A note on quasilinear wave equations in two space dimensions

In this paper, we give an alternative proof of Alinhac's global existence result for the Cauchy problem of quasilinear wave equations with both null conditions in two space dimensions[S. Alinhac, The null condition 
 for quasilinear wave equations in two space dimensions I, Invent. Math. 
 145 (2001) 597--618]. The innovation in our proof is that when applying the vector fields method to do the generalized energy estimates, we don't employ the Lorentz boost operator and only use the general space-time derivatives, spatial rotation and scaling operator.

In the initial conditions of (u, u t ), ε is a positive small parameter and u 0 , u 1 ∈ C ∞ c (R 2 ). Without loss of generality, we can assume that u 0 and u 1 are supported in |x| ≤ 1. We always assume that the following symmetric conditions hold: The set g = (g αβγ ) and h = (h αβγδ ) are said to satisfy the null conditions if for any x ∈ R 2 , ω 0 = −1, ω i = x i /r(i = 1, 2), r = |x|, we have and h αβγδ ω α ω β ω γ ω δ = 0, respectively. In Alinhac [2], for the Cauchy problem (1)-(2), the following global existence result is proved.
1.1. Notation. Denote the spatial rotation and the scaling operator by respectively. We define the vector fields Z = (∂, Ω, S) = (Z 1 , · · · , Z 5 ). For multiindices a = (a 1 , · · · , a 5 ), we denote Z a = Z a1 1 · · · Z a5 5 . b ≤ a means for each i = 1, · · · , 5, b i ≤ a i . It is easy to verify that we have the following commutation property Z, ∂ = ∂. We also use the vector fields(the so called "good derivatives") where ω 0 = −1, ω i = x i /r(i = 1, 2), r = |x|. Denote T = (T 0 , T 1 , T 2 ). It is easy to verify that for r ≥ t/2, where σ = (1 + |σ| 2 ) 1/2 . The energy associated to the linear wave operator is and the higher order energies are We will also use the following weighted L 2 norm To simplify the exposition, we truncate the nonlinearity at the cubic level, but this entails no loss of generality since the higher order terms have no essential influence on the discussion of the global existence of solutions with small amplitude. The nonlinear equation which we consider can be written as the following form where 2. Some preliminaries.
Combination of Lemma 2.1 and the decay estimates (10) gives Lemma 2.2. If g = (g αβγ ) and h = (h αβγδ ) satisfy the null conditions, then for any r ≥ t/2, Lemma 2.3. Let u be a solution of (14) and assume that the null conditions (6), (7) hold for the nonlinearities (15), (16) respectively. Then for any multi-indices a = (a 1 , · · · , a 5 ), we have where each N 1d is a quadratic nonlinearity of the form (15) satisfying the null condition (6) and each N 2e is a cubic nonlinearity of the form (16) satisfying the null condition (7).
Now we give the higher order version of Lemma 2.4.

