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The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, I: 3D Euler 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 first part of our three papers, we will confirm this physical phenomenon for the compressible inviscid fluids by obtaining the exact lower and upper bound on the density function.

Suppose that the movement of fluid in the ball is governed by the 3D compressible isentropic Euler * Gang Xu (gxu@ujs.edu.cn) and Huicheng Yin (huicheng@nju.edu.cn) are supported by the National Natural Science Foundation of China (No.11025105, No.11571177) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
The purpose of this paper is to prove the following theorem.
In the following, we give some remarks on the above theorem.
for large t can be expected by the conservation of mass.

Remark 1.2. When the fluid is governed by the compressible Navier-Stokes equations and the cor-
responding boundary condition is given by u(t, x) = R ′ (t)x R(t) for (t, x) ∈ ∂Ω 0 , the corresponding result was obtained in [25]. The case for a rarefied gas in Ω 0 governed by the Boltzmann equation was also obtained in [26]. Remark 1.3. If the initial density contains vacuum, the local well-posedness of the compressible Euler system have been extensively studied, cf. [3], [7][8], [11][12], [16], [18] and the references therein. In general, classical solutions will blow up in finite time as shown in [3] and [23]. For the problem considered in this paper, the vacuum is the time asymptotic state so that the classical solution exists globally in time. Remark 1.4. When the initial velocity u 0 (x) is close to a linear field, the authors in [10] and [19] proved the global existence of smooth solution to the Cauchy problem of the compressible Euler system. And this is different from the case considered in this paper. Remark 1.5. If the ball S 0 is pulled outwards rapidly, namely, when the number L is large, part of the region inside the ball may become vacuum in finite time (see [6] and so on). Remark 1.6. Note that (ρ(t, x),û(t, x)) = ( 1 R 3 (t) , Lx R(t) ) with R(t) = 1 + Lt is a special solution to (1.1)- (1.2). In fact, Theorem 1.2 gives the stability of this special solution. In addition, the smallness of L in Theorem 1.1 is only used to prove the local existence of the solution to (1.1)- (1.2) and to obtain the smallness of (ρ(1, x) − 1, u(1, x)) in (6.11).
Remark 1.8. The linearized operator of the quasilinear wave equation (1.10) around the special expanding solution has the approximate form of On the other hand, if one considers the Cauchy problem of (1.1) for initial data as a small perturbation of a uniform constant density ρ 0 and velocity (0, 0, q 0 ), that is, (1.18) does not fulfill the "null-condition" introduced in [4] and [13]. Therefore, according to the results obtained in [1][2], [5], [20] and [27], classical solution to (1.18) (1.10) together with a fixed wall condition comes from the rarefaction property of fluid.
We now give some remarks on the proof of Theorem 1.2. Since the local solvability of problem (1.10) together with (1.16)-(1.17) is known as long as the vacuum does not appear, cf., for example [17], the proof of Theorem 1.2 is based on the continuation argument.
First of all, note that the linearized operator cf. (3.2), is different from the corresponding one in [24] which is The key ingredients in the analysis for the global existence and pointwise estimate are to obtain some weighted energy estimates by choosing some appropriate multiplier and anisotropic weights. For this, we need to 1. show that the solution does not contain vacuum both on the boundary and inside the region; 2. obtain the different time decay rates of the density and velocity of the solution when it tends to vacuum state at infinite time; 3. fully use the Neumann-type boundary condition (1.16) on Φ by applying the material derivative ∂ r on the solution, and estimate the radial derivatives and angular derivatives of Φ, together with some weighted Sobolev interpolation inequalities given in [15].
The rest of the paper is organized as follows. In the next section, we will give some basic properties of the background solution and some preliminary weighted Sobolev interpolation inequalities. In Section 3, we will reformulate problem (1.10) together with (1.16)-(1.17) by decomposing its solution as a sum of the background solution and a small perturbation so that its linearization can be studied clearly. In Section 4, we will establish a uniform weighted energy estimate for the corresponding linear problem, where an appropriate multiplier is constructed. In Section 5, the uniform higher order weighted estimates ofΦ are obtained by the analysis on the radial derivatives and angular derivatives ofΦ, where the domain decomposition technique is applied. In the last section, we complete the proof of Theorem 1.2 by applying the Sobolev embedding theorem and the continuation argument, and then Theorem 1.1 follows from Theorem 1.2 and the local existence result on the problem (1.1)-(1.2).

