Stability of fluids in spacetimes with decelerated expansion

Abstract

We prove global existence for the relativistic Euler equations on expanding spacetimes under the weakest admissible assumptions on the spacetime expansion rate and the fluid sound speed. In contrast to all previous results on the fluid stabilization in expanding spacetimes, we consider decelerated spacetimes, which introduce new, highly singular terms into the fluid equations that cannot be analyzed using existing methods. To control these terms, we introduce a new energy method that interpolates decay between the density and velocity variables, yielding optimal decay. In a complementary paper, we present numerical evidence supporting the sharpness of this method. In particular, we conjecture that global existence fails under weaker assumptions on the expansion rate and sound speed.