On the stability of relativistic perfect fluids with linear equations of state p=Krho where 1/3

Abstract

For 1/3<K<1, we consider the stability of two distinct families of spatially homogeneous solutions to the relativistic Euler equations with a linear equation of state p=Krho on exponentially expanding FLRW spacetimes. The two families are distinguished by one being spatially isotropic while the other is not. We establish the future stability of nonlinear perturbations of the non-isotropic family for the full range of parameter values 1/3<K<1, which improves a previous stability result established by the second author that required K to lie in the restricted range (1/3, 1/2). As a first step towards understanding the behaviour of nonlinear perturbations of the isotropic family, we construct numerical solutions to the relativistic Euler equations under a mathbb {T}{}^2-symmetry assumption. These solutions are generated from initial data at a fixed time that is chosen to be suitably close to the initial data of an isotropic solution. Our numerical results reveal that, for the full parameter range 1/3<K<1, the density gradient frac{partial _{x}rho }{rho } associated to a nonlinear perturbation of an isotropic solution develops steep gradients near a finite number of spatial points where it becomes unbounded at future timelike infinity. This behavior of the density gradient was anticipated by Rendall (Ann Henri Poincaré 5(6):1041–1064, 2004), and our numerical results confirm his expectations.