On the existence of solutions to the relativistic Euler equations in 2 spacetime dimensions with a vacuum boundary

We prove the existence of a wide class of solutions to the isentropic relativistic Euler equations in 2 spacetime dimensions with an equation of state of the form $p=K{\rho }^2$ that have a fluid vacuum boundary. Near the fluid vacuum boundary, the sound speed for these solutions are monotonically decreasing, approaching zero where