On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls

In this paper, we are concerned with the global smooth solution problem for 3-D compressible isentropic Euler equations with vanishing density in an infinitely expanding ball. It is well-known that the classical solution of compressible Euler equations generally forms the shock as well as blows up in finite time due to the compression of gases. However, for the rarefactive gases, it is expected that the compressible Euler equations will admit global smooth solutions. We now focus on the movement of compressible gases in an infinitely expanding ball. Because of the conservation of mass, the fluid in the expanding ball becomes rarefied meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. We will confirm this interesting phenomenon from the mathematical point of view. Through constructing some anisotropy weighted Sobolev spaces, and by carrying out the new observations and involved analysis on the radial speed and angular speeds together with the divergence and rotations of velocity, the uniform weighted estimates on sound speed and velocity are established. From this, the pointwise time-decay estimate of sound speed is obtained, and the smooth gas fluids without vacuum are shown to exist globally.

With respect to the initial data excluding vacuum where ρ 0 (x) > 0, (ρ 0 (x), v 0 (x)) ∈ C ∞ (R 3 ), the author in [30] proves that the classical solution (ρ, v) of (1) with (2) blows up in finite time when the initial velocity v 0 is locally supersonic. The analogous result is also extended into the case of 2-D compressible Euler equation (see [27]). In addition, for the 2-D equations (1), when the rotationally invariant data are a perturbation of size ε > 0 of a rest state, S. Alinhac in [2] establishes that the smooth solution (ρ, v) blows up and the lifespan T ε satisfies lim ε→0 ε 2 T ε = τ 2 0 > 0. Recently, under the suitable conditions of initial data, the authors in [6]- [7] and [23] prove that the multidimensional compressible Euler equations generally form the shocks as well as blow up in finite time due to the compressions of gases.
When the initial data (2) contain vacuum, the authors in [9]- [10], [18]- [19], [22] and [26] prove that (1) has a local solution (ρ, v) under some distinct restrictions on the initial data. In the general case, the local solution blows up in finite time (see [32]). Recently, the author in [31] constructs a class of expanding global affine solutions (ρ, v) with physical vacuum condition and finite degrees-of-freedom by solving the related nonlinear ODEs. For different adiabatic exponent γ, authors in [15] and [29] establish the global existence and stability of these solutions with small perturbation, where the physical vacuum boundary condition on the vacuum surface played the essential roles in deriving the weighted energy estimates and establishing the existence of the global solutions.
When the state equation of Chaplygin gases in (1) is given by p = A − B ρ , where A > 0 and B > 0 are constants, so far there are many fundamental results on (1) with (2). For examples, if the Chaplygin gases are isentropic and irrotational, then the 2-D or 3-D compressible Euler equations can be changed into a second order quasilinear wave equation by introducing a velocity potential function Φ with ∇Φ = v. In this case, the related null conditions hold, then it follows from the results in [3], [5] and [20] that the small perturbed smooth solution (ρ, v) of (1) exists globally. In addition, for the full Euler equations (1), when the solution is symmetric and small perturbed, the authors in [11]- [13] and [16] have established the global existence of smooth solution.
When the initial states in (2) force particles to spread out, roughly speaking, v 0 (x) is close to a linear field, which means lim |x|→∞ |v 0 (x)| = ∞, the authors in [14] and [28] have proved the global existence of smooth solution to (1).
Remark 3. For the compressible Navier-Stokes equations, when the initial density may vanish at infinity, and the smallness assumption on the total energy is posed, the authors in [17] establish the global existence and uniqueness of classical solutions. Our Theorem 1.1 is a somewhat analogous result to [17] for the compressible Euler equations.
) with R(t) = 1 + Lt is a special solution to (1) with (4). In fact, Theorem 1.2 below will show 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) with (4) and to obtain the smallness of (ρ(1, x) − 1, v(1, x)).

Remark 5.
