Remarks on a system of quasi-linear wave equations in $3$D satisfying the weak null condition

We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature of the new proof lies in that it never uses the Lorentz boost operator in the energy integral argument. The proof presented here has an advantage over the former one in that the assumption of compactness of the support of data can be eliminated and the amount of regularity of data can be lowered in a straightforward manner. A recent result of Zha for the scalar unknowns is also refined.

Remark 1.2. In Section 3 of [7], thanks to compactness of the support of initial data together with the finite speed of propagation, the proof of Theorem 1.1 was able to employ the standard local existence theorem in solving locally (in time) the Cauchy problem with data given at t = 0 and in continuing the local solutions to a larger strip, though some partial differential operators with "weight" (see just below) were naturally used. We should remark that the constant ε in the above theorem is independent of the "radius" of the support of given data (u i (0), ∂ t u i (0)) = (f i , g i ) (i = 1, 2), that is, R * := inf r > 0 : supp {f 1 , g 1 , f 2 , g 2 } ⊂ {x ∈ R 3 : |x| < r} .
We note that the proof of Theorem 1.1 fully exploits the Lorentz invariance in the sense that it uses all of the operators ∂ α , Ω ij , L k , and S. When it comes to the Cauchy problem for a nonrelativistic system satisfying the weak null condition (see, e.g., (2.8) of [10]) or the initial-boundary value problems in a domain exterior to an obstacle (see, e.g., [19], [11], [16]), the use of L k should be avoided. The purpose of this paper is to revisit the Cauchy problem for (1.1) and prove global existence without relying upon L k . Moreover, we also aim at eliminating compactness of the support of data and lowering the amount of regularity of data. To state the main theorem precisely, we set the notation. As in [5], we define N 1 (u(t)) := E 1 (u(t)), N 2 (u(t)) := where, for a = (a 1 , a 2 , a 3 ) and b = (b 1 , b 2 , b 3 ), ∂ a x := ∂ a 1 1 ∂ a 2 2 ∂ a 3 3 , Ω b := Ω b 1 12 Ω b 2 13 Ω b 3 23 . We also define for a pair of time-independent functions (v(x), w(x)) D(v, w) (1.8) Here, we have set Λ := x · ∇, which can be regarded as a time-independent analogue of S. Theorem 1.3. Under the same assumption as in Theorem 1.1, there exists ε ∈ (0, 1) such that if f 1 , f 2 ∈ L 6 (R 3 ) and D(f 1 , g 1 ) + D(f 2 , g 2 ) < ε, then the Cauchy problem for (1.1) with data (u i , ∂ t u i ) = (f i , g i ) (i = 1, 2) given at t = 0 admits a unique global solution u(t, x) = (u 1 (t, x), u 2 (t, x)) satisfying for all T > 0, with a constant C > 0 independent of T .
For the definition of N (u), G T (u), and L T (u), see (3.15), (4.12), and (4.13), respectively. Note that we limit the number of occurrence of S and ∂ t to 1 in the definition of N 4 (u(t)). With this, there exist a couple of advantages. Firstly, thanks to ∂ t Su = ∂ t u + x · ∇∂ t u at t = 0, we can bypass the burdensome calculation of ∂ j t u i (0, x) (i = 1, 2, j = 2, 3, 4) when computing D(f 1 , g 1 ) + D(f 2 , g 2 ). Compare this with the fact that we must successively calculate ∂ j t u i (0, x) (i = 1, 2, j = 2, 3, 4) with the help of the equation (1.1) when computing W 4 (u 1 (0)) + W 4 (u 2 (0)) appearing in Theorem 1.1. In this connection, by the standard way we can easily find a sequence (We remark that the corresponding procedure becomes rather complicated when we employ W 4 (see (1.6)), as in [6], to measure the size of data.) We are naturally led to proving Theorem 1.3 first for compactly supported smooth data (because the proof of global existence becomes easier for such initial data), and then we use this helpful property to complete its proof by passing to the limit of a sequence of compactly supported (for any fixed time) smooth solutions. See Section 8. Secondly, when the initial data is radially symmetric about x = 0, we easily see the size condition in Theorem 1.3 is satisfied whenever the norm with the low weight x := 1 + |x| 2 is small enough. Note that, thanks to its low weight, we can allow such an oscillating and slowly decaying data as g 1 (x) = x −d sin x with d > 5/2. Note that by setting H 11,αβ 2 = 0 for all α, β and choosing the trivial data u 2 (0, x) = ∂ t u 2 (0, x) = 0 and thus considering the trivial solution u 2 (t, x) ≡ 0, we can go back to the wave equation for the scalar unknowns (1.11) ✷u + G αβγ (∂ γ u)(∂ 2 αβ u) + H αβ (∂ α u)(∂ β u) = 0, t > 0, x ∈ R 3 and thus obtain: Theorem 1.4. Suppose the symmetry condition G αβγ = G βαγ . Also, suppose the null condition: there holds Let δ, η and µ be sufficiently small positive constants. Then, there exists ε ∈ (0, 1) such that if f ∈ L 6 (R 3 ) and D(f, g) ≤ ε, then the Cauchy problem (1.11) with initial data (f, g) given at t = 0 admits a unique global solution u(t, x) satisfying ess sup t>0 N 4 (u(t)) (1.13) See the beginning of the next section for the definition ofZ. This improves Theorem 1.1 of the second author [25] which says global existence of solutions to (1.11) in the absence of the semi-linear term H αβ (∂ α u)(∂ β u) for small data with higher regularity than is assumed in Theorem 1.4. Since we no longer assume compactness of the support of initial data, Theorem 1.4 is also an improvement of global existence results of [2] and [7] for (1.11) (see [2, p. 94] and [7, Theorem 1.5]).
