Relativistic perfect fluids near Kasner singularities

Abstract

We establish the existence of a stable family of solutions to the Euler equations on Kasner backgrounds near the singularity with the full expected asymptotic data degrees of freedom and no symmetry or isotropy restrictions. Existence is achieved through transforming the Euler equations into the form of a symmetric hyperbolic Fuchsian system followed by an application of a new existence theory for the singular initial value problem. Stability is shown to follow from the existence theory for the (regular) global initial value problem for Fuchsian systems that was developed in [16]. In fact, for each solution in the family, we prove the existence of an open set of nearby solutions with the same qualitative asymptotics and show that any such perturbed solution agrees again with another solution of the singular initial value problem. All our results hold in the regime where the speed of sound of the fluid is large in comparison to all Kasner exponents. This is interpreted as the regime of stable fluid asymptotics near Kasner big bang singularities.

Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions $n\geq 4$

“…In the previous section, we derived, in Theorems 3.1 and 3.2, a Fuchsian formulation of the gauge reduced conformal Einstein-Yang-Mills equations. In this section, we quickly review the global existence theory for symmetric hyperbolic Fuchsian equations developed in [4], which is an extension of the Fuchsian existence theory from [21]; see also [3] for related results. This theory relies on the Fuchsian system satisfying a number of structural conditions, which we recall for the convenience of the reader.…”

“…2 The Yang-Mills curvature Fab is globally defined when viewed as taking values in the adjoint bundle. 3 In fact, we will only make use of the future half [0, ∞) × Σ of M n given by [0, ∞) × Σ. 4 See the references [13, 28] for a more detailed introduction to de Sitter spacetime.…”

Localized big bang stability for the Einstein-scalar field equations

“…As we discuss in more detail in Section 1.6, the approach we take to establishing stability in this article is broadly the same. It is worth pointing out that the Fuchsian approach to establishing the global existence of solutions to systems of hyperbolic equations is a very general method and has recently been employed to establish a variety of stability results in the following articles [12,24,39,43,44,45,48,49,60]. 1.6.…”

“…The virtue of this Fuchsian formulation is that we can now appeal to the existence theory developed in the articles 2 [12,13] to conclude, for suitably small choice of initial data u 0 at t = t 0 > 0, that there exist a unique solution of (1.37) that is defined all the way down to t = 0 and satisfies u| t=t0 = u 0 . The Fuchsian existence theory also yields energy and decay estimates that provide uniform control over the behaviour of solutions in the limit t ց 0.…”

“…The proof amounts to an application of the Fuchsian global existence theory established in [13,47]. The actual result that we employ is Theorem A.2 together with Remark A.3 from [12], which is a slight generalization of the global existence result from [13].…”