ON GLOBAL AXISYMMETRIC SOLUTIONS TO 2D COMPRESSIBLE FULL EULER EQUATIONS OF CHAPLYGIN GASES

. For 2D compressible full Euler equations of Chaplygin gases, when the initial axisymmetric perturbation of a rest state is small, we prove that the smooth solution exists globally. Compared with the previous references, there are two diﬀerent key points in this paper: both the vorticity and the variable entropy are simultaneously considered, moreover, the usual assumption on the compact support of initial perturbation is removed. Due to the appearances of the variable entropy and vorticity, the related perturbation of solution will have no decay in time, which leads to an essential diﬃculty in establishing the global energy estimate. Thanks to introducing a nonlinear ODE which arises from the vorticity and entropy, and considering the diﬀerence between the solutions of the resulting ODE and the full Euler equations, we can distinguish the fast decay part and non-decay part of solution to Euler equations. Based on this, by introducing some suitable weighted energies together with a class of weighted L ∞ - L ∞ estimates for the solutions of 2D wave equations, we can eventually obtain the global energy estimates and further complete the proof on the global existence of smooth solution to 2D full Euler equations.

By our knowledge, so far this conjecture has not been solved yet even for the small initial data. As illustrated in Page 89 of [25], the above conjecture is mainly of mathematical interest but its resolution would elucidate both the nonlinear nature of the conditions requiring linear degeneracy of each wave field and also might isolate the fashion in which the shock wave formation arises in quasilinear hyperbolic systems. In this paper, we focus on the global solution problem of A. Majda's Conjecture when the initial axisymmetric perturbation in (4) is small.
Next, we give a brief survey on some remarkable works related to Theorem 1.1. In two or three space dimensions, it is well known that the smooth solution (ρ, u, S) to problem (1) with the state equation of polytropic gases (2) will generally blow up in finite time. For examples, • for a special class of initial data (ρ(0, x), u(0, x), S(0, x)), T. Sideris [27] proved that the smooth solution (ρ, u, S) of (1) in two and three space dimensions can develop singularities in finite time.
Up to now there are also many interesting results on the multidimensional compressible Euler equations of Chaplygin gases. For examples, • when the Chaplygin gases are isentropic and irrotational in two or three space dimensions, (1) can be written as a second order quasilinear wave equation of the velocity potential function Φ(t, x), where u = ∇ x Φ. In 3D case, this quasilinear wave equation satisfies the null condition (its definition see [5] and [18]); in 2D case, the nonlinear equation verifies both the first and the second null condition (see their definitions in [3]). Consequently, according to the results in [5], [18] and [3], we know that the small perturbed smooth solution (ρ, u) of (1) exists globally when 3D or 2D Chaplygin gases are isentropic and irrotational.
• when the Chaplygin gases are spherically symmetric and non-isentropic, P. Godin in [11] proved the global existence of smooth symmetric solution to the 3D non-isentropic compressible Euler equations under small perturbation of a rest state. Similar result in two space dimensions was established in [4].
• when the 3D Chaplygin gases are isentropic and irrotational, the authors in [22] proved the global existence of small perturbed smooth solutions in the exterior domain with slip boundary condition u · n = 0, where n stands for the unit outer normal of the boundary.
• when the 2D Chaplygin gases are axisymmetric and isentropic, if the rotationally invariant initial data are a compactly supported perturbation of a rest state, we show that the small perturbed smooth solution with non-zero vorticity exists globally in [14].
Next we give the comments on the proof of Theorem 1.1. When the first and the second null conditions of 2D quasilinear wave equations are both satisfied, S. Alinhac in [3] have established the global existence of small data smooth solution with compact support in the space variable by the "ghost weight" technique. Based on this, the global existence to 2D isentropic and irrotational Euler equations of Chaplygin gases (1) is a direct corollary of [3]. In addition, for the symmetric initial data, by introducing a new "ghost weight" and by utilizing the both null conditions and the variable entropy, the authors in [4] have established a global weighted energy estimate and obtained the global existence of smooth solutions for 2D full compressible Euler system under the small initial perturbation with compact support. In [14], we obtain the global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity under the small perturbation with compact support. In the present paper, for the nonisentropic and rotational case, we will study the global existence of (7) with non-zero vorticity and non-compact initial data. On the one hand, due to the strong effect of the term (1+h)g 2 r (including the both entropy and voticity) in Q 2 , the time-decay of (v, f, g, h) becomes worse for problem (7). To overcome this essential difficulty, motivated by [14], we seek a suitable transformation v(t, r) =ṽ(t, r)+G(t, r) so that the unknown functionṽ will admit a better decay in time meanwhile the function G has some required "good" properties in the process of deriving energy estimates on (ṽ, f, g, h). For this purpose, we delicately choose G to satisfy the nonlinear ODE:

