13.2 Proof of decay of the energy

Let us briefly review the idea behind the proof of Theorem 4 [75Jump To The Next Citation Point, Section 4, pp. 668–670]. We shall do so by considering a simple example. Consider, in the polarized case, a solution of the form P (t,𝜃) = x(t)cos𝜃. With this ansatz, Equation (51View Equation) is equivalent to
¨x + 1x˙+ x = 0. t

13.2.1 Toy model

Let us simplify the above equation further and consider

2 ¨x + 2a˙x + b x = 0,

where a and b are constants and a > 0 and 2 2 b > a. Clearly, this equation is of no interest in itself; there is no need to develop methods for analyzing an equation, which can be solved explicitly. However, considering this equation from a different perspective might lead to the development of methods that can be used in a more general situation. Since energies have turned out to be very useful in the analysis of systems of nonlinear wave equations, let us consider

1 2 2 2 H = 2(x˙ + b x ).

We know that this quantity decreases exponentially as e−2at. Let us try to prove this statement without explicitly solving the equation. A natural first step is to differentiate:

dH--= − 2a ˙x2. dt

Clearly, H decreases. However, this is only a qualitative statement. In order to obtain a quantitative statement, let us introduce

Γ = axx˙.

We shall refer to this quantity as a correction term. It is important to note that this object has the following two properties. First

| | | | | | |a-| |a|1- 2 2 2 |a| |Γ | = |b ||bx ˙x| ≤ |b|2 (˙x + bx ) = |b|H.

As a consequence, there are constants c1, c2 > 0 such that

c1H ≤ H + Γ ≤ c2H;

recall that |a ∕b| < 1. Second,

d(H--+-Γ-) dt = − 2a (H + Γ ).

Combining these two properties, we obtain H ≤ K exp (− 2at). This estimate is optimal.

13.2.2 Polarized case

In the case of the polarized Gowdy equation (51View Equation) it is possible to carry out a similar argument. In fact, the correction

1 ∫ Γ = -- (P − ⟨P ⟩)Ptd𝜃 2t S1

can be used to prove that H ≤ K ∕t [75Jump To The Next Citation Point, Section 4, pp. 668–670] for the details.

13.2.3 General case

Ideas similar to the above can be used in the general Gowdy case, though there are additional complications. Due to the nonlinear character of the problem, it is necessary to divide the proof of the decay of the energy into two parts. The first part involves proving that if the energy is small initially, then the energy decays as 1∕t. The second step consists of proving that the energy converges to zero. The small data result is, just as above, based on the introduction of a certain correction Γ with the properties that

K-- |Γ | ≤ t H

and that

d(H--+-Γ ) 1- 1- K-- 3∕2 dt ≤ − t(H + Γ ) − tΓ + t H . (56 )
Combining these two inequalities, it is possible to conclude that −1 H = O (t ), assuming H to be small enough initially.

One way to take the step from small data to large data is to prove that H converges to zero. What is known a priori is that

dH 1 ∫ ---- = − -- (P2t + e2PQ2t)d𝜃. (57 ) dt t S1
Just as before, this implies that H decays, but not that H converges to zero. However, it does prove that the right-hand side of Equation (57View Equation) is integrable. This information might not seem so useful. However, if it were possible to prove the integrability of H ∕t to the future, we would be allowed to conclude that H converges to zero (recall that H is monotonically decreasing). This would then finish the result. In order to take the step from integrability of the right-hand side of (57View Equation) to the integrability of H ∕t, we need to prove that
∫ t ∫ ∫ t ∫ 1- (P 2− P 2)d𝜃ds, 1- e2P (Q2 − Q2 )d 𝜃ds t0 s S1 t 𝜃 t0 s S1 t 𝜃

are bounded to the future. Note that these expressions are far from arbitrary. It is natural to integrate by parts twice and to use the equations (it is of some interest to note that the order in which one considers the two expressions is very important). Doing so leads to the desired conclusion; see [75Jump To The Next Citation Point] for the details.

It is of interest to note that the results concerning the decay rate can be generalized to a larger class of spacetimes [81].

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