DONGBING ZHA
On the region r ≤ 1, by the Sobolev embedding The higher order version of Lemma 2.6 is the following: 2.3. Weighted decay estimates. In this section, we will give the control of the weighted L 2 norm M k (u(t)).
Proof. See Klainerman and Sideris [10], Lemma 3.1. Note that their proof is obvious valid for all spatial dimension n ≥ 2.
Next we estimate the nonlinear terms on the right-hand side of (41).
We first estimate the first collection on the right hand side of (43). On the region r ≤ t/2, noting that t − r ∼ t , we have Let m = [ k−2 2 ] + 1 = k − 4. We have either |b| ≤ m or |c| ≤ m. In the first case, (44) can be estimated as follows using (29): Otherwise, we use (31) to estimate (44) by On the region r ≥ t/2, it follows from Lemma 2.2 and Lemma 2.3 that For the first term on the right hand side of (47), when |b| ≤ m, it follows from (40) that When |c| ≤ m, by (39), we have For the second and third term on the right hand side of (47), similar dichotomy argument can derive the following bound: Now we estimate the second collection on the right hand side of (43). We must estimate terms of the form Noting that (t + r) ≤ C r t − r , we have that In the first case, owing to (29), we have the upper bound In the second and the third case, the combination of (29) with (31) gives the upper bound be a solution of system (14) and k ≥ 10.
Define µ = k − 3, and assume that is sufficiently small. Then for 0 ≤ t ≤ T , Proof. See the proof of Lemma 7.3 in Sideris and Tu [16].
3. Proof of Theorem 1.1. In this section we shall complete the proof of Theorem 1.1. Assume that u(t) is a local smooth solution of the initial value problem (14)- We will prove that for any t ∈ [0, T ], it holds where the constant A 1 , A 2 , c will be determined later. There are two key steps in the proof. First, under the assumption (i) for the lower order energy, we will prove higher order energy estimate (ii). Second, under the assumption (i) and (ii), we will show that (i) holds with A 1 replaced by A 1 /2. To accomplish this bootstrap argument, we will derive a pair of coupled differential inequalities for the lower order energy E µ (u(t) and the (modified) higher order energy E k (u(t).
3.1. High order energy. Following Alinhac [2], we will use the ghost weight energy method. Let σ = t − r, q(σ) = arctan σ, q (σ) = 1 As we are now treating quasilinear system, for the right hand side of (58), |a| = k−1, special attention should be paid to terms with b = a or c = a for the first collection of terms and d = a for the second collection of terms. So we can rewrite (58) as By the integration by parts argument, we have Since q is bounded, there exists c > 1 such that We have It follows from the symmetric conditions (4) and the integration by parts argument that where the symbol η γν = diag[1, −1, −1]. Similarly, via the symmetric conditions (5), we can get Define the perturbed energy Noting that ||∂u|| L ∞ ≤ CE 1/2 3 (u(t)), for small solutions, by (63) we have Noting the symmetric condition (4), we have Returning to (59), we have derived the following energy identity: The second term on the left hand side of (69) is called the ghost weight energy, which play a key role in the control of the highest generalized derivatives in the nonlinearity after applying the null condition. Now we can estimate the terms on the right hand side of (69). All terms corresponding to the cubic nonlinearity are bounded above by Note that t ≤ C r t − r . Let m = [ k 2 ] + 1. It follows from (30) that Similarly, we have To estimate the terms on the right hand side of (69) corresponding to the quadratic nonlinearity, we need to exploit the null condition. So we will separate integrals over the regions r ≤ t/2 and r ≥ t/2.
Inside the cone. On the region r ≤ t/2, all terms on the right hand side of (69) corresponding to the quadratic nonlinearity are bounded above by Since r ≤ t/2, we have t − r ∼ t . For the first part of (73), it follows from (29) and (31) that It follows from Lemma 2.10 that we have achieved an upper bound of the form For the second part in (73), by (29) we have Away from the origin. Now we consider the terms on the right hand side of (69) corresponding to the quadratic nonlinearity on the region r ≥ t/2. For Denote m = [ k−1 2 ] + 1. It follows from (31), (39) and (40) that ||∂Z a uZ b+1 u∂ 2 Z c u|| L 1 (r≥t/2) The second term can be handled using (29): The final term can be estimated using (29), (31), (39) and (40): It follows from (19), the Cauchy-Schwartz inequality, (10), (29), (31) and (56) that ||g αβγ e −q(σ) ∂γZ a u∂α∂tu∂ β Z a u|| L 1 (r≥t/2) ≤ C||T Z a u∂ 2 u∂Z a u|| L 1 (r≥t/2) + C||∂Z a uT ∂u∂Z a u|| L 1 (r≥t/2) The first term on the right hand side of (81) can be absorbed into the ghost weight energy term on the left hand side of (69). Now we estimate the terms involve q (σ) = t−r −2 . Owing to (19), the Cauchy-Schwartz inequality, (10), (29) and (40), we have 3.2. Low order energy. Following the general energy method, we have Inside the cone. On the region r ≤ t/2, the right hand side of (83) is bounded above by Since r ≤ t/2, we have t − r ∼ t . It follows from (30) that a typical term in the first part of (84) can be estimated as the following: Noting that b + c ≤ a, |a| ≤ µ − 1, µ = k − 3, k ≥ 20. We have either |b| + 3 ≤ µ and |c| + 2 ≤ k, or |b| + 3 ≤ k and |c| + 2 ≤ µ. By Lemma 2.10, we have achieved an upper bound of the form Similar argument gives the following upper bound for the second part of (84): Away from the origin. On the region r ≥ t/2, we must use the null condition. An application of Lemma 2.2 gives We still need to squeeze out an additional decay factor of t −1/2 . Since r ≥ t/2, we have r ≥ C t . It follows from (31), (39), (56) and (57) that The second term can be handled similarly by using (29): ||∂Z a u∂Z b u∂Z c+1 u|| L 1 (r≥t/2) |b|+3 (u(t))E 1/2 |c|+2 (u(t)) ≤ C t −1/2 E µ (u(t))E 1/2 k (u(t)).
3.3. Conclusion of the proof. Now we have get a pair of coupled differential inequalities for the lower order energy E µ (u(t) and the modified higher order energy E k (u(t): d dt E µ (u(t)) ≤ C 2 t −3/2 E µ (u(t)) E 1/2 k (u(t)) + C 2 t −2 E µ (u(t)) E k (u(t)).