13.1 Energy decay

In order to analyze the general case, it is of interest to consider the energy
∫ 1 2 2 2P 2 2 H = 2- [Pt + P 𝜃 + e (Qt + Q 𝜃)]d𝜃. (53 ) S1
The numerical studies of Berger and Moncrief [8] indicate that this quantity should decay as 1∕t. Using Hölder’s inequality, this would imply that ℓ = O (t− 1∕2). That this is the behavior, which actually occurs was later established in [75Jump To The Next Citation Point, Theorem 1.1, p. 660]:

Theorem 4 Consider a solution to Equations (11View Equation)–(12View Equation). Then there is a T ≥ 1 and a K such that for all t ≥ T, the energy H defined by Equation (53View Equation) satisfies

K-- H (t) ≤ t . (54 )

In fact, this result can be improved somewhat to [75Jump To The Next Citation Point, Theorem 1.6, p. 664]:

Theorem 5 Consider a solution to Equations (11View Equation)–(12View Equation). Then if H is given by Equation (53View Equation), there is a K, a T > 0 and a constant cH such that

K |tH (t) − cH | ≤ --- (55 ) t
for all t ≥ T. Furthermore, if cH is zero, the solution is independent of 𝜃, and in that case, t2H (t) is constant.

Note that, in some respects, this result leads to conclusions that, on an intuitive level, are somewhat contradictory. First, since H converges to zero, it is clear that the spatial variation of the solution, i.e., ℓ, converges to zero. Consequently, it seems natural to expect the solution to behave as a spatially homogeneous solution to the equations. Thus, consider a non–spatially-homogeneous solution and a spatially homogeneous solution, which is supposed to approximate it, and let Hsol and Hhom denote the corresponding energies. Then, due to Theorem 5,

H − H H − H lim --sol-----hom- = ∞, lim --sol----hom-= 1, t→ ∞ Hhom t→ ∞ Hsol

even though both limits should be zero if the solution is well approximated by a spatially homogeneous solution.


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