For the n-dimensional (n = 2, 3) steady supersonic Euler equations where v 2 > c(ρ) = p (ρ) for n = 2, or v 3 > c(ρ) for n = 3 (in this case, (8) is hyperbolic with respect to x 2 -direction for n = 2 or x 3 -direction for n = 3), motivated by the methods in the proof of Theorem 1.1, we can establish that the global smooth supersonic polytropic gases only with vacuum state at infinity exist in an n-D infinitely long divergent nozzle (see [34]). For the irrotational and isentropic polytropic gases in a 3-D infinitely long divergent nozzle, we have shown this phenomenon in [33]. For the clarity of readers, we give the following pictures on the supersonic flows in nozzles (one can also see more detailed physical backgrounds in [8]):   We now give some illustrations on Theorem 1.1 from the physical point of view. When the ball B 0 is pulled outwards slowly (i.e., the constant L > 0 is small in R 0 (t)), due to the conservation of mass, the gases in the expanding ball will become rarefied and eventually tends to a vacuum state at infinite time, meanwhile there are no appearances of vacuum domains in any part of the expansive ball. This phenomenon can be strictly verified from the pointwise time-decay estimate ρ ∼ 1 R 3 (t) in (6) of Theorem 1.1. For different models (including compressible Euler equations, Navier-Stokes equations and Boltzmann equation, respectively), such physical phenomenon has already been verified as follows: (i) When viscosities of gases are neglected and the gases are isentropic and irrotational, the movement of the gases can be described by the potential flow equation, which is a second order quasilinear wave equation. In [35], we have proved the same results as in Theorem 1.1.
(ii) When viscosities of gases are considered, and the movement of the gases is described by the compressible Navier-Stokes equations, we have established the existence of global smooth gases and obtained the same result as in (6) of Theorem 1.1 (see [36]).
(iii) When microcosmic factors of fluid particles are considered, and the movement of the gases is described by the Boltzmann equation, in [37] we show that the Boltzmann equation has a global solution and the macroscopical density ρ of gases globally satisfies (6).
In this paper, we will remove the restriction of irrotational assumption for (1), and show the global existence and stability in Theorem 1.1.
Let us comment on the proof of Theorem 1.2. Since it follows from [24] that the local well-posedness of problem (9) with (10)-(11) is known as long as the vacuum does not appear, we will use the continuous induction method to prove Theorem 1.2. To achieve this objective, we need to establish the global energy estimates of (σ, u) with suitable weights, where σ is degenerate at infinity time. Inspired by our former paper [35], at first, we choose some suitable multipliers to derive the "weak" weighted energy estimates on (σ,u), where (σ,u) is the solution to the linearized equations of (9) (see (21)- (22) in Section 3), and the "weak" weighted energy estimate is referred to that the resulting weighted estimates on (σ,u) are weaker than the really required energies E T (σ,u) and S T (σ,u) (see their definitions in (27)-(28)) due to the weaker time-decay weights. Secondly, we decompose (21)-(22) into a system of rotations curlu and a coupled degenerate hyperbolic system of (∇σ, divu). To estimate (curlu, ∇σ, divu), we will take the following measures: • Since the main part of the system on curlu is a linear ordinary differential equation (see (43) in Section 5), by choosing a suitable multiplier we can obtain the weighted energy estimates on curlu.
• For the coupled degenerate hyperbolic system of (∇σ, divu) (see (79)-(80) in Section 5), by choosing some multipliers we can establish the uniform weak weighted energy estimates of (∇σ, divu) and its derivatives. The key ingredients in the step are to look for suitable anisotropic weights, and to find the available boundary conditions of higher order derivatives of (∇σ, divu) on the boundary ∂Ω.
• Based on some key observations and delicate analysis on (21)- (22), we can derive the required weighted energy estimates of (∇σ, divu). Indeed, from the linearized momentum equation (22), if the weighted energy estimates ofσ and its derivatives are obtained, then we can take ∇σ as the known quantity. In this case, the main linear part of (22) is regarded as an ODE ofu, and then we can find a new multiplier to re-estimateu. The advantage of this doing is that we can avoid to utilize the integration by parts for the spatial derivatives ofu and overcome the difficulty arisen by the slip boundary condition (10) and by the lack of boundary value ofu on ∂Ω.
