On the stability of relativistic perfect fluids with linear equations of state $$p=K\rho $$ where $$1/3<K<1$$

Abstract

AbstractFor $$1/3<K<1$$ 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 $$p=K\rho $$ p = K ρ 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 $$1/3<K<1$$ 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 $$\mathbb {T}{}^2$$ 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 $$1/3<K<1$$ 1 / 3 < K < 1 , the density gradient $$\frac{\partial _{x}\rho }{\rho }$$ ∂ x ρ ρ 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.