The operators L k together with the other elements of Γ played an essential role in the proof of Theorem 1.1. Namely, the use of all the elements of Γ was crucial for the purpose of getting time decay estimates for local solutions with the help of the inequality of Klainerman [12] and its H 1 -L q version due to Ginibre and Velo [4]. Since we avoid the use of the operators L k , some good substitutes for these inequalities are necessary. In fact, there already exist two major ways of obtaining time decay estimates without relying upon L k . One is to use point-wise decay estimates for homogeneous and inhomogeneous wave equations (see, e.g., [24]). The other is to use the Klainerman-Sideris inequality [13] in combination with some Sobolev-type inequalities with weights such as t − r , r 1/2 t − r . As in [25], we proceed along the latter approach to compensate for the absence of L k in the list of the available differential operators and intend to combine the ghost weight method of Alinhac with the Klainerman-Sideris method. Actually, such an attempt of combining these two methods has been already made in [25]. With the help of some observations in [5] and [7], we adjust the machinery thereby assembled in [25], in order to reduce the amount of regularity of initial data, and also to discuss the system (1.1) violating the standard null condition but satisfying the weak null condition. We hope that in the future, this machinery will be useful in discussing the Cauchy problem for a nonrelativistic system satisfying the weak null condition or the initial-boundary value problems in a domain exterior to an obstacle. This paper is organized as follows. In the next section, we prove some basic inequalities. In Section 3, we consider the bound for the weighted L 2 norm of the second or higher-order derivatives of local solutions. Sections 4-5 and 6-7 are devoted to the energy estimate and the space-time L 2 estimate for local solutions, respectively. In Section 8, we complete the proof of Theorem 1.3 by the continuity argument.
holds with the new coefficients {G αβγ i } also satisfying the null condition. Also, the equality holds with the new coefficients {H αβ i } also satisfying the null condition.
For the proof, see, e.g., [2, p. 91]. It is possible to show the following lemma essentially in the same way as in [2, pp. 90-91]. Together with it, we will later exploit the fact that for local solutions u, the special derivatives T i u have better space-time L 2 integrability and improved time decay property of their L ∞ (R 3 ) norms.
We also need the following inequality.
In our proof, the trace-type inequality also plays an important role. (For the proof, see, e.g., (3.16) of [20].) Lemma 2.6. There exists a positive constant C such that if v = v(x) decays sufficiently fast as |x| → ∞, then the inequality

holds.
We also need the space-time L 2 estimates for the variable-coefficient operator P defined as 3), and suppose the symmetry condition h αβ = h βα . We have the following: Lemma 2.7 (Theorem 2.1 of [6]). For 0 < µ < 1/2, there exists a positive constant C such that the inequality holds for smooth and compactly supported (for any fixed time) functions u(t, x).
See also [17] for an earlier and related estimate. The estimate (2.24) was proved by the geometric multiplier method of Rodnianski (see Appendix of [23]). At first sight, the above estimate may appear useless for the proof of global existence, because of the presence of the factor (1 + T ) −2µ . Combined with Lemma 2.5 and the useful idea of dyadic decomposition of the time interval (see (4.49) below), the estimate (2.24) actually works effectively for the proof of global existence with no use of L j and with limitation of the occurrence of S to 1 in the definition of N 4 (u(t)).
