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Linear Systems of Wave Equations on Cosmological Backgrounds with Convergent Asymptotics
 
Hans Ringström KTH Royal Institute of Technology , Stockholm, Sweden and Institut Mittag-Leffler, Stockholm, Sweden
A publication of the Société Mathématique de France
Linear Systems of Wave Equations on Cosmological Backgrounds with Convergent Asymptotics
Softcover ISBN:  978-2-85629-926-5
Product Code:  AST/420
List Price: $105.00
AMS Member Price: $84.00
Please note AMS points can not be used for this product
Linear Systems of Wave Equations on Cosmological Backgrounds with Convergent Asymptotics
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Linear Systems of Wave Equations on Cosmological Backgrounds with Convergent Asymptotics
Hans Ringström KTH Royal Institute of Technology , Stockholm, Sweden and Institut Mittag-Leffler, Stockholm, Sweden
A publication of the Société Mathématique de France
Softcover ISBN:  978-2-85629-926-5
Product Code:  AST/420
List Price: $105.00
AMS Member Price: $84.00
Please note AMS points can not be used for this product
  • Book Details
     
     
    Astérisque
    Volume: 4202020; 512 pp
    MSC: 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.

    Readership

    Graduate students and researchers.

  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 4202020; 512 pp
MSC: 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.

Readership

Graduate students and researchers.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.