Clearly, the situation is more complicated in the nonpolarized case. Consequently, it is natural to start by asking questions concerning the rough behavior of solutions. It is also natural to carry out numerical simulations. This was done in [8], a paper, which played an important role in the further development of the subject. In the polarized case, the difference between the solution and its spatial average converges to zero. Does the same phenomenon occur in the general case? This is perhaps too complicated a question to start with, but it is of interest to know how the spatial variation of the solution evolves with time. In order to define what is meant by the spatial variation, recall that Equations (11) – (12) can be viewed as wave-map equations in which the metric of the target is given by Equation (20). Furthermore, at each point in time, the solution defines a loop in hyperbolic space. The natural definition of the spatial variation of the solution at one point in time is the length of this loop with respect to the hyperbolic metric (20). In other words, it is natural to ask how

evolves with time. In the case of polarized Gowdy, we know that

asymptotically; see Equation (52) and the adjacent text. Consequently, in that case, but there is no better estimate.

13.1 Energy decay

13.2 Proof of decay of the energy

13.2.1 Toy model

13.2.2 Polarized case

13.2.3 General case

13.3 Asymptotic ODE behavior

13.3.1 Conserved quantities

13.3.2 Interpreting the conserved quantities as ODEs for the averages

13.4 Geometric interpretation of the asymptotics

13.5 Concluding remarks

13.6 Geodesic completeness

13.2 Proof of decay of the energy

13.2.1 Toy model

13.2.2 Polarized case

13.2.3 General case

13.3 Asymptotic ODE behavior

13.3.1 Conserved quantities

13.3.2 Interpreting the conserved quantities as ODEs for the averages

13.4 Geometric interpretation of the asymptotics

13.5 Concluding remarks

13.6 Geodesic completeness

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