The following was proved by Klainerman and Sideris, and will be used in the proof of Proposition 3.4 below. By setting t = 0 in (2.25), we get the simple inequality M 2 (v(0)) ≤ C KS N 2 (v(0)) which, together with Proposition 3.4, will be used in the proof of Proposition 8.1 below. [13]). There exists a constant C KS > 0 such that the inequality

Bound for M 4 (u(t))
Since the second order quasi-linear hyperbolic system (1.1) can be written in the form of the first order quasi-linear symmetric hyperbolic system (see, e.g., (5.9) of Racke [18]), the standard local existence theorem (see, e.g., Theorem 5.8 of [18]) applies to the Cauchy problem for (1.1). To begin with, we assume that the initial data are smooth, compactly supported, and small so that (3.6), and (3.18) for the constants ε * 1 , ε * 2 , and ε * 3 , respectively. See (8.5) for A, and see (8.1) for C 0 and C 1 . See (8.8) and the inequality following it for C 2 and C 3 . Note that ε 0 is independent of R * (see Remark 1.2).
We know that a unique, smooth solution to (1.1) exists at least for a short time interval, and it is compactly supported at any fixed time by the finite speed of propagation.
Before entering into the energy estimate in the next section, we must refer to an elementary result concerning point-wise estimates for u 1 and u 2 . It compensates for the absence of Lemma 3.1. There exists a constant ε * 1 > 0 depending on the coefficients of (1.1) with the following property: whenever smooth solutions u = (u 1 , u 2 ) to (1.1) satisfy holds.
There exists a constant ε * 2 > 0 depending on the coefficients of (1.1) with the following property: whenever smooth solutions u = (u 1 , u 2 ) to (1.1) satisfy For the proof of Lemmas 3.1 and 3.2, we have only to repeat essentially the same argument as in the proof of Lemma 3.2 of [5]. We thus omit the proof.
Also, it follows from (3.5) and (3.3) that Moreover, we also have from (3.7) and (3.8) with |a| = 1 for i = 1, 2. These inequalities will be frequently used in the following.
Proof. We prove Proposition 3.4. Corollary 3.5 is an immediate consequence of it, because C u(t) M(u(t)) can be absorbed into the left-hand side of (3.17) for small u(t) . We use (3.11), (3.12) with |a| ≤ 2, d = 0. Obviously, it suffices to explain how to bound M 2 (Z a u i (t)) for |a| = 2, i = 1, 2.
We first bound M 2 (Z a u 1 (t)). In view of the Klainerman-Sideris inequality (2.25), our task is to bound the L 2 (R 3 ) norm of the 2nd, 3rd, . . . , and 8th terms on the left-hand side of (3.11) for |a| = 2, d = 0. In fact, it is enough to bound the 5th and 8th terms for |a ′ | + |a ′′ | = 2 because the others can be handled similarly. For any fixed t ∈ (0, T ), we bound their L 2 norm over the set {x ∈ R 3 : |x| ≤ (t + 1)/2} and its complement, separately. Let χ 1 (x) be the characteristic function of this set, and we set χ 2 (x) := 1 − χ 1 (x). Recall that we now have |a ′ | + |a ′′ | ≤ 2 at the 5th term on the left-hand side of (3.11). For |a ′ | ≤ 1, we get by (3.8) and (2.21) where, by virtue of the weight |x| appearing on the left-hand side of (2.21), we have used the Hardy inequality at the last inequality. For |a ′ | = 2 (therefore |a ′′ | = 0), we get by (3.3) and the Hardy inequality For the 8th term on the left-hand side of (3.11), we get, assuming |a ′ | ≤ |a ′′ | (therefore, |a ′ | ≤ 1) without loss of generality in the same way as above.
Next, let us consider the estimate over the set {x ∈ R 3 : |x| > (t + 1)/2} for any fixed t ∈ (0, T ). Recall that χ 2 (x) = 1 − χ 1 (x). Since the coefficientsĜ αβγ satisfy the null condition thanks to Lemma 2.1, we can first use Lemma 2.2 to get and then we use Lemma 2.3 to get It suffices to show how to treat the 3rd and 4th terms on the right-hand side of the second last inequality, because the other terms can be estimated in a similar way.
If |a ′ | ≤ 1, then we obtain If |a ′ | = 2, then we use the L 4 ω norm as above to obtain We thus conclude that Similarly, the coefficientsH αβ satisfy the null condition and thus we can use (2.10) to get the inequality for |a ′ | + |a ′′ | ≤ 2 in the same way as above.
