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The Stabilizing Effect of Spacetime Expansion on Relativistic Fluids With Sharp Results for the Radiation Equation of State

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Abstract

In this article, we study the 1 + 3-dimensional relativistic Euler equations on a pre-specified conformally flat expanding spacetime background with spatial slices that are diffeomorphic to \({\mathbb{R}^3.}\) We assume that the fluid verifies the equation of state \({p = c^{2}_{s} \rho,}\) where \({0 \leq c_{s} \leq \sqrt{1/3}}\) is the speed of sound. We also assume that the reciprocal of the scale factor associated with the expanding spacetime metric verifies a c s −dependent time-integrability condition. Under these assumptions, we use the vector field energy method to prove that an explicit family of physically motivated, spatially homogeneous, and spatially isotropic fluid solutions are globally future-stable under small perturbations of their initial conditions. The explicit solutions corresponding to each scale factor are analogs of the well-known spatially flat Friedmann–Lemaître–Robertson–Walker family. Our nonlinear analysis, which exploits dissipative terms generated by the expansion, shows that the perturbed solutions exist for all future times and remain close to the explicit solutions. This work is an extension of previous results, which showed that an analogous stability result holds when the spacetime is exponentially expanding. In the case of the radiation equation of state p = (1/3)ρ, we also show that if the time-integrability condition for the reciprocal of the scale factor fails to hold, then the explicit fluid solutions are unstable. More precisely, we show the existence of an open family of initial data such that (i) it contains arbitrarily small smooth perturbations of the explicit solutions’ data and (ii) the corresponding perturbed solutions necessarily form shocks in finite time. The shock formation proof is based on the conformal invariance of the relativistic Euler equations when \({c^{2}_{s} = 1/3,}\) which allows for a reduction to a well-known result of Christodoulou.

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  1. Department of Mathematics, Massachusetts Institute of Technology, Room 2-163, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Jared Speck

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  1. Jared Speck
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Correspondence to Jared Speck.

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Communicated by C. Dafermos

The author gratefully acknowledges support from NSF Grant # DMS-1162211, from a Solomon Buchsbaum Grant administered by the Massachusetts Institute of Technology, and from an NSF All-Institutes Postdoctoral Fellowship administered by the Mathematical Sciences Research Institute through its core Grant DMS-0441170.

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Speck, J. The Stabilizing Effect of Spacetime Expansion on Relativistic Fluids With Sharp Results for the Radiation Equation of State. Arch Rational Mech Anal 210, 535–579 (2013). https://doi.org/10.1007/s00205-013-0655-3

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