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8 1. THE LOCAL THEORY OF THE CAUCHY PROBLEM 10j(ε + ε ) if γj ≤ C0. If we choose ε0 so small that 10J+1 (ε0 + ε0) ≤ C0, the condition γj ≤ C0 always holds, so that, since w S(Ij) + Dβw Lq I j Lx q ≤ 3γj, we obtain the desired estimates for u S(I) + Dβu LqLxq I . The second estimate follows from the first one using a similar argument, which concludes the proof. Some useful corollaries of the Perturbation Theorem: Corollary 1.13. Let K ⊂ ˙ 1 × L2 be so that K is compact. Then ∃T + K 0,T − K 0 such that ∀(u0,u1) ∈ K, T+ (u0,u1) ≥ T + K , T−(u0,u1) ≤ T − K : Choose A = sup(u 0 ,u1) (u0,u1) ˙ 1 ×L2 , A = C(A), with C as in the “local theory of the Cauchy problem”, M = 1, ε0 = ε0 (1, 1,A) as in the Perturba- tion Theorem, ε1 ≤ min(ε0, 1). Cover K by balls B ((u0,j,u1,j) ε1) , 1 ≤ j ≤ J, by compactness of K. Find T + j , T − j so that uj S(T − j ,T + j ) ≤ 1 and set T + k = min1≤j≤J T + j , T − K = max1≤j≤J T − j . Then, if (u0,u1) ∈ B ((u0,j,u1,j) ε1), for some j, by the Perturbation Theorem, the solution exists in T − K , T + K , as desired. Corollary 1.14. If (u0, u1) ∈ ˙ 1 ×L2, (u0, u1) ˙ 1 ×L2 ≤ A, u is the solution in (T− (u0, u1) , T+ (u0, u1)), and (u0,n,u1,n) → (u0, u1) in ˙ 1 × L2, then T− (u0, u1) ≥ lim T− (u0,n,u1,n) , T+ (u0, u1) ≤ lim (T+ (u0,n,u1,n)) and ∀t ∈ (T− (u0, u1) , T+ (u0, u1)) , (un(t),∂tun(t)) → (u(t),∂tu(t)) in ˙ 1 × L2. In fact, if I ⊂⊂ I, I = (T− (u0, u1) , T+ (u0, u1)), we have that sup t∈I (u(t),∂tu(t)) ˙ 1 ×L2 ≤ C(A, I ), u S(I ) ≤ M(A, I ). If ε0 = ε0 (M, C, 1) and n is so large that (u0,n − u0,u1,n, −u1) ˙ 1 ×L2 ≤ 1, S(t) (u0,n − u0,u1,n − u1) S ≤ ε0, we have that un exists on I and sup t∈I (un(t) − u(t),∂tun(t) − ∂tu(t)) ˙ 1 ×L2 ≤ C(I ) (u0,n − u0,u1,n − u1) α ˙ 1 ×L2 and the claim follows. Remark 1.15. Using Remark 1.10 and the Perturbation Theorem, if (u0,u1) and I are as in the “local theory of the Cauchy problem”, and supp(u0,u1) ⊂ B (0,b), then u(x, t) ≡ 0 on {(x, t) : |x| b + t, t ≥ 0,t ∈ I}. Hence, if (u0,u1),I are as in the “local theory of the Cauchy problem”, using the Perturbation Theorem, we can approximate the solution u on I by regular, compactly supported solutions. Similar statements hold for t 0. A useful Lemma, in connection with these facts is: Lemma 1.16. Assume that (v0,v1) ∈ ˙ 1 × L2, (v0,v1) ⊂ K, K compact in ˙ 1 × L2. Let Ψ be radial, Ψ ∈ C∞(R3), 0 ≤ Ψ ≤ 1, Ψ ≡ 0 for |x| 1, Ψ ≡ 1 for |x| 2, ΨM(x) = Ψ( x M ). Let vM be the solution with initial data (ΨMv0, ΨMv1).