Dominant contribution to the asymptotic velocity

10.5 Dominant contribution to the asymptotic velocity

It is important to note that, not only does the kinetic energy density converge pointwise, but in the limit, the only term contributing is $P 2τ$ . In fact, the following result holds (see [79

, Proposition 1.3, p. 983]):

Proposition 1 Consider a solution to Equations (16 )–(17 ) and let $𝜃0 ∈ S1$ . Then, either $Pτ(τ,𝜃0)$ converges to $v∞ (𝜃0)$ or to $− v∞ (𝜃0)$ . If $Pτ(τ,𝜃0) → − v∞ (𝜃0)$ , then $(Q1, P1) = Inv(Q, P )$ has the property that $P1τ(τ,𝜃0) → v∞ (𝜃0)$ . Furthermore, if $v∞ (𝜃0) > 0$ , then $Q1 (τ,𝜃0)$ converges to $0$ .

Similar to what we have already seen for false spikes, see Section 9.3, we see that by applying an inversion we can always obtain a non-negative limit for $P τ$ .