10.6 Value of the asymptotic velocity as a criterion for the existence of expansions

Not only is the asymptotic velocity a geometric quantity (from the wave-map perspective), not only can it be used as an indicator for curvature blow up, it can also be used as a criterion to determine whether asymptotic expansions exist or not. There are many results of this form, see, e.g., [12Jump To The Next Citation Point7683Jump To The Next Citation Point]. However, we shall only describe some of them, beginning with [79Jump To The Next Citation Point, Proposition 1.5, p. 984] (note that this result was essentially obtained in a previous paper [74]):

Proposition 2 Let (Q,P ) be a solution to Equations (16View Equation)–(17View Equation) and assume 0 < v∞ (𝜃0) < 1. If Pτ(τ,𝜃0) converges to v∞ (𝜃0), then there is an open interval I containing 𝜃0, ∞ va,ϕ, q,r ∈ C (I,ℝ ), 0 < va < 1, polynomials Ξk for all k ∈ ℕ and a T such that for all τ ≥ T

∥Pτ(τ,⋅) − va∥Ck(I,ℝ) ≤ Ξk(τ)e− ατ, (45 ) ∥P (τ,⋅) − p(τ,⋅)∥ k ≤ Ξ (τ)e− ατ, (46 ) ∥ ∥C (I,ℝ) k ∥e2p(τ,⋅)Qτ(τ,⋅) − r∥Ck(I,ℝ) ≤ Ξk(τ)e− ατ, (47 ) ∥ ∥ ∥∥e2p(τ,⋅)[Q (τ,⋅) − q] +-r--∥∥ ≤ Ξ (τ)e− ατ (48 ) ∥ 2va ∥Ck(I,ℝ) k
where p(τ,⋅) = va ⋅ τ + ϕ and α > 0. If P τ(τ,𝜃0) converges to − v∞ (𝜃0), then Inv (Q,P ) has expansions of the above form in the neighborhood of 𝜃0.

It is worth noting that the above proposition proves that if 0 < v∞(𝜃0) < 1, then v∞ is smooth in the neighborhood of 𝜃0. In other words, knowledge concerning v∞ at one point can sometimes yield conclusions in the neighborhood of that point; see [79Jump To The Next Citation Point, Remark 1.6, p. 985].

Equation (48View Equation) essentially has the same content as Equation (38View Equation). In order to see this, define the object inside the norm on the left-hand side of Equation (48View Equation) to be w&tidle;. Then

[ ] −2p r Q = q + e − ----+ &tidle;w . 2va

Using the above expansions and equations, expressions for the higher-order time derivatives can be derived.

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