Background solution and some preliminaries
In this section, we analyze the background solution to (1.10) with (1.16)-(1.17) when the initial data (1.17) areΦ In this case, the density ρ(x) and velocity u(t, One can easily check that (2.2) has a solution Next, we include the weighted Sobolev interpolation inequality from [15] that will be used in Lemma 2.4. Lemma 2.1 Suppose s, τ, p, α, β, q, a are real numbers, and j ≥ 0, m > 0 are integers, satisfying is not a non-negative integer.
There exists a positive constant C such that the following inequality holds for all v ∈ C ∞ 0 (R n ):

6)
if and only if the following conditions hold: where T > 0 is a constant, y = (t, x) and |y| = t 2 + |x| 2 , then (i) (2.6) still holds under the restriction (2.5) and (2.7)-(2.9), moreover, the constant C on the right hand side of (2.6) is independent of T .
Proof. (i) The proof is completely parallel to that of Lemma 2.1 (one can check the details in [15]), then we omit it here. . And this completes the proof of the corollary.
Proof. LetẼ be an extension operator defined by Note that H − 1 L < T + LR(T ) and T > 1 hold. This means that for T < t ≤ H − 1 L and 0 ≤ j ≤ 3, In addition, Consequently, one has that for (t, x) ∈ D S , This yields for |α| ≤ 3 Next, we construct the extension operator E starting from the operatorẼ. In terms of the geometric property of D T , it is convenient to use the spherical coordinates. Let x 0 = t + 1 L , and Denoting byũ(s, ω) = u(sω) with ω = (ω 0 , ..., ω 3 ). LetĒ be an extension operator defined by  (−1) k ν j = 1 for k = 0, 1, 2, 3.
Z i f · Z i g for any C 1 smooth functions f and g.
As a direct consequence of Remark 2.1, we have the following weighted inequalities which will be used often in Section 5.
, then there exists a generic positive constant C independent of T such that Proof. Let E be the extension operator given in Lemma 2.2. Then we have (i)

whereΦ(t, r) is given in (2.4).
Proof. The quasilinear equation (1.10) is strictly hyperbolic with respect to t. Thus, by the standard Picard iteration as in [17], one can derive that Lemma 3.1 holds.
For later analysis, we use Z-fields to rewrite LΦ as follows: On the lateral boundary ∂Ω of Ω,Φ satisfies In addition, we have the following initial data ofΦ from (1.17) 2) has the following form at the t−axis: which is strictly hyperbolic but degenerate as t → ∞. On the other hand, the operator where the corresponding linear operator is respect to the semilinear wave equations where µ > 0 and α > 0 are suitable constants, there have been extensive works on the global existence or blowup results for different nonlinear function f (u), cf. [9], [21][22] and the references therein. However, for the critical exponent α = 1 and near the critical value µ = 1, there are still some open questions on the blowup or global existence of solutions to (3.7). Here, (3.6) corresponds to the critical case of (3.7) when L = 1 and γ is close to 4 3 .

The first-order weighted energy estimate and reformulation of (3.2)-(3.3)
In this section, we derive the weighted energy estimate of ∇ t,xΦ for the linear part (3.3) together with (3.4)-(3.5). Set satisfy the boundary condition (3.4) and initial data condition (3.5).
constant depending only on the initial data, and δ > 0 is a small fixed constant.
can be obtained as shown in Sections 5 and 6. On the other hand, c 2 (ρ(t)) = γR(t) −3(γ−1) holds. Therefore, in order to guarantee the absence of vacuum for any finite , where the non-negative functions a(t, r) and b(t, r) will be determined later, then In view of the boundary condition (3.4), we have To guarantee I ≥ 0, it requires that on the boundary r = R(t), b(t, R(t)) = La(t, R(t)). In this case, Next, we consider II. To fulfill Thus, combining (4.3) and (4.6) yields On the other hand, byĉ 2 = γR(t) −2(γ−1) , we have It follows from Lemma 3.1 that Finally, we deal with III. In fact, we only need to choose a(t, r) ≡ a(t). In this case, direct computation yields In order to have III > 0, we choose From this, we naturally set Substituting (4.4), (4.8)-(4.9) and (4.13) into (4.2) yields (4.14) This together with Remark 2.2 give (4.1), and it completes the proof of the lemma. Note that the material derivative D t plays a crucial role in the energy estimate (4.1). Then it is necessary that we rewrite equation (3.2) and formula (3.3) by using the operator D t as follows: Especially, in the spherical coordinates, (4.15) has the form Here, we point out that the precise expression ofḟ is useful in the higher order energy estimates ofΦ, cf. Setion 5.

