Formation of singularities for the Relativistic Euler equations

This paper contributes to the study of large data problems for $C^1$ solutions of the relativistic Euler equations. In the $(1+1)$-dimensional spacetime setting, if the initial data are away from vacuum, a key difficulty in proving the global well-posedness or finite time blow-up is coming up with a way to obtain sharp enough control on the lower bound of the mass-energy density function $\rho$. First, for $C^1$ solutions of the 1-dimensional classical isentropic compressible Euler equations in the Eulerian setting, we show a novel idea of obtaining a mass density time-dependent lower bound by studying the difference of the two Riemann invariants, along with certain weighted gradients of them. Furthermore, using an elaborate argument on a certain ODE inequality and introducing some key artificial (new) quantities, we apply this idea to obtain the lower bound estimate for the mass-energy density of the (1+1)-dimensional relativistic Euler equations. Ultimately, for $C^1$ solutions with uniformly positive initial mass-energy density of the (1+1)-dimensional relativistic Euler equations, we give a necessary and sufficient condition for the formation of singularity in finite time, which gives a complete picture for the ($C^1$) large data problem in dimension $(1+1)$. Moreover, for the (3+1)-dimensional relativistic fluids, under the assumption that the initial mass-energy density vanishes in some open domain, we give two sufficient conditions for $C^1$ solutions to blow up in finite time, no matter how small the initial data are. We also do some interesting studies on the asymptotic behavior of the relativistic velocity when vacuum appears at the far field, which tells us that one can not obtain any global regular solution whose $L^\infty$ norm of $u$ decays to zero as time $t$ goes to infinity.