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Well-posedness of the Einstein-Euler system in asymptotically flat spacetimes: the constraint equations


Brauer, Uwe and Karp, Lavi (2011) Well-posedness of the Einstein-Euler system in asymptotically flat spacetimes: the constraint equations. Journal of Differential Equations, 251 (6). pp. 1428-1446. ISSN 0022-0396


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This paper deals with the construction of initial data for the coupled Einstein-Euler 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 Lichnerowicz-York scaling method (Choquet-Bruhat 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

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