Let us turn our attention to the bound for M 2 (Z a u 2 (t)), |a| ≤ 2. Naturally, we may focus on the 2nd, 4th, 6th, and 7th terms on the left-hand side of (3.12) whose coefficients do not necessarily satisfy the null condition. We will show how to treat the 4th and 6th terms, because the 2nd and 7th terms can be handled similarly. For the 3rd , 5th, and 8th terms whose coefficients satisfy the null condition, we have only to proceed as we did in the treatment of M 2 (Z a u 1 (t)), thus we may omit the details.
Let us resume with the estimate of the 4th and 6th terms. Recall that Lemma 2.2 has played no role in (3.20)-(3.22) and it has played an essential role in (3.23)-(3.31). Since we can no longer use Lemma 2.2, our task is to consider their bound over the set {x ∈ R 3 : |x| > (t + 1)/2}. If |a ′ | ≤ 1, then we get by (3.8) If |a ′ | = 2, then we get by (3.3) Similarly, we obtain Summing up, we have finished the proof of Proposition 3.4.
We have finished the estimate for χ 1 J 1,2 .
When |a ′ | ≤ 2 and |a ′′ | = 0, we obtain We also get in the same way as in (4.44) We are ready to complete the proof of Proposition 4.1. First, we note that owing to (4.16), the function g(t − r) is bounded, which means that the function e g(t−r) appearing in (4.1) satisfies c ≤ e g(t−r) ≤ C for some positive constants c and C. Second, we must mention how to deal with rather troublesome terms t 0 τ −(1/2)−2µ+2δ L(u 1 (τ ))L(u 2 (τ ))dτ, (4.47) t 0 τ −1+η+δ G(u 2 (τ ))dτ, (4.48) which naturally come from the integration of such terms as in (4.21) and (4.33) with respect to the time variable. As in [22, p. 363], the idea of dyadic decomposition of the interval (0, t) plays a useful role. Without loss of generality, we suppose T > 1. For any t ∈ (1, T ), we see Here, and later on as well, we abuse the notation to mean t by 2 N +1 . Because δ is a sufficiently small positive number, we are able to obtain the desired estimate. Also, , because δ and η are sufficiently small positive numbers. The estimate of the integration from 0 to 1 is much easier, thus we omit it. Integrating (4.1) over (0, t) × R 3 , we are now able to obtain (4.14). The proof of Proposition 4.1 has been finished.

Energy estimate of u 2
This section is devoted to the energy estimate of u 2 . We will show: Proposition 5.1. The following inequality holds for smooth local solutions to (1.1) u = (u 1 , u 2 ), as long as they satisfy (3.18) for some time interval (0, T ) : The rest of this section is devoted to the proof of this proposition. As in the previous section, we have only to deal with the highest-order energy. In the same way as in (4.1), we get + ∇ · {· · · } + e gq + e g (J 2,1 + J 2,2 + · · · + J 2,5 ) = 0.
Here, g = g(t − r) is the same as in (4.16),q = q 3 − (1/2)g ′ (t − r)q 4 , and Recall that we have dealt with χ 1 q and χ 1 J 1,1 , . . . , χ 1 J 1,5 without relying upon the null condition. Therefore, it is possible to handle χ 1q and χ 1 J 2,1 , . . . , χ 1 J 2,5 as before. We may thus focus on the estimate of χ 2q and χ 2 J 2,1 , . . . , χ 2 J 2,5 . For the estimate of χ 2q , it suffices to show how to handle the terms with the coefficients {G 12,αβγ 2 }, because the coefficients {G 22,αβγ 2 } satisfy the null condition and thus we are able to treat all the terms with the coefficients {G 22,αβγ 2 } in the same way as before. Using the first equation in (1.1) to represent ∂ 2 t u 1 as ∆u 1 + (higher-order terms) and then use (3.3) to represent ∂ 2 t u 1 appearing in these higher-order terms, we obtain (If we employed (3.3) directly, it would meet with the troublesome factor t −1+3δ on the right-hand side above. This is the reason why we have used the first equation in (1.1) to represent ∂ 2 t u 1 as ∆u 1 + (higher-order terms).) Here, we have used the assumption that u is small, so that we have u 2 ≤ u . In the same way, we get It is easy to show We have finished the estimate of χ 2q . We next deal with χ 2 J 2,1 , . . . , χ 2 J 2,5 . We may focus on χ 2 J 2,1 , χ 2 J 2,3 , and χ 2 J 2,4 because the coefficients {Ĝ αβγ } and {H αβ } satisfy the null condition and it is therefore possible to handle χ 2 J 2,2 and χ 2 J 2,5 in the same way as before. Let us first deal with χ 2 J 2,1 . When |a ′′ | + d ′′ = 2 (and thus |a ′ | + d ′ ≤ 1), we get by (3.8)-(3.10) Note that we have again used (3.10) to handle χ 2 ∂ 2 tZ a ′′ Su 2 (t) L 2 (R 3 ) (|a ′′ | = 1) along with |x|∂Su 2 (t) L ∞ ≤ t δ u(t) and smallness of u(t) as follows: On the other hand, when |a ′′ | + d ′′ ≤ 1 (and thus |a ′ | + d ′ ≤ 2), we get It is easy to obtain a similar estimate for χ 2 J 2,3 and χ 2 J 2,4 . Using the basic fact that the integration of (1 + τ ) −1+2δ from 0 to t is O(t 2δ ) for large t, we can now complete the proof of Proposition 5.1. The proof has been finished.