FEI HOU AND HUICHENG YIN
with G(t, ∞) = 0. On the other hand, all the proofs in [3], [4] and [14] heavily depend on the compactness of the initial data, namely, such a crucial Hardy type inequality x) ⊂ {x : r = |x| ≤ M + t} should be used. However, for the non-compact support case in Theorem 1.1, such a kind of Hardy type inequality does not hold, and we have to use other methods to overcome this essential difficulty (see Lemma 5.1). To this end, we carefully observe the following phenomena: by the transport equations of g and h in (7), g and h are expected to behave like r − with suitably large number > 0; near the light cone {r = t}, the flow will be "almost" irrotational and isentropic for large time since the vorticity curl u = ∂ r g + 1 r g and the perturbed entropy h = B(S) − B(S) decay like t − . Therefore, we can take the measure as follows: by introducing the radial velocity potential ϕ (see (96) in Section 4.3), we get a nonlinear wave equation which approximately satisfies the null conditions. Through utilizing a class of weighted L ∞ -L ∞ estimates of solution φ to the 2-D linear wave equation φ = F (t, x) in [15] instead of the usual L 2 -L ∞ estimates used in [3] and so on (one can also see (49) which is derived by Sobolev embedding's Theorem), we can get a better time-decay rate of ϕ (see (108)). Based by these, we will manage to derive the suitable global energy estimate for the solution to problem (7).
As in [14], the vector fields Γ ∈ {∂ t , X := t∂ t + r∂ r } will be utilized. In addition, in order to distinguish the external and internal energy of solution to problem (7), we will introduce the auxiliary weighted energy near the light cone and away form the light cone {r = t} where t = (1 + t 2 ) 1 2 , χ 0 (s) and χ 1 (s) are smooth cutoff functions satisfying 0 ≤ χ 0 ≤ 1 with for s ∈ R. Moreover, both |χ 0 | 2 χ0 and |χ 1 | 2 χ1 are bounded. Here, we should pay more attention on the possible singularity near r = 0 in X k (t). Note that (∂ r + 1 r )Γ a f (t, r) = div Γ a u, (∂ r + 1 r )∂ r Γ bṽ (t, r) = ∆Γ bṽ and x r ∂ r (∂ r + 1 r )Γ b f (t, r) = ∇ div Γ b u, then each term in X k (t) is finite for any fixed time t ≥ 0 and for smooth solution (v, f, g, h) of (7). On the other hand, in order to derive the L ∞ estimate of ∂ r g and h by Sobolev embedding theorem W 1,3 (R 2 ) ⊂ L ∞ (R 2 ) instead of W 2,2 (R 2 ) ⊂ L ∞ (R 2 ) (the latter one needs one more space derivatives), we also introduce the W 1,3 higher order norm of (w, h) as follows where w(t, r) := curl u(t,x) ρ(t,r) = (1 + v(t, r))(∂ r + 1 r )g(t, r). As in [24], it is easy to know that w(t, r) satisfies which is the same as the equation of h in (7). Combining the preparations above with some rather involved analysis and delicate observations, the global energy estimates of (v, f, g, h) can be eventually established and then the proof of Theorem 1.1 is completed by the continuation method. This paper is organized as follows: In Section 2, we list or derive some basic results including the Sobolev-type embedding inequalities and Hardy-type inequalities. In Section 3, by the null condition structures of the nonlinearities, some good estimates for the L ∞ and L 2 norms of v + f and its derivatives near the light cone will be established for the smooth solution (v, f, g, h) of (7). On the other hand, the pointwise estimates for the smooth solution (v, f, g, h) of (7) and the improved decay rates of v + f near the light cone by the weighted L ∞ -L ∞ estimates are derived in Section 4. In Section 5, we obtain the estimates for the auxiliary weighted energy X k (t) and Y k (t) defined by (12) and (13). In Section 6, by using S. Alinhac's "ghost weight" technique together with some other suitable multipliers, the L 2 estimates for (v, f ) in the energy E k (t) are obtained, where E k (t) := |a|≤k [ Γ a v(t, r) L 2 + Γ a f (t, r) L 2 + r Γ a h(t, r) L 2 + Γ a g(t, r) L 2 ]+ |b|≤k−1 r (∂ r + 1 r )Γ b g(t, r) L 2 . The L 2 estimates for (g, h) in E k (t) and W 1,3 estimates for (w, h) are established in Section 7. The proof of Theorem 1.1 is finished in Section 8 by the local existence of smooth solution to problem (7) and the continuity argument method.
2. Some preliminaries. At first, it follows from direct computation that for any smooth functions φ(t, r) and ψ(t, r), X(φ∂ r ψ) = (X + 1)(φ∂ r ψ) = φ∂ r Xψ + Xφ∂ r ψ, Through the paper, we always assume that for fixed integer N ≥ N 0 = 11, where the large positive constants M and M will be chosen later.
Lemma 2.1. For any smooth function φ(t, x), the following Sobolev inequalities hold where and below A ≤ C B 1 + B 2 means A ≤ C(B 1 + B 2 ) with the generic positive constant C which is independent of t, ε, M and M .
where the definition of χ 0 has been given in (14).
Proof. The first two inequalities (22)- (23) has been proved in [14,Lemma 2.4]. For the proof of (24), it is easy to get that

