σ muj [∂/∂t + div(u·)] on (3.23) j , summing with respect to j, and integrating over Ω, we obtain where we have utilized (1.1) 1 and the boundary condition u · n = 0.
Next, it follows from the slip boundary condition u · n| ∂Ω = 0 that for u ⊥ −u × n, For the first boundary termJ 1 1 , denoting h u · (∇n + ∇n tr ), we have by (3.16), Lemma 2.11, and Gagliardo-Nirenberg's, Young's, and Hölder's inequalities, where in the last inequality we have used the following key observation due to [6] − (3.29) It follows from (2.11) and (2.16) that (3.31) ∂Ω (u · ∇n · u)F ds ≤ C F L 4 (∂Ω) u 2 For the second boundary termJ 2 1 , Similarly, for the last boundary termJ 3 1 , (3.35) Together with these estimates of the three boundary terms above, and by (3.25), we conclude that (3.36) By (3.14), a direct computation shows that which together with (2.8) yields where in the second inequality we have used Proof. Multiplying (1.1) 2 by u t and integrating over Ω, a direct calculation gives Hence, Therefore, it follows from Gronwall's inequality, Lemmas 2.7 and 3.1 that (3.41) holds.

Proof. By Lemma 3.3, it indicates that
Due to (2.20), (2.18), (2.19) and Lemma 3.6, Finally, we shall give the estimate of the last term on the right hand side of (3.54). By (3.55), it is easy to check that (3.56) Next, we deduce from (2.20), (3.41) and (3.9) that if C 0 ≤ ε 1 , then It is time to derive a uniform (in time) upper bound for the density, which plays a crucial role in deriving all the higher order estimates and thus extending the local classical solution to be a global one. The method we are going to employ can be found in [29] [30].
Lemma 3.8 Let (ρ, u) be a smooth solution of (1.1)-(1.5) on Ω×(0, T ] satisfying (3.9) and ∇u 0 L 2 ≤ M , then there exists a positive constant ε as described in Theorem 1.1 depending only on µ, λ, γ, a,ρ, ρ ∞ , Ω and M such that, Proof. Denote then (1.1) 1 can be rewritten as (3.58) In order to finish the proof, by Lemma 2.3, it is sufficient to check that the function b(t) must verify (2.3) with some suitable constants N 0 , N 1 .

A priori estimates (II): higher order estimates
From now on, we assume that the initial energy C 0 always satisfies (3.61) and the positive constant C may depend on besides µ, λ, ρ ∞ , a, γ,ρ, Ω and M, where g ∈ L 2 (Ω) is given by the compatibility condition (1.10). This section is devoted to some necessary higher order estimates of the smooth solution (ρ, u) which will make sure that the local existence of classical solution can be extended globally in time. The methods used are mainly from [30]. For completeness, we still write it down in detail.
Hence, by Lemmas 4.1 and 4.2, which gives T 0 P tt 2 L 2 dt ≤ C. One can deal with ρ t and ρ tt similarly. Thus (4.21) is proved.