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Brauer, Uwe and Karp, Lavi (2011) Wellposedness of the EinsteinEuler system in asymptotically flat spacetimes: the constraint equations. Journal of Differential Equations, 251 (6). pp. 14281446. ISSN 00220396

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Official URL: http://www.sciencedirect.com/science/journal/00220396
Abstract
This paper deals with the construction of initial data for the coupled EinsteinEuler system. We consider the condition where the energy density might vanish or tend to zero at infinity, and where the pressure is a fractional power of the energy density. In order to achieve our goals we use a type of weighted Sobolev space of fractional order.\par The common LichnerowiczYork scaling method (ChoquetBruhat and York, 1980 [9]; Cantor, 1979 [7]) for solving the constraint equations cannot be applied here directly. The basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources. This problem has been addressed, although in a different context, by Dain and Nagy (2002) [11]. We show that if the matter variables are restricted to a certain region, then the Einstein constraint equations have a unique solution in the weighted Sobolev spaces of fractional order. The regularity depends upon the fractional power of the equation of state.
Item Type:  Article 

Uncontrolled Keywords:  Gravitational field; Matter variables; Makino variable; Equation of state; Compressible Euler equations; Fractional weighted Sobolev spaces 
Subjects:  Sciences > Mathematics > Differential equations 
ID Code:  14478 
Deposited On:  30 Jan 2012 08:27 
Last Modified:  12 Dec 2018 15:07 
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