AXISYMMETRIC SOLUTIONS OF 2D EULER EQUATIONS 1443
Applying (22) to the last term in (25) yields (24). This completes the proof of Lemma 2.2. Lemma 2.3. For a ∈ N 2 0 , then the following weighted Sobolev type inequalities hold where the definition of χ 1 has been given in (14).
Proof. At first, we show lim r→∞ ψ(t, r) = 0, where ψ(t, r) = r + t r − t |χ 1 Γ a f (t, r)| 2 . Indeed, direct calculation implies that for large r, s > 0, where we have used the fact of R + t + R − t ≤ C R on supp χ 1 (R/ t ) and R ≈ t on supp χ 1 (R/ t ). Thus, we have proved the existence of lim r→∞ ψ(t, r).
and the definition of G is given in (11).
Next, we deal with (2). Similarly to the proof of (1), we see that In addition, by χ 0 ≤ 0 in (14), we know that χ 0 r t ≤ χ 0 r t holds for r ≤ r.
which yields (2). The proof of (3)-(4) is the same as in Lemma 2.5 of [14], here we omit it. Finally, we turn our attention to (5). According to (21), we achieve If h is replaced by w, then (31) is also true. This completes the proof of Lemma 2.4.
Next we focus on the estimates of G in (11).
is a smooth solution of (7), let G(t, r) be defined in (11) and suppose that (19) holds. Then for a , a, b ∈ N 2 0 with |a | ≤ N , |a| ≤ N − 1 and |b| ≤ N − 3, the following inequalities hold (1) lim r→∞ r 2 Γ a G(t, r) = 0 for any fixed t ≥ 0; Proof. Proof of (1): At first, we show that lim r→∞ r 2 ∂ a1 t G(t, r) = 0 for any integer a 1 ≤ N . Indeed, according to the definition of G and lim r→∞ G(t, r) = 0, we arrive at Applying ∂ a1 t on both sides of (32) yields For fixed t ≥ 0 and r ≥ t , by applying Cauchy-Schwartz inequality we have where we have used the fact of we know that the L ∞ norms on the right hand side of (34)-(35) are bounded for fixed t ≥ 0. Hence, the boundedness of the integrations ∞ r · · · dr on the right hand side of (34)-(35) yields that lim r→∞ r 2 ∂ a1 t G(t, r) = 0. Now we prove (1) for all a ∈ N 2 with |a | ≤ N by induction method on a 2 and therefore assume that lim r→∞ r 2 ∂ a1 t X a2 G(t, r) = 0 holds for any a 1 +a 2 ≤ N and fixed t ≥ 0. Note that for a 1 + a 2 ≤ N − 1, According to the induction assumption, we have lim r→∞ J 1 1 (r) = 0. By the same treatments as in (34)-(35), we obtain the boundedness of the integration ∞ t |J 2 1 (r)|dr, and further derive lim r→∞ J 2 1 (r) = 0 as in (28). Therefore, we have proved lim Proof of (2) and (3): According to (1), it holds that Γ a G(t, r) = ∞ r ∂ r Γ a G(t, r )dr , which yields the first inequality in (2). Next, we focus on the proof for the second inequality in (2). Applying the commutation identities 1446 FEI HOU AND HUICHENG YIN (18), we get the following equation for the higher order derivatives of G Thereafter, we arrive at for |a| = 0, and Now, we deal with each term in the second line of (37). By the definition of G and Cauchy-Schwartz inequality, we achieve where we have used (24) in the last inequality.
In addition, set J cde Then, by Cauchy-Schwartz inequality and Hardy inequality (24) again, we have and Next we treat the term 1 r Γ c g L ∞ in the right hand side of (41). Similarly to the proof of (2) in Lemma 2.4, we achieve which also yields the first inequality in (3). (21), then we obtain On the other hand, by the definition of w, we see that At last, we treat the third line in (37). Due to |c| < |a| ≤ N − 1, by assumption (19) we get c+d=a, c<a,d<a In addition, it is easier to prove (2) for |a| ≤ N −5. Therefore, for all |a| ≤ N −1, Collecting (37)-(46), we obtain (2) and (3) for sufficiently small ε 0 > 0.
Proof of (4): The proof of (4) is similar to that for (2). Indeed, by taking the L ∞ norm on both sides of (36), we arrive at Consequently, (4) is derived for small ε > 0.
Proof of (5): We conclude from (1) that Note that for c + d = a with |a | ≤ N and N ≥ N 0 = 11, then |c| ≤ N − 6 or |d| ≤ N − 6 holds. This, together with (36), yields that Thus we complete the proof of Lemma 2.5.
3. The estimates of (v, f ) near the light cone {r = t}. Benefit from the inherent null condition structure near the light cone for the system in (7), some time-decay estimates for the L ∞ or L 2 norms of the smooth solution (v, f, g, h) to (7) will be established in this section.
is a smooth solution of (7), then the following estimates for the L ∞ and L 2 norms of v + f hold Proof. ActingΓ a on two sides of the first two equations in (7) and applying (18), Direct computation shows that wherẽ Next we treat each term in the expression of (52). By χ 1 = χ 1 (χ 0 + χ 1 ) and v =ṽ + G, we arrive at where we have used (19) In addition, the treatment on the term Γ b h∂ t Γ c f in the first line of (52) is much easier. Indeed, by applying (5) The term in the second line of (52) can be analogously treated as in (53): To overcome the loss of the space regularity in the term Γ b hΓ c f ∂ r Γ d f since ∂ r is not a "good derivative", we will repeatedly make full use of the scaling operator as follows (57) Based on this, we get Consequently, substituting (53)-(58) into (51) yields (47). The proof of (48) is similar. Indeed, from |b| + |c| ≤ |a | ≤ N − 1, one has |b| ≤ N − 2 or |c| ≤ N − 3. Therefore, we see that This, together with (47), yields (48).
Finally, by direct computation and Cauchy-Schwartz inequality, we achieve which implies (49). This completes the proof of Lemma 3.1.

