7.3 Asymptotics of the solution to the polarized T3-Gowdy equations

Let us briefly illustrate how one can obtain conclusions concerning solutions to Equations (29View Equation) – (31View Equation) similar to those obtained for the VTD system. Define the energies as
∑ ∫ Ek = 1- [(∂j ∂τP)2 + e−2τ(∂j+1P )2]dšœƒ 2j≤k−1 S1 šœƒ šœƒ

for k ≥ 1. Differentiating with respect to τ, integrating by parts and using Equation (29View Equation) leads to the conclusion that the energies Ek are decaying. Combining this observation with Sobolev embedding leads to the conclusion that, regardless of the choice of j, j ∂šœƒ∂τP is bounded to the future. Consequently, regardless of the choice of j, ∂jšœƒP does not grow faster than linearly to the future. Inserting this information into Equation (29View Equation) and integrating yields two smooth functions v and Ļ• such that

P(τ,šœƒ) = v(šœƒ)τ + Ļ• (šœƒ ) + u(τ,šœƒ ), Pτ(τ,šœƒ) = v(šœƒ) + w (τ,šœƒ ), (35 )
where u and w are functions such that they and all their spatial derivatives are O (τ e−2τ). In other words, the leading-order behavior is given by the solution to the VTD equations, and the VTD equations, in their turn, are obtained by essentially dropping the spatial derivatives. The above analysis should be compared with [50Jump To The Next Citation Point, Theorem III.5, pp. 102–103].
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