• From the resulting estimates on third-order derivatives ofσ, we can re-estimate the derivatives ofu up to second order. Based on this and Sobolev imbedding theorem, we can derive the better decay rate ofu than that in [35] and obtain the estimates of E T (σ,u) and S T (σ,u).
In [35], we assume that the gases are isentropic and irrotational. For this situation, the Euler equations can be simplified to a second order quasi-linear hyperbolic equation of potential Φ, whose linearized part admits the degenerate form we can conveniently derive some neat boundary conditions of ∂ αΦ (|α| ≤ 4). From this, the higher order weighted energy estimates ofΦ can be successfully obtained and further establish an analogous result to Theorem 1.1. However, in the present paper, we have to treat the really Euler equations, which are more involved and difficult for treating the rotations and deriving boundary conditions of higher order derivatives of (σ, u). The authors in [15] establish the global existence of classical solutions to 3-D compressible Euler equations with the expanding physical vacuum boundary, where it plays an crucial role in deriving the weighted energy estimates of solutions by the vacuum boundary value (ρ ≡ 0 on the vacuum surface). Compared with [15], in our paper, the complicated boundary conditions of (∂ α σ, ∂ α u) should be concerned more carefully since only the fixed boundary condition (10) (a linear combination of u 1 , u 2 and u 3 ) is known. Moreover, more delicate observations on the different time-decay properties of radial speed and angular speeds are required. The paper is organized as follows. In Section 2, we derive some estimates on the background solution, and list the Sobolev interpolation inequality and an elliptic estimate of ∇u by divu and curlu. In Section 3, we reformulate problem (9) with (10)-(11) by decomposing its solution as a sum of the background solution and a small perturbation (σ,u). In addition, the required weighted Sobolev norms are introduced in this section. In Section 4, we establish a uniform weak weighted energy estimate for the resulting linear problem, where an appropriate multiplier is constructed. In Section 5, at first, we give a main and basic weighted energy result (Theorem 5.1), whose proof will be completed in the whole Section 5-7. To prove Theorem 5.1 partly, in Section 5, we will decompose the linearized equations (21)-(22) into a system of curlu and a coupled hyperbolic system of (∇σ, divu). Subsequently the higher-order weighted energy estimates of curlu are established. Meanwhile, the uniform zero-order and first-order weak weighted estimates of (∇σ, divu) are derived by delicate analysis on some radial derivatives and tangent derivatives of (∇σ, divu), where the domain composition techniques are applied near and away from |y| = 0 respectively. Based on this, the weak weighted estimates on first-order and second-order derivatives of (σ,u) can be obtained. In addition, due to the lengthy proofs on Lemma 5.3 and Lemma 5.5, we will put them in Section 6. In Section 7, at first we derive the uniform weak weighted energy estimates on the third-order derivatives of (σ,u) in terms of the resulting higher-order boundary conditions of (σ,u). In addition, by utilizing the obtained estimates on the higher-order derivatives, we can get the uniform weighted energy on the lower order derivatives ofu. Synthesizing these conclusions, the weighted energy estimates ofu in E T (σ,u) and S T (σ,u) are derived. In last section, we complete the proof of Theorem 1.2 by applying the Sobolev embedding theorem and the continuation argument, and Theorem 1.1 follows from Theorem 1.2 immediately.
One can directly check that (13) has a solution This yieldsσ(t, r) =ρ . Under the transformation y = x R(t) , we haveû(t, y) =v(t, x) = Ly. Introduce the basis of the smooth vector fields tangent to the sphere S 2 as As in Lemma 4.4 of [21], it follows from direct computation that Lemma 2.1.
Z i f · Z i g for any C 1 smooth functions f and g.
(iv) |Zv| ≤ r|∇ y v| for any C 1 smooth function v, here and below Z ∈ {Z 1 , Next we cite the following two results.
Lemma 2.2 (Gagliardo-Nirenberg Inequality, see [1]). Let Ω be any bounded domain in R 3 with smooth boundary. Then where the constant C > 0 depends only on s and |Ω|.
The proof of Lemma 2.3 can be found in [4], we omit it here.