4.1.
The pointwise estimates of (v, f, g, h) away from the light cone. In this subsection, the pointwise estimates away from the light cone for the space or time derivatives of smooth solution (v, f, g, h) will be achieved through the structure of the hyperbolic system (7).
is a smooth solution of (7), then for small ε 0 > 0, the following inequalities hold Proof. Proof of (59): At first, it is easy to know that (ṽ, f ) satisfies According to (18), we have

FEI HOU AND HUICHENG YIN
Direct computation yields Similarly, we also have Applying the Leibniz's formula and (18), we achievẽ We now deal with the term ∂ t Γ a G in (68). Taking the time-derivative ∂ t on equation (11) yields that where we have used the equations of g and h in (7). For a = (a 1 , a 2 ) ∈ N 2 0 , by (18), acting ∂ a1 t (X + 2) a2 on (70) yields Similarly to (37), we obtain Here we point out that the second line in (72) does not appear when |a| = 0. In addition, from inequality (27) and Lemma 2.5, we see that Applying (24) to the last integration in (72), we arrive at For the last term in the last line of (74), we conclude from |e| ≤ |a| ≤ N − 3, r∂ r = X − t∂ t and the fourth equation in (65) that ∞ 0 | r r∂ r Γ e h| 2 rdr | r r∂ r Γ e h| 2 rdr.
We now treat the other nonlinearities in (68)-(69). It suffices only to deal with Γ b f ∂ r Γ c G, Γ c (ṽ + G)∂ r Γ bṽ and Γ b h∂ t Γ c f since the treatments on the other left terms in (68)-(69) are similar or much easier. For the lower order case |a| ≤ N − 6, by using assumption (19) directly, we obtain

