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

In fact, this result can be improved somewhat to [75, Theorem 1.6, p. 664]:

Theorem 5 Consider a solution to Equations (11)–(12). Then if is given by Equation (53), there is a , a and a constant such that

for all . Furthermore, if is zero, the solution is independent of , and in that case, is constant.Note that, in some respects, this result leads to conclusions that, on an intuitive level, are somewhat contradictory. First, since converges to zero, it is clear that the spatial variation of the solution, i.e., , 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 and denote the corresponding energies. Then, due to Theorem 5,

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

http://www.livingreviews.org/lrr-2010-2 |
This work is licensed under a Creative Commons License. Problems/comments to |