Higher order weighted energy estimates ofΦ
In this section, we will derive the higher order energy estimates ofΦ to (3.1) with (3.4)- (3.5). For this, we need to take care of the difficulties coming from the Neumann boundary condition (3.4), the asymptotic degeneracy of some coefficients in (4.16), and the different decay rates of D tΦ and ∇ xΦ . (3.5), and assume with some δ > 0, Then for sufficiently small ε > 0 and 0 ≤ k ≤ 2, we have , C > 0 is independent of M , and the domains Ω T , S T are defined in Section 4.
In order to prove Theorem 5.1, we will apply the induction on k in (5.3)-(5.4) to establish the following estimates respectively: (i) D t S kΦ and ∇ x S kΦ with S = S l 1 0 Z l 2 (S 0 = R(t)D t ) and 1 ≤ k = l 1 + l 2 ≤ 3 (in this case, all the tangent derivatives of ∇ xΦ up to the third order are estimated, where the tangent derivative means the one of boundary B T ); (ii) D t S 1Φ and ∇ x S 1 ∇ xΦ with S 1 = r∂ r (in this case, together with the case k = 1 in (i), all the second order derivatives ∇ 2 t,xΦ are analyzed); (iii) D t SS 1Φ , ∇ x SS 1Φ , D t S 2 1Φ and ∇ x S 3 1Φ (in this case, together with the case k = 2 in (i), all the estimates of third order derivatives ∇ 2 t,xΦ are given); and ∇ x S 3 1Φ (in this case, together with the case k = 3 in (i), all the fourth order derivatives ∇ 4 t,xΦ are estimated). These estimates will be given in Lemma 5.2-Lemma 5.5 respectively. Firstly, we establish the tangent derivative estimates ofΦ under the suitable induction assumption. Set S 0 = R(t)D t and S m = S l 1 0 Z l 2 with m = l 1 + l 2 , which are tangent to the boundary ∂Ω. Then we have Especially, for m = 0, the following estimate holds Remark 5.2 It is noted that the normal derivatives ofΦ are included on the right hand side of (5.5) and (5.6), which implies that we have not obtained the close estimates on the tangent derivative estimates oḟ Φ. However, since the coefficients of normal derivatives ofΦ in (5.5) are small, then together with some subsequent normal derivative estimates, we can derive (5.3).
This, together with Theorem 4.1 and (3.5), yields We divide the estimate (5.5) into three cases: We now derive an explicit representation of LS m 0Φ for later use. By direct computation, we have By induction, for 1 ≤ m ≤ 3, we obtain where B 2mΦ contains the (m+1)−th order (the highest order) derivatives ofΦ, but B 1mΦ only includes the terms ∇ α t,xΦ with |α| ≤ m (the lower order derivatives ofΦ) and the term ∇ m+1 t,xΦ with small coefficients.
In addition, from equation (4.15), we have that for 0 ≤ m ≤ 3 where Based on the above preparation, we now treat Ω T LS m 0Φ · MS m 0Φ dtdx on the right hand side of (5.8) as follows. This procedure is divided into five parts.
Note that for C 1 -smooth functions g, one has Then In addition, for m ≤ 3, direct computation yields On the other hand, by the expressions off ij andf 0i , and (5.7) on B T , we observe that Thus, by (5.14) together with (5.13) and (5.15), it follows from the expressions off ij ,f 0i ,f 0 and assumption (5.1) that where we have used It follows from the expressions off ij ,f 0i ,f 0 and (5.1) that This derives that for m ≤ 2 First, we treat the case of Ω T I m 3 · MS m 0Φ dx with m ≤ 2. For m ≤ 2, as in Part 2, it follows from the expression of f i and assumption (5.1) that which yields for m ≤ 2 Next we deal with Ω T I 3 3 · MS 3 0Φ dtdx. Note that the difficult terms in I 3 3 are the those that include the products of third order derivatives ofΦ because there are no related weighted L ∞ estimates in (5.1). For the convenience, we decompose I 3 3 into J 1 and J 2 by using S 2 0 = LR(t)D t + R(t) 2 D 2 t , where only J 2 contains the products of third order derivatives ofΦ. Namely, where and By assumption (5.1) and the expressions off 0i ,f ij ,f 0 , direct computation yields On the other hand, by the expression of f ij , we further decompose J 2 into J 2 = J 21 + J 22 so that only J 22 contains the product terms of third order derivatives ofΦ. More precisely, Finally, we estimate Ω T |J 22 · MS 3 0Φ |dtdx. To overcome the difficulty induced by the lack of weighted L ∞ estimates of |∇ 3 t,xΦ | in J 22 , we will use the interpolation inequalities in Corollary 2.1 and Lemma 2.4. In fact, by (5.1) and the expression of J 22 , it is sufficient to estimate some typical terms in 0Φ |dtdx as follows: .