FEI HOU AND HUICHENG YIN
Right now, we assume (59) holds for the lower order case |a| ≤ N − 6. While for the higher order case |a| ≤ N − 3, it always holds that |b| ≤ N − 8 or |c| ≤ N − 6.
Then, we see that Finally, we deal with Γ b h∂ t Γ c f . Since |b| ≤ |a| ≤ N − 3, applying (5)   Proof of (60): At first, we start to prove (60). For this purpose, it follows from the fourth equation in (65) that Then, by Lemma 2.3-2.4, (57) and (59) we conclude that Combining this with the smallness of ε 0 > 0, one has where we have used the fact of |a | ≤ N − 4. Analogously, we can get the estimate of w in (60) since equation (16) is the same as for h. By the analogous but much easier treatment, we can achieve the estimate of |∂ t Γ a g(t, r)| in (60). Proof of (61): Finally, we take the estimate of | r ∂ r ∂ t Γ a G(t, r)|. Acting ∂ t on both sides of (36) yields From (79), one has This, together with (59), (60) and Lemma 2.3-2.5, yields Therefore, we finish the proof of (61). Based on these, Lemma 4.1 is proved.

4.2.
The pointwise estimates of (v, f ) near the light cone.
is a smooth solution of (7), then for small ε 0 > 0, the following estimates of weighted L ∞ norms hold Proof. Similarly to (66)-(67), we achieve

4.3.
Better decay rates of v + f near the light cone. At first, we consider the following 2D linear Cauchy problem where x ∈ R 2 , and = ∂ 2 where and We then have the following results (see Section 3 of [15]) where · W k,p is the standard Sobolev norm.
In order to apply Lemma 4.3 and Lemma 4.4, we need to find out the inherent wave equation from (7). To this purpose, we introduce the velocity potential function: Here, we point out that ϕ(t, r) is well defined (see Remark 3 below). In this case, we have Lemma 4.5. If (v, f, g, h) is a smooth solution of (7), then ϕ fulfills the following wave equation ϕ = F(t, r), where the nonlinearity F is and H is defined below by (104). In addition, Proof. At first, we seek the initial data of ϕ(t, r). Obviously, (99) holds if we set t = 0 in (96). In addition, from the second equation in (7), we arrive at Integrating (101) with respect to the variable r yields where we have used the definitions of v = B(S) ρ − 1 and h = B(S) − 1. Let t = 0 in (102). Then (100) is derived.
Next, we look for the equation which is fulfilled by ϕ(t, r). According to the second equation in (62), we achieve Let H(t, r) := ∞ r (h∂ t f + hf ∂ r f − G∂ rṽ )(t, r )dr with H(t, ∞) = 0 and Then integrating (103) with respect to the variable r yields By acting ∂ t on both sides of (105) and using the first equation in (62), we get On the other hand, by the second equation in (62) again, we find that Substituting (107) into (106) yields   (v, f, g, h) is a smooth solution of (7), there exists a positive constant C which is independent of t, ε, M and M such that Proof. According to (105), we deduce that Applying Γ a on both sides of (109) yields where C a b are some real constants that only depend on the multi-indices a and b. In addition, it is easy to check that Substituting (111) into (110) derives On the other hand, actingΓ a on both sides of (104) yields By the similar analysis as in Lemma 2.5, we can prove lim r→∞ Γ a H(t, r) = 0. Consequently, we conclude from (113) that Next we only deal with Γ b G∂ r Γ cṽ adn Γ b h∂ t Γ c f in (114) since the other term Γ b hΓ c f ∂ r Γ d f is more easily treated. Applying Lemma 2.5 and 4.1-4.2 yields that Analogously, through replacing E |b|+1 (t) by W |b| (t) in the last line of (115), we arrive at In addition, from the proofs of (115) and (116), we know that (117) also holds for |a| ≤ N − 3. Collecting (112) with (117) and Lemma 2.3-2.5 implies Finally, we deal with the last term |χ 1 Γ c Γ b ϕ| in (118). Let Γ a = ΓΓ a and we turn to the pointwise estimate for the top-order term Γ a ϕ in (118). According to (18) and Lemma 4.5, it is easy to find that Γ a ϕ satisfies: where the constants C a b , C a bc , C a bcd may vary from line to line. By applying Lemma 4.3-4.4 with 1 = 3 8 and 2 = κ = 1 8 to (119), we obtain (120) Next we treat the nonlinearity |F a (t , r)| in the second line of (120). For convenience, we denote t by t below.
∂ t Γ a H: By using (117) directly with 1 + |a | ≤ N − 3 and choosing M 2 ε 0 ≤ 1, we then arrive at (121) It is convenient to divide this term into several parts by the cutoff function (14) as follows: Applying Lemma 2.4-2.5 and 4.1 to the first term in the second line of (122) directly derives In addition, from (26), (49) and (80), we see that By using (49) and (59) to the first term in the third line of (122), we achieve On the other hand, we conclude from Lemma 2.3-2.5 and 4.1-4.2 that

