Asymptotics of the solution to the polarized T3-Gowdy equations

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 (29

) – (31

) similar to those obtained for the VTD system. Define the energies as

∑ ∫ Ek = 1- [(∂j ∂τP)2 + e−2τ(∂j+1P )2]d𝜃 2j≤k−1 S1 𝜃 𝜃

for $k ≥ 1$ . Differentiating with respect to $τ$ , integrating by parts and using Equation (29 ) 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 $j$ , $j ∂𝜃∂τP$ is bounded to the future. Consequently, regardless of the choice of $j$ , $∂j𝜃P$ does not grow faster than linearly to the future. Inserting this information into Equation (29 ) and integrating yields two smooth functions $v$ and $ϕ$ such that

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 [50

, Theorem III.5, pp. 102–103].

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

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