Formation of singularities for the relativistic Euler equations

Abstract

This paper contributes to the study of large data problems for C1 solutions of the relativistic Euler equations. First, in the (1+1)–dimensional spacetime setting, if the initial data are strictly away from the vacuum, a key difficulty in considering the singularity formation is coming up with a way to obtain sharp enough control on the lower bound of the mass–energy density function ρ. For this reason, via an elaborate argument on a certain ODE inequality and introducing some key artificial (new) quantities, we provide one time-dependent lower bound of ρ of the (1+1)-dimensional relativistic Euler equations, which involves looking at the difference of the two Riemann invariants, along with certain weighted gradients of them. Ultimately, for C1 solutions with uniformly positive initial mass–energy density of the corresponding Cauchy problem, we give a necessary and sufficient condition for the singularity formation in finite time. Second, for the (3+1)–dimensional relativistic fluids, under the assumption that the initial mass–energy density vanishes in some open domain, we give a sufficient condition for C1 solutions to blow up in finite time, no matter how small or smooth the initial data are. Moreover, we present some interesting study on the asymptotic behaviour of the relativistic velocity, which shows that one cannot obtain any global regular solution whose L∞ norm of u decays to zero as time t goes to infinity.