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

Abstract

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. 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.

On the well-posedness of the vacuum Einstein’s equations

“…The idea to solve both the evolution and the constraint equations in the weighted Sobolev spaces of fractional order H s,δ has previously appeared in [2,3] (and for integer order in [24]), but for the Einstein-Euler systems. Here, we generalize the classical result of Hughes, Kato, and Marsden [17] to the weighted Sobolev spaces of fractional order and improve the regularity index by one.…”

“…The following theorem was proved under various assumptions in [1,3,6,[8][9][10]12,22]. THEOREM C. Let m be an integer greater or equal to one, − 3 2 < δ < − 1 2 .…”

“…Klainerman and I. Rodnianski [20] succeeded in improving regularity below the critical index 3 2 . THEOREM B.…”

Local existence of solutions to the Euler–Poisson system, including densities without compact support

“…We prove this simple proposition since a proof is not found in the standard literature such as [BK11,BK15,Tri76a].…”

“…The properties of the weighted Sobolev spaces H s,δ are presented in Section 3. For the proofs of those properties, we refer to [BK11,BK15,Tri76a,Tri76b], except Lemma 1, which is new and crucial for the proof of the nonlinear power estimate, Proposition 12. In section 4 we establish the main mathematical tools, including the local existence and well posedness of symmetric hyperbolic systems in the H s,δ weighted spaces, two energy type estimates of the solutions to hyperbolic systems, the elliptic estimate for the Poisson equation and two non-linear estimates.…”