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

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., [12

, 76 , 83

]. However, we shall only describe some of them, beginning with [79

, 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 (16 )–(17 ) 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 [79 , Remark 1.6, p. 985].

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