The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, II: 3D Navier-Stokes equations

We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In this paper, as the second part of our three papers, we will confirm this physical phenomenon for the compressible viscid fluids by obtaining the exact lower and upper bound on the density function.

Remark 2. The solution obtained in Theorem 1.1 does not contain vacuum state in any finite time. The corresponding phenomena for the inviscid fluid governed by the Euler equations, and the rarefied gas governed by the Boltzmann equation were also confirmed in [31,34] respectively. Remark 3. Note that the energy dissipation rate is not integrable in time because the ball is pulling outwards continuously. That is, . This is different from those cases studied in the references [1,2,3,6,7,8,13,14,18,19,20,24,29] in various settings.

Remark 4.
If the ball is pulled out rapidly, then it seems that the classical solution will blow up in finite time and vacuum may be formed from physical point of view. Hence, the assumption on the smallness of |R (k) (t)| (1 ≤ k ≤ 4) for 0 ≤ t ≤ 1 and the initial perturbation in Theorem 1.1 is necessary for the global existence of the classical solution. Note that, for the compressible Euler flow, if the gas is rapidly expanding, then the vacuum state forming in finite time is confimed in [4].
, hx R(t) ) with R(t) = 1 + ht is a special solution to (1)-(2) when the initial-boundary value condition (3) is neglected. Then in the case of small perturbation, the asymptotic stability of solution (ρ(t, x), u(t, x)) to (ρ(t, x),ū(t, x)) is a direct consequence of Theorem 1.1. Now let us review some related works. There have been extensive studies on the global existence and behavior of solutions to the Navier-Stokes equations in various settings. For results in one-dimensional case, see [9,26,32,33] and the references therein. For multi-dimensional problems with constant viscosity coefficients, known results include the local existence of classical solution in [25] in the absence of vacuum, and the local existence of strong solutions in [1,2,3] when the initial density may vanish in some open sets. The global existence of classical solution was first obtained in [22,23] for the initial data close to a non-vacuum state and then these results were generalized in other settings, for example in [5,27]. Moreover, the author in [10,11,12,13] studied the global existence with discontinuous initial data. Recently, for the case when the initial density may vanish in some region, under the smallness assumption on the total energy, the authors in [14] established the global existence and uniqueness of classical solutions. For large initial data with the finite total energy, the global existence of weak solutions was established by P.L. Lions in [20] (see also [6,8], [15], [24]) for different assumptions on the adiabatic constant γ. In addition, the authors in [7] obtained the global existence of weak solution to the compressible barotropic Navier-Stokes system in a time dependent domain with slip boundary condition. For the equations (1)-(2) (or including the energy equation) with the viscosity coefficients depending on the density, existence of weak and classical solutions to the initial value problem and the initial-boundary value problem have also been extensively studied, cf. [16,17], [19], [21], [28], [32], [33] and the references therein.
Finally in the introduction, let us give some discussion on the proof of Theorem 1.1. In order to prove Theorem 1.1, we will take a coordinate transformation of the variable (t, x) so that the moving domain Ω becomes a fixed cylindrical domain [0, +∞) × S 0 under the new coordinates (τ, y) = (t, x R(t) ). Correspondingly, the system (1)-(2) becomes a system consists of a first order hyperbolic equation for the density ρ(τ, y) and three degenerate parabolic equations for the velocity u(τ, y) with degenerate coefficients, cf. the system (9) in § 2. To obtain the local and global existence of the smooth solution to the degenerate system (9), we will choose new unknown functions φ(τ, y) = R 3 (τ )ρ(τ, y) and v(τ, y) = u(τ, y) − R (τ )y instead of the density ρ and the velocity u so that the uniform positive lower and upper bounds of φ can be derived. The proof is then based on a delicate weighted energy method. For obtaining the weighted energy estimate on (φ, v), some suitable weights will be chosen for the systems (10)- (11) in § 2 and (51)-(52) in § 3. In particular, for the system (51)-(52), we obtain a uniform weighted energy estimate on (φ, v) by careful estimation and involved analysis in the interior and boundary regions separately. The rest of the paper is arranged as follows. In § 2, we prove the local existence of the solution to (1)- (2). Although its proof is standard, we still give some detail for readers' easy reference. In § 3, we derive some uniform weighted inequalities by some subtle estimation. Based on the analysis in § 3, the uniform energy estimate will be given in § 4 and then the proof of Theorem 1.1 will be given in the last section.
The following notations will be used throughout this paper: V (τ, ·) dτ , and Dg represents ∂ k g for any k = 1, 2, 3.
2. Local existence. To study the solvability of the problem (1)-(2) with (3), we at first transform the variables (t, x) such that the moving domain Ω becomes a fixed cylindrical domain [0, +∞) × S 0 . For this, set Then the system (1)-(2) in the coordinates (τ, y) ∈ [0, ∞) × S 0 takes the form of From now on, ∇ and div are taken with respect to the variable y.
Theorem 2.1. Under the assumptions of Theorem 1.1, there exists some constant T * > 0 such that system (10)-(11) with (12) has a unique solution (φ, v) which satisfies Remark 6. Since problem (10)-(11) with (12) is equivalent to problem (1)-(2) with (3), by Theorem 2.1 we know that problem (1)-(2) with (3) has a local solution Remark 7. Without loss of generality and based on the smallness on h and the perturbation, we choose T * > 1 so that sup To prove Theorem 2.1, we first state some useful estimates on linear parabolic equations with suitable initial-boundary conditions, cf. [28].