FEI HOU AND HUICHENG YIN
Here, we only need to deal with 1 Γ b (1 +ṽ)∂ t Γ c G: By applying Lemma 2.4-2.5 and inequality (59) directly, we find that According to Lemma 2.3-2.4 and 4.1, we deduce that In addition, by Lemma 2.3-2.5 and inequality (47), we obtain This term can be dealt with as in (126).
We conclude from Lemma 2.3-2.5 that In addition, from Lemma 2.5 and 4.2, we see that At last, we turn to the initial data in the first line of (120). It is not hard to get that where the last inequality follows from In addition, by taking the divergence on the second equation of (4), we find that Then we conclude from the equation of ∂ t div u in (135), (96) and (101) that Substituting (10), (121)-(136) into (120) yields (108). Thus we complete the proof of Lemma 4.6.

5.
Auxiliary energy estimates of (v, f, g, h). In this section, we mainly establish the estimates for the auxiliary energies X k (t) and Y k (t) of the smooth solution (v, f, g, h) to (7).

5.1.
Auxiliary energy of (v, f, g, h) near the light cone.
Lemma 5.1. For any smooth function φ(t, x) with bounded norms φ(t, x) L 2 (R 2 ) and χ 1 ∇φ(t, x) L 2 (R 2 ) , where the definition of χ 1 is given in (14), then the following Hardy type inequality holds Proof. According to the boundedness of φ(t, x) L 2 (R 2 ) and χ 1 ∇φ(t, x) L 2 (R 2 ) , it holds that for each fixed t, Indeed, for fixed t, then χ 1 |x| t ≡ 1 for large r. In this case, one has that for large r, s ≥ 1, This means that lim r→∞ S 1 rφ 2 (t, rω)dω exists. If lim r→∞ S 1 rφ 2 (t, rω)dω = 0, then there exists a positive constant C(t) such that for large r, lim holds.
On the other hand, χ 1 |x| t ≡ 0 holds for any r ≤ 1 4 ≤ 1 4 t and t ≥ 0. Thereafter, we conclude from the integration by parts and (138) that This, together with the Cauchy-Schwartz inequality, yields the proof of Lemma 5.1.

Remark 4. For any
in the left hand side of (137). The only difference in the proof lies in that the function "arctan(r − t)" is replaced by .
is a smooth solution of (7), then for small ε 0 > 0, the following auxiliary weighted energy inequality holds Proof. For a ∈ N 2 0 with |a| ≤ N − 1, set In view of (82), it is not hard to see that At first, we deal with the term Γ b (v + f )∂ r Γ c v inΓ a Q 1 andΓ a Q 2 of (83)-(84). Then, we obtain b+c=a Note that r + t ≤ C r − t + C t holds. Applying Lemma 5.1 to where we have used the Cauchy-Schwartz inequality and (48). Recalling that |b| + |c| ≤ |a| ≤ N − 1 and N ≥ N 0 = 11, if |b| > N − 6, then we have |c| + 2 ≤ 6 ≤ N − 5. Therefore, combining (140)-(141) with Lemma 2.3-2.5 derives b+c=a Now, we turn to the treatment of 1 r Γ b vΓ c f inΓ a Q 1 . It is easy to get that where we have used |b| ≤ |a| ≤ N − 1.