7. Space-time L 2 estimates of u 2 In this section, we consider the space-time L 2 estimates of u 2 . We can prove: Proposition 7.1. The following inequality holds for smooth local solutions to (1.1) u = (u 1 , u 2 ), as long as they satisfy (3.18) for some time interval (0, T ) : We have only to repeat essentially the same argument as in Section 6. We thus omit the proof.

Proof of Theorem 1.3
So far, we have proved that local solutions to (1.1) defined for (t, x) ∈ [0, T ) × R 3 with compactly supported smooth data satisfy for suitable constants C 0 , C 1 > 0, provided that See (3.18) for ε * 3 . In order to get the key a priori estimate (see (8.11) below), we must show that u(t) is small (at least for a short time interval), whenever N 4 (u 1 (0)) + N 4 (u 2 (0)) is small enough. (See (8.10) below.) Since initial data belong to C ∞ 0 (R 3 ) × C ∞ 0 (R 3 ) and the uniqueness theorem of C 2 -solutions and its corollary in [9, p. 53] apply to the system (1.1), smooth local solutions satisfy where R > 0 is a constant such that u i (0, x) = ∂ t u i (0, x) = 0 (i = 1, 2) for |x| ≥ R.
For the constant A, see (8.5) above.
To complete the proof of Theorem 1.3, we must relax the regularity of data and eliminate compactness of the support of data. Naturally, we employ the standard mollifier and cut-off idea (see, e.g., [3, p. 12] and [8, p. 122]). Then, we easily see that, for any (f i , g i ) (i = 1, 2) satisfying f 1 , f 2 ∈ L 6 (R 3 ) and D(f 1 , g 1 ) + D(f 2 , g 2 ) ≤ ε 0 /2, there exists a sequence (f i,n , g i,n ) ∈ C ∞ 0 (R 3 ) × C ∞ 0 (R 3 ) (n = 1, 2, . . . ) such that (8.12) (We must keep in mind that this procedure becomes rather complicated when we employ W 4 (see (1.6)), as in [7], to measure the size of data.) Thanks to (8.12), we know that the Cauchy problem (1.1) with data (u i (0), ∂ t u i (0)) = (f i,n , g i,n ) (i = 1, 2) admits a unique solution, which is denoted by u n (t, x) = (u 1,n (t, x), u 2,n (t, x)), for every large n. Also, we have (8.14) N T (u n ) + G T (u n ) + L T (u n ) ≤ C i=1,2 D(f i,n , g i,n ) ≤ Cε 0 for all T > 0, with a constant C > 0 independent of n and T . Furthermore, owing to (8.14) and M(u n (t)) ≤ CN (u n (t)) for 0 < t < ∞ (see (8.6)), we obtain by the same argument as (in fact, essentially simpler argument than) in Sections 4-7, with a few obvious modifications sup t>0 N 1 (u 1,m (t) − u 1,n (t)) + sup t>0 t −δ N 1 (u 2,m (t) − u 2,n (t)) (8.15) for sufficiently large m, n, with a constant C independent of m, n. (When showing (8.15), we are supposed to choose ε 0 smaller than before, if necessary.) We thus see by the standard argument that u n = (u 1,n , u 2,n ) has the limit that is the solution to (1.1) with the data (f i , g i ) (i = 1, 2) given at t = 0. The proof of Theorem 1.3 has been completed.
Acknowledgements. The first author was supported in part by JSPS KAK-ENHI Grant Number JP15K04955 and JP18K03365. The second author was supported by the Fundamental Research Funds for the Central Universities (No.17D110913) and Shanghai Sailing Program (No.17YF1400700).