HUICHENG YIN AND LIN ZHANG
We now ready to prove Theorem 2.1.
Proof of Theorem 2.1. To prove Theorem 2.1, we construct an approximate solu- (10)-(11) with (12) by solving the linearized problem (14)- (16). For this, we first construct be the solution to the following heat equation Here the choice of c 0 in (44) and (18) is possible because c 0 depends only on ρ 0 H 3 and u 0 H 3 . LetṼ = V 0 . Then by Lemma 2.3-2.6, there exists a unique strong solution (Φ 1 , V 1 ) to the linearized problem (14)- (16), which satisfies all the estimates in (43). Therefore, an induction argument allows us to construct the approximate solution (Φ k , V k ) for all k ≥ 1: assume that V k−1 has been defined for k ≥ 1, and let (Φ k , V k ) be a unique solution to problem (14)-(16) withṼ = V k−1 . It follows from (43) that there exists a uniform positive constant C depending on c 0 such that for all k ≥ 1 DefineΦ Then we haveΦ k+1 τ Multiplying (46) byΦ k+1 and integrating over S 0 yield d dτ In addition, one has Multiplying (48) byV k+1 and integrating over S 0 , we have that, for small η > 0, Combining (47) with (49) and applying the Gronwall inequality yield Choosing η > 0 and T * suitably small, then Therefore, we conclude that the sequence {(Φ k , V k )} converges to a limit (φ, v) in C([0, T * ], H 2 ) and (φ, v) solves problem (10)-(11) with (12). Moreover, (φ, v) satisfies sup Hence, we complete the proof of Theorem 2.1 for the local existence of the solution to problem (10)-(11) with (12).
3. Some uniform weighted energy estimates. In this section, we will establish some a priori energy estimates on the solution to (10)-(11) with (12). Denote by w = φ − 1,τ = R 2 (t) and s = 3 4 (γ − 1), then it follows from (10)-(11) that for 2h∂τ and We will derive a series of basic energy estimates on (w, v). Set f 0 = − w τ divv and dϕ dτ = 2hϕτ + 1 τ v · ∇ϕ. Then we have Lemma 3.1 (Weight L 2 -estimate of (w, v)). For small h > 0 and any t ≥τ 0 , one has Remark 8. For large t, (53) still holds even if the assumption on the smallness of h is removed. Indeed, this can be easily seen from (57) below.

Proof. Computing
and Then by substituting these estimates and (72)-(73) into (71), and combining with (70), we have here we have used the fact that Integrating (74) over (τ 0 , t) with respect to the variableτ , one has Set η 1 = η 2 2 , we then complete the proof of Lemma 3.3.
Here, we point out that the property v ∈ H 1 0 ([0, T ] × S 0 ) plays a crucial role in deriving the lower order energy estimates in Lemma 3.1-Lemma 3.4 by using integration by parts. However, in order to derive the higher order energy estimates on (w, v), we need to cope with the fact that ∇v = 0 may not hold on the boundary [0, T ] × ∂S 0 so that the direct integration by parts for higher order derivatives of v does not work. Motivated by the method used in [23], we will estimate (w, v) in the interior and the boundary regions separately.
We first derive the energy estimates on (w, v) in the interior region.
Next we study the weighted energy estimates on (w, v) near the boundary [0, T ]× ∂S 0 . For this, as in [30], it is convenient to use the spherical coordinates ∇div as ∇div in the spherical coordinates.
We now turn to the estimates on the derivatives of w in normal direction.
For later use in next section, we state the following estimate that can be easily obtained by the standard regularity theory of the second order elliptic equation. 4. Global energy estimates. In this section, based on the estimates obtained in §3, we will establish the global energy estimates of the solution (w, v) to (51)-(52). For this, we first define that for t 2 ≥ t 1 ≥τ 0 and k = 2, 3, On the other hand, it follows from (149) hτ s ( 1 τ v · ∇v)τ 2 2 dτ ≤ Ch −1 N 2 3 (τ 0 , t).