10 Asymptotic Velocity, General T3-Gowdy

The results we described in Section 8 and 9 consisted of constructions of solutions with certain types of asymptotics. However, considering the formulation of the strong cosmic-censorship conjecture, it is of interest to obtain conclusions given assumptions, which are phrased in terms of the initial data. The first question to ask is if there is a condition on initial data, which ensures the existence of asymptotic expansions. In [76Jump To The Next Citation Point], such a condition was established. Being phrased in terms of L2-based energies, the condition is rather technical. However, it does have the advantage of applying to higher dimensional analogues of the equations (as opposed to much of the analysis to be described below). Other, less technical, conditions were later established [12Jump To The Next Citation Point74Jump To The Next Citation Point]. Even though these results are of interest, they still only describe a part of the dynamics (as the construction of spikes, see Section 9, demonstrates). The question then arises concerning how to proceed. Considering the analysis in the polarized case, the asymptotic expansions (37View Equation) – (38View Equation) and the spikes, it is clear that the velocity v plays a central role. Thus, it is natural to try to prove that it is possible to make sense of the concept of a velocity under more general circumstances.

 10.1 Existence of an asymptotic velocity
 10.2 Relevance of the asymptotic velocity to the issue of curvature blow up
 10.3 Interpretation of the asymptotic velocity as a rate of convergence to the boundary in hyperbolic space
 10.4 Two dimensional version of the asymptotic velocity
 10.5 Dominant contribution to the asymptotic velocity
 10.6 Value of the asymptotic velocity as a criterion for the existence of expansions

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