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

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 = K ? 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 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 ?x?/? 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 Ren-dall (Ann Henri Poincar & eacute; 5(6):1041-1064, 2004), and our numerical results confirm his expectations.