Proof. (1) is a symmetrizable hyperbolic system if the density ρ is bounded below away from zero. Then by making use of the linearized iteration in Theorem 2.1 (a)-(b) of [24] together with the linear energy estimates in [12], Lemma 3.1 holds.
Corollary 1. There exists T 0 > 0, such that (9) with (10)-(11) has a local solution ). Moreover, following estimates hold We now linearize (9) with (10)- (11). Letσ(t, y) = σ(t, y) −σ(t, y) andu(t, y) = u(t, y) −û(t, y). Then (9) can be reformulated as where On the lateral boundary ∂Ω,u(t, y) satisfies In addition, we have the following initial data of (σ,u) from (11) For the later convenience, we introduce following notation of radial velocityu r and angular velocitiesu z = (u z1 ,u z2 ,u z3 ) This yields     u In order to state our results later on, we now introduce the following anisotropy weighted energy norms with the positive constant µ = 3(γ − 1) and and the constant δ > 0 will be chosen later in a suitable range of values. By the definitions of E T (σ,u) and S T (σ,u), ∇ 2u z admits more rapid time-decay rate than ∇ 2u . Indeed, if S T (σ,u) ≤ Cε 2 holds uniformly for T > 0, then by Sobolev imbedding theorem, |u z | ≤ CεR(t) −µ+ δ 2 and |u| ≤ CεR(t) −µ+δ for any t > 0. Here we point out that the more rapid time-decay rate ofu z thanu will play an important role in deriving the energy estimates of the third order radial derivatives of (σ,u) (one can see more details in Section 7). 4. Weak weighted energy estimate of (σ,u). In this section, we derive the weak weighted energy estimate of (σ,u) for the linear part (21)-(22) with (23)- (24).
, Ω T and Γ T are defined in (27) and (28) respectively. Then we have (23) and (24). Then for 1 < γ < 5 3 , there exist multipliers where C > 0 is a generic positive constant, and 0 < δ < 2 − µ = 5 − 3γ is a small but fixed constant which will be determined later. (29) is necessary due to the following two reasons: On the one hand, to guarantee the positivity of III in (30) can be obtained as shown in Section 8.
where M > 0 is a fixed constant, ε > 0 is sufficiently small, and 0 where C > 0 is a generic positive constant independent of M and T .
From Lemma 4.2, we have got the weak weighted energy estimate ofu since we need to further improve the estimate ofu with the higher time-decay rate by comparing with the norms ofu in (27)- (28). This will be done in Section 7.

5.
Main energy estimates and weak weighted energy estimates on some derivatives of (σ,u). In this section, at first, we give a main and basic weighted energy estimate (σ,u), which will be shown in the whole Section 5-Section 7. On the other hand, we will derive the weak weighted energy estimates on some derivatives of (σ,u).

Remark 7.
Here we point out that the range of γ is reduced from (1, 5 3 ) in Theorem 4.1 to (1, 15 11 ), the reason comes from the requirement of better weighted energy estimates on curlu in Lemma 5.2) (see (47)).
Due to the slip boundary condition (23) and the lack of boundary values of the derivatives of (σ,u), the weighted energy estimates in Theorem 4.1 can not be applied directly to the higher order derivatives of (σ,u). To overcome this difficulty, we will decompose equations (21)-(22) into a system of curlu and a coupled hyperbolic system of (∇σ, divu). For the system of curlu (see (43) below), its main part can be regarded as the ODEs, then we can derive the weighted energy estimates of curlu directly. For the coupled hyperbolic system of (∇σ, divu) (see (79)-(80)), the weak weighted energy estimates in Theorem 4.1 can be applied. Based on Lemma 2.3, we can establish all the weak weighted energy estimates of (∇σ, ∇u). Same idea can be used to treat the second and third order derivatives of (σ,u). To prove Theorem 5.1, we will take the following estimates: (see Lemma 5.2) (in this case, all desired estimates on the derivatives of curlu are obtained) (see Lemma 5.4) (in this case, together with case (i) and (ii), Lemma 2.3 and Lemma 4.2, all the weak weighted estimates on the first order derivatives of (σ,u) are obtained) (iv) Tangent derivatives of divu and ∇σ: Radial derivatives of divu and ∇u: (in this case, together with (i)-(ii) and Lemma 4.2, all the weak weighted estimates on the second order derivatives of (σ,u) are obtained) (v) Second order tangent derivatives of divu and ∇σ: First order tangent derivatives of S 1 divu(0 ≤ l ≤ 1) and S 1 ∇σ: Second order radial derivatives of divu and ∇u: S 2 1 divu, S 2 1 ∇σ. (in this case, together with (i)-(iii) and Lemma 4.2, all the weak weighted estimates on the third order derivatives of (σ,u) are obtained) Later on (iv) and (v) will be established in Lemmas 5.5-5.7 of Sections 5-6 and Lemmas 7.1-7.5 of Section 7 in turn. From (i)-(v), we can eventually complete the proof of Theorem 5.1 in Section 7.