By analogous but easier analysis for the term Γ
Next, we deal with the terms Γ b h∂ t Γ c f and Γ b hΓ c f ∂ r Γ d f inΓ a Q 2 . Since the treatment on the cubic nonlinearity Γ b hΓ c f ∂ r Γ d f is much easier than the quadratic one Γ b h∂ t Γ c f , here we only estimate Γ b h∂ t Γ c f . It follows from direct computation that b+c=a Finally, for the term 1 Thus, collecting (142)-(146) leads to Therefore, for small ε > 0, we complete the proof of (139).

5.2.
Auxiliary energy of (v, f, g, h) away from the light cone.
Lemma 5.3. For any integer k ∈ N with k ≤ N and multi-index a ∈ N 2 0 with |a| ≤ N − 1, if (v, f, g, h) is a smooth solution of (7), then for small ε 0 > 0, the following weighted energy inequalities hold Proof. We divide the proof of Lemma 5.3 into two parts.
Part I: The estimates including the first order space derivatives ∂ r Γ aṽ (t, r) and According to (66) and (67), we have At first, we deal with the term Γ b (ṽ+G)(∂ r + 1 r )Γ c f inΓ aQ 1 of (68). If |b| ≤ N −6, we find that where we have used (19), (27) and Lemma 2.4-2.5. For |b| > N −6, due to |b|+|c| = |a| ≤ N − 1 and N ≥ N 0 , one has |c| ≤ 4 ≤ N − 3. Therefore, we achieve The treatments for 1 andΓ aQ 2 are analogous. Then one has Now we deal with the terms in the last line of (151). By an analogous analysis for the multi-index b and c as in (150), we achieve Next we turn our attention to the last term ∂ t Γ a G L 2 in (151). From Lemma 2.5 we know that lim r→∞ r 2 ∂ t Γ a G(t, r) = 0. Thus, direct computation yields Then we achieve (153) Instead of r∂ r ∂ t Γ a G L 2 in (153), we will deal with the more general one r ∂ r ∂ t Γ a G L 2 . Indeed, taking L 2 norms on both side of (79) derives Then similarly to (152), we obtain b+c=a, c<a and In addition, according to the fourth equation in (65) and (57), we deduce that where we have used (23), Lemma 2.3-2.5 and Lemma 4.1. In addition, the treatment of ∂ t Γ c g L 2 is similar. Consequently, we achieve This yields Inserting (155), (156) and (158) into (154), we arrive at This, together with (149), (151), (152) and (157), yields that for all |a| ≤ N − 1, Part II: The estimates including the second order space derivatives ∂ r (∂ r + 1 r )Γ a f and (∂ r + 1 r )∂ r Γ a ṽ. Let |a | ≤ N − 2. From (65), we easily get We now treat the last two nonlinearities ∂ rΓ a Q 1 and (∂ r + 1 r )Γ a Q 2 in (160). It is not hard to check that and (162) Applying (23) to the first term 1 r ∂ r Γ cṽ in the second line of (161), we achieve Next we start to treat each term in the last line of (163).
We conclude from Hardy inequality (23), Lemma 2.4-2.5 and 4.1 that 1472 FEI HOU AND HUICHENG YIN With the help of (57), we deduce that This, together with (163) and (164), yields With an analogous analysis to χ 0 ∂ rΓ a Q 1 L 2 , we achieve Substituting (165) and (166) into (157) yields Thus, collecting Part I and Part II completes the proof of Lemma 5.3.
is a smooth solution of (7), then for small ε 0 > 0, there exists a positive constant C which is independent of t, t , ε, M and M such that the following elementary energy inequality holds where I := e q r(1 +ṽ) with the smooth function q = q(r − t) satisfying q (s) = s − 3 2 and lim s→−∞ q(s) = 0.
Proof. Multiplying (50) by e q rΓ a v and e q rΓ a f respectively, we obtain We now treat the nonlinearities containing top-order derivative ∂ r Γ a v and ∂ r Γ a f in the right hand side of (169). To this end, (83)-(84) can be rewritten as follows: Direct computation yields that Substituting this into (169), we have where I is defined by (168).
Multiplying (170) First, we deal with the third line of (171). From Lemma 6.1-6.2, we know that where we have used that By (108) and Lemma 6.2, we achieve Next, we treat the fourth line of (171). It is easy to calculate that Applying Lemma 6.2 again, we obtain On the other hand, by Lemma 6.1, we obtain Finally, we turn our attention to the last line of (171). Applying the integration by parts to the fifth line of (171), we conclude from the equation of h in (7) Therefore, substituting (172)-(176) into (171) derives (167). This completes the proof of Lemma 6.3.

