Softcover ISBN:  9782856299265 
Product Code:  AST/420 
List Price:  $105.00 
AMS Member Price:  $84.00 
Softcover ISBN:  9782856299265 
Product Code:  AST/420 
List Price:  $105.00 
AMS Member Price:  $84.00 

Book DetailsAstérisqueVolume: 420; 2020; 512 ppMSC: Primary 35; 53; 58; 83
The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalized, converges. The model examples are the Kasner solutions. The main result of the article is optimal energy estimates. However, the author also derives asymptotics and demonstrate that the leading order asymptotics can be specified (also in situations where the asymptotics are not convergent).
It is sometimes argued that if the factors multiplying the spatial derivatives decay exponentially (for a system of wave equations), then the spatial derivatives can be ignored. This line of reasoning is incorrect; the author gives examples of equations such that: (1) the factors multiplying the spatial derivatives decay exponentially, (2) the factors multiplying the time derivatives are constants, (3) the energies of individual modes of solutions asymptotically decay exponentially, and (4) the energies of generic solutions grow as \(\mathrm{exp}[\mathrm{exp}(t)] as t\rightarrow\infty\).
When the factors multiplying the spatial derivatives grow exponentially, the Fourier modes of solutions oscillate with a frequency that grows exponentially. To obtain asymptotics, the author fixes a mode and considers the net evolution over one period. Moreover, he replaces the evolution (over one period) with a matrix multiplication. He cannot calculate the matrices explicitly; he approximates them. To obtain the asymptotics theb author needs to calculate a matrix product where there is no bound on the number of factors, and where each factor can only be approximated. Nevertheless, he obtains detailed asymptotics. In fact, it is possible to isolate an overall behaviour (growth/decay) from the (increasingly violent) oscillatory behaviour. Moreover, then author is also in a position to specify the leading order asymptotics.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
ReadershipGraduate students and researchers.

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The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalized, converges. The model examples are the Kasner solutions. The main result of the article is optimal energy estimates. However, the author also derives asymptotics and demonstrate that the leading order asymptotics can be specified (also in situations where the asymptotics are not convergent).
It is sometimes argued that if the factors multiplying the spatial derivatives decay exponentially (for a system of wave equations), then the spatial derivatives can be ignored. This line of reasoning is incorrect; the author gives examples of equations such that: (1) the factors multiplying the spatial derivatives decay exponentially, (2) the factors multiplying the time derivatives are constants, (3) the energies of individual modes of solutions asymptotically decay exponentially, and (4) the energies of generic solutions grow as \(\mathrm{exp}[\mathrm{exp}(t)] as t\rightarrow\infty\).
When the factors multiplying the spatial derivatives grow exponentially, the Fourier modes of solutions oscillate with a frequency that grows exponentially. To obtain asymptotics, the author fixes a mode and considers the net evolution over one period. Moreover, he replaces the evolution (over one period) with a matrix multiplication. He cannot calculate the matrices explicitly; he approximates them. To obtain the asymptotics theb author needs to calculate a matrix product where there is no bound on the number of factors, and where each factor can only be approximated. Nevertheless, he obtains detailed asymptotics. In fact, it is possible to isolate an overall behaviour (growth/decay) from the (increasingly violent) oscillatory behaviour. Moreover, then author is also in a position to specify the leading order asymptotics.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and researchers.