In addition, for reader's convenience, we list the following main boundary conditions on B T of some higher order derivatives of (σ,u), which will be derived in details and utilized later on.
(i) The boundary condition of ∇σ: The boundary condition of divu: The boundary condition of S 1 ∆σ: We next derive the equation system of curlu. Take the curl-operator to (22), and denote by ω = curlu = (ω 1 , ω 2 , Proof. Acting 1 R(t) S k1 0 ∇ k2 on two sides of equation (43) yields Multiplying R(t) 2µ−δ S k1 0 ∇ k2 ω i to (45) i and integrating over Ω T , we have that by the integration by parts In order to keep the positivity of last line in (46), one needs 1 − 2µ−δ 2 > 0. This yields γ < 4 3 + δ 6 . By δ < min{5 − 3γ, γ−1 2 }, we have Thus, it follows from (46) and Lemma (3.1) that Next, we treat the remainder terms in the right hand of (48). Direct computation yields In addition, it follows from (35) that In the end, we estimate the last term . We divide this process into six cases for different k 1 and k 2 .
In this case, Then it follows from (35) and Hölder inequality that This, together with (48)-(50), yields one then has In addition, by (35) and Hölder inequality we arrive at and Then we have
In this case, In addition, it follows from (35) that Analogously to the treatment as in (72), we have Thus, together with (48)-(50), we get Lemma 5.2 for case (5).
In this case, the treatments are very similar to those in cases (4) and (5), so we omit it here.

Lemma 5.3 (Estimates on higher derivatives (S
. Under the assumptions of Theorem 5.1, for 1 ≤ k ≤ 3, then for µ = 3(γ − 1), Proof. Its proof is lengthy and involved, we put it in Section 6.
Lemma 5.4 (Estimates on 1 st order derivatives (∇σ, divu)). Under the assumptions of Theorem 5.1, then for µ = 3(γ − 1), Proof. Acting ∂ i to (21) and computing where At first, we derive the boundary conditions of (∇σ, divu) on B T . Multiplying (22) i by y i and summarizing them from i = 1 to i = 3, we then arrive at To apply the boundary condition (82), we change (80) into the following new form

GANG XU AND HUICHENG YIN
As in Theorem 4.1, we choose a little different multipliers to get Thanks to boundary condition (82), it follows from (35), (41) and direct computation that In addition, by the similar computation as in Lemma 5.3, one has Next we derive the estimates on the second order derivatives of (σ,u). By Lemma 5.2, we have obtained the estimate of (∇ω, S 0 ω), and from Lemma 5.3, we get the estimate of (S 2 0u , S 2 0σ ). In order to establish the energy estimate on the second order derivatives of (σ,u), it is required to estimate (S 0 divu, ∇divu, ∇ 2σ ). Due to the restriction of boundary condition, we can't estimate ∇divu and ∇ 2σ directly. To overcome this difficulty, we will estimate the tangent derivatives (Zdivu, Z∇σ) and the radial derivatives (S 1 divu, S 1 ∇σ) with S 1 = 3 i=1 y i ∂ i = r∂ r , respectively.
Next we focus on the estimates on the radial derivatives ∇S 1σ and S 1 divu.