Direct computation yields
Then it follows from the expressions off ij ,f 0i and (5.1) that which gives that for m ≤ 2, and for m = 3, For m = 1, it follows from the expressions off ij ,f 0i ,f 0 and (5.1) that This yields Therefore, it follows from the expressions off ij ,f 0i ,f 0 and (5.1) that This implies Next, we deal with Ω T |K 2 32 · MZ 2Φ |dtdx. By Hölder inequality and Lemma 2.4, we have for 0 < δ ≤ γ − 1 that (5.50) Combining (5.47) and (5.49) gives (5.51) Next, we consider the case of m = 3. Due to the lack of L ∞ assumptions on the third order derivatives ofΦ in (5.1), we will decompose K 3 3 as follows where K 3 31 is linear with respect to the third and fourth order derivatives ofΦ, K 3 32 contains the products of two third order derivatives ofΦ. K 3 31 and K 3 32 have the following expressions: and Then it follows from the expressions off ij ,f 0i ,f 0 and (5.1) that We now turn to estimate Ω T |K 3 32 ·MS 3 0Φ |dtdx. Again, to overcome the lack of weighted L ∞ estimates of |∇ 3 t,xΦ | in K 3 32 , we use the interpolation inequalities in Corollary 2.1 and Lemma 2.4. In fact, by (5.1) and the expression of K 3 32 , it suffices to estimate the following typical terms in D T |K 3 32 · MZ 3Φ |dtdx: . .

(iii) and (vi) forΦ)
≤ Cε . Since the estimation for this case is similar to the Case (1) and Case (2), we omit the detail. Combining the Cases (1)-(3), we complete the proof of Lemma 5.1. Based on Lemma 5.1, we now derive a series of estimates on the higher order derivatives ofΦ. For Proof of Lemma 5.2. Noting LS 1Φ = S 1 LΦ − 2ĉ 2 ∆Φ, then it follows from Theorem 4.1 that Since S 1 and Z are expressed in terms of , similar to the proofs of (5.43)-(5.45) and (5.47), we have In addition, thanks to (5.4) for the case of S m = S 0 , it follows This, together with (5.62) and Lemma 5.1 for m = 2 and Remark 2.2, yields the following estimate Proof of Lemma 5.3. In order to derive the third order radial derivative estimate, we need to study a higher order boundary condition on B T . Differentiating (4.16) with respect to r and applying (5.7) yield on B T For convenience, we rewrite (5.65) as follows where χ(s) is a smooth cut-off function On the other hand, direct computation yields As in (4.2), we have In view of (5.66) and divergence theorem, we have And it follows from (5.2) that Next we deal with Ω T (S 2 1 − S 1 )ḟ · M(S 2 1 − S 1 )Φdtdx. Similar to (5.12), we have where Similar to (5.14), one has On the other hand, by the expressions off ij ,f 0i and (5.7), direct observation yields on (5.78) Similar to (5.71)-(5.74), we obtain Then, similar to (5.45) and (5.51), it follows Next, we will give the estimates on ∇ 4 t,xΦ in Lemma 5.4-Lemma 5.6.