Remark 5.
The multiplier function e q(r−t) in (169) is called the "ghost weight" by S. Alinhac in [3], which was first used to treat the global small data solution problem for the 2D quasilinear wave equation when both the null conditions are fulfilled.
6.2. The treatment on I near the light cone. In this subsection, we deal with t 0 χ 1 Idrdt.
is a smooth solution of (7), then for small ε 0 > 0, there exists a positive constant C which is independent of 1476 FEI HOU AND HUICHENG YIN t, t , ε, M and M such that the following inequality holds where Q a (t ) := Proof. Rewrite the nonlinearity I defined by (168) By using Lemma 6.1-6.2 to I 11 directly yields t 0 Now we deal with I 12 . For |a| ≤ N , we conclude from Lemma 6.1-6.2 that b+c=a, c<a b+c=a, |b|≤N −6,c<a While for |a| ≤ N − 5, we obtain b+c=a, c<a The treatment on the other term ∂ r Γ c (v + f )Γ a vΓ b f in I 12 is the same. Finally, we treat I 13 . For N ≥ N 0 = 11 and b+c = a with |a| ≤ N , it always holds that |c| ≤ N − 7 or |b| ≤ N − 5. In the former case, by applying Cauchy-Schwartz inequality, we see that While in the latter case |b| ≤ N − 5 or the lower-order energy case |a| ≤ N − 5, we conclude from (108) that b+c=a, c<a,|b|≤N −5 Collecting (179) f, g, h) is a smooth solution of (7), then for small ε 0 > 0, the following inequality holds where Q a (t ) is defined by (178).
Proof. At first, by v =ṽ + G, the nonlinearity I defined by (168) can be rewritten as I = I 01 + I 02 + I 03 , Next, we deal with I 01 . Note that |c| ≤ N − 3 or |b| ≤ N − 3 always holds for b + c = a with |a| ≤ N . If |a| ≤ N , by using Lemma 5.3 and 6.2 to the first term Γ a f Γ b h∂ t Γ c f in I 01 directly, we have b+c=a, c<a While |a| ≤ N − 5, we arrive at b+c=a, c<a In addition, for the term Γ a f Γ b hΓ c f ∂ r Γ d f in I 01 , we have b+c+d=a, d<a Next, we only treat the term Γ a f Γ b v∂ r Γ cṽ in I 02 since the other left terms can be analogously estimated. In this case, it always holds that |b| ≤ N − 6 or |c| ≤ N − 3.
Therefore, for |a| ≤ N , we obtain b+c=a, c<a While, for |a| ≤ N − 5, we achieve b+c=a, c<a Finally, we turn to the treatment of I 03 . At first, we estimate the last term Γ a vΓ b f ∂ r Γ c G in I 03 . In fact, it is not hard to check that for |b| ≤ N − 3 or the lower order case of |a| ≤ N − 5, While in the remaining case of |b| > N −3, it holds |c| ≤ 2 ≤ N −6. Consequently, we have Next we estimate the other two terms Γ a f Γ a [g 2 (1 + h)] and Γ a f Γ b v∂ r Γ c G in I 03 . For Γ a = ∂ a1 t X a2 with a 1 ≥ 1, let ∂ t Γ a = Γ a with |a | = |a| − 1 ≤ N − 1. It is easy to find that Subsequently, we achieve (220) For J 1 22 , applying integration by parts with respect to t, one has J 1 22 ≤ C r 2 t 2 |Γ a f ∂ r h| 2 rdr + t 0 r 2 t|Γ a f ||∂ r h| 2 (|r∂ r Γ a f | + |Γ a f |)rdrdt The treatment of the term J 11 22 is the same as for J 21 (218), then Analogously for J 12 22 , we obtain