Lemma 5.6 (Estimates on 2 nd order radial derivatives (∇S 1σ , S 1 divu)). Under the assumption of Theorem 5.1, then for µ = 3(γ − 1), Proof. To derive the estimates on the radial derivatives of ∇σ and divu, we need to derive a boundary condition of divu on B T . Multiplying (79) i by y i and then summarizing them, we have By the boundary condition (82), we know S 1σ = 1 γ |u| 2 on B T . Substituting this into (90) yields Acting S 1 to (79)-(80), we have where As in (84), one has that where

By the expression of G and (94), direct computation yields
As in (162) for the proof of Lemma 5.5, we also have Next, we deal with Ω T E ·M(S 1 divu)dtdy in (95). It follows from the expression of E that Similar to the treatment in (163) in the proof of Lemma 5.5, we get We now handle the left term in (97). From (80), one has Substituting (99) into the second term in (97) and using Lemma 5.5 derive Combining (98) and (100) yields In addition, it follows from the expression of G and the a prior bound (35) that This means On the other hand, from the expression of B, one has It follows from direct computation that and Collecting (101) and (104)-(105) yields Finally, we deal with the term K 4 . Since K 4 is expressed as by (102), (35) and (41), one gets Thus, combining (96) and (106)-(107), we finish the proof of Lemma 5.6.
The vectors S 1 and Z have been used in our energy estimates. However, as shown in Lemma 2.1, ∂ i (1 ≤ i ≤ 3) are equivalent to S 1 and Z only for |y| = 0. To get the energy estimates of (σ,u), we will take the domain decomposition techniques. For this purpose, we choose a cut-off function χ(s) as follows (108) Lemma 5.7 (Estimates on (∇ 2σ , ∇divu) near |y| = 0). Under the assumptions of Theorem 5.1, then for µ = 3(γ − 1), Proof. By direct computation, we have As in the treatment of (84), we have +L(∂ kσ , ∂ k divu) · χ(|y|)M(∂ k divu) dtdy

GANG XU AND HUICHENG YIN
Note that the function χ (|y|) has a compact support away from |y| = 0, then by Lemma 2.1, the last term on the right hand side of (112) can be estimated as in Lemmas 5.4-5.6. On the other hand, by a similar argument for Ω T (158), we can obtain Then (109) follows from (112)-(113) and Lemmas 2.1, 5.2, 5.5 and 5.6.
Corollary 3. By Lemma 2.3, Corollary 2, Lemma 5.2, 5.5 and 5.6, one can arrive at Proof. Acting 1 R(t) S k 0 R(t) to equations (21) and (22) yields Motivated by the proof of Theorem 4.1, we will choose the modified multipliers )u i to derive Lemma 5.3. Note that S 0 is tangent to boundary B T , one then has By direct computation yields Next, we treat A 0 k,i , we will divide this process into three parts.

It follows from the integration by parts and direct computation that
This, together with (35), yields Note that Substituting (125) into (124) yields Here we especially point out that due to the second line in (126) is negative, then we can remove it in the right hand of (126). In addition, From now on and by the induction method, we assume that all the estimates of the derivatives up to (k − 1)−order are known before estimating the k-order derivatives. Then we have Next, we estimate II 2 and II 3 in (127). If k = 1, then II 2 = II 3 = 0; If k = 2, by (35), (41) and Hölder inequality, then If k = 3, we also apply (35), (41) and Hölder inequality to get Thus, by (126)-(130), we arrive at Part 3. Estimate of Ω T A 0 k,3 · M 0 S k 0σ dtdy. It follows from the definition of A 0 k,3 and direct computation that By the similar method for (128), we have In addition, we obtain that If k = 1, then l = 1 holds. Thus it follows from (41) and Hölder inequality that If k = 2, then l = 1 or l = 2 holds. By (35), (41), Hölder inequality and (134), then If k = 3, then l = 1, 2, 3 hold, and we have Since there exists a first order derivative term S 0u in III 1 2 , by (35), (41) and Hölder inequality we have

GANG XU AND HUICHENG YIN
To treat III 2 2 , we still use the similar method as in (72). For this purpose, it follows from (116) that which yields Thus In addition, by Hölder inequality and Gagliardo-Nirenberg inequality, as in (72), we know Thus, it follows from (134)-(141) that Analogously, by the same method as in (136)-(141), we can obtain Then combining (132)-(133) with (142)-(143) yields Together with (123), (131) and (144), we have Next, we estimate At first, it follows from the integration by parts, (23) and (35) that In addition, For k = 1, 2, the integrand in the right hand of (147) includes the first order derivative term |S 0u | or |∇u|, then it follows from (35), (41) and Hölder inequality that For k = 3, one has Since the integrand in A 1 has the first order derivative term |S 0u | or |∇u|, as in (148), we have In addition, the integrand in A 2 includes the two second order derivative terms and one third order derivative term, then as in the treatments of (72) and (140)-(141), one can arrive at Thus, collecting (147)-(151) yields By (146) and (151), one gets Combining (121), (145) with (153), we complete the proof of Lemma 5.3.
Next we start to prove Lemma 5.5.

GANG XU AND HUICHENG YIN
with B and B i being given in (79)-(80). We now derive the boundary condition of Zσ. Noticing that Z is tangent to the boundary B T and [S 1 , Z] = 0 holds, then it follows from (82) that Analogously to (85)-(86), one has Next, we compute Ω T 3 i=1B i ·M 0 (γ∂ i Z 1σ + y i Z 1 |u| 2 ) +B ·M(Z 1 divu) dtdy in (158). At first, direct computation yields whereB i1 (i = 1, 2, 3) only includes the third-order derivative terms. It follows from the divergence theorem that By the boundary condition (23), the second line of (159) is 0. This, together with the a prior bound (35), yields SinceB i2 only includes the second order error terms, it follows from (35) and direct computation that Thus, collecting (160) and (161) yields Analogously to (159)-(162), we also have Substituting (162) and (163) Replacing Z 1 by Z 2 or Z 3 , we also have the similar estimate to (164), then Part 2. Estimates on (S 0 ∇σ, S 0 divu).
Multiplying R(t) on (5.76)-(5.77) and then taking ∂ t on the resulting equations, we arrive at where Noting that S 0 is tangent to the boundary B T , then it follows from (82) that Similarly to (84)-(85), one has Here we point out that except the term µR(t) −µ ∂ i (divu), all the other terms in D i admit the analogous properties forB i in (158). We now firstly estimate Then In addition, S 0 |u| 2 is a second order error term, then it is easy to get Then by the similar method as in (161)-(165), and together with (171)-(172), we obtain Substituting this into (169) yields Collecting (165) and (174), then Lemma 5.5 is proved.
7. Weighted estimates on the third-order derivatives of (σ,u) and proof of Theorem 5.1. In this section, we will derive the estimates on all third-order derivatives of (σ,u). In fact, we have already got the estimates on some thirdorder derivatives of (σ,u) in Section 5, which include the derivatives S k 0 ∇ 2−k curlu (0 ≤ k ≤ 2), S 3 0u and S 3 0σ . Next, we still need to estimate the following left terms: • second order tangent derivatives of ∇σ and divu, i.e., ∇(S k 0 Z 2−kσ ), S k 0 Z 2−k divu (0 ≤ k ≤ 2); • first order tangent derivatives of ∇S 1σ and S 1 divu, i.e., ∇(S m 0 Z 1−m S 1σ ), • second order radial derivatives of ∇σ and divu, i.e., S 2 1 ∇σ, S 2 1 divu. In subsequent Lemma 7.1, we will give the estimates on the second order tangent derivatives of (∇σ, divu) and the first order tangent derivatives of (∇S 1σ , S 1 divu).
Lemma 7.1. Under the assumptions of Theorem 5.1, then for µ = 3(γ − 1), Proof. Since S 0 and Z are tangent to the boundary B T , then from (82) and (91), we have the following boundary conditions on B T Based on this, completely analogously to the proof of Lemmas 5.5 and 5.6, we can complete the proof of Lemma 7.1. Here we omit the details.