The lifespan of 3D radial solutions to the non-isentropic relativistic Euler equations

Abstract

This paper investigates the lower bound of the lifespan of three-dimensional spherically symmetric solutions to the non-isentropic relativistic Euler equations, when the initial data are prescribed as a small perturbation with compact support to a constant state. Based on the structure of the hyperbolic system, we show the almost global existence of the smooth solutions to Eulerian flows (polytropic gases and generalized Chaplygin gases) with genuinely nonlinear characteristics. While for the Eulerian flows (Chaplygin gas and stiff matter) with mild linearly degenerate characteristics, we show the global existence of the radial solutions, moreover, for the non-strictly hyperbolic system (pressureless perfect fluid) satisfying the mild linearly degenerate condition, we prove the blowup phenomenon of the radial solutions and show that the lifespan of the solutions is of order $$O(\epsilon ^{-1})$$ , where $$\epsilon $$ denotes the width of the perturbation. This work can be seen as a complement of our work (Lei and Wei in Math Ann 367:1363–1401, 2017) for relativistic Chaplygin gas and can also be seen as a generalization of the classical Eulerian fluids (Godin in Arch Ration Mech Anal 177:497–511, 2005, J Math Pures Appl 87:91–117, 2007) to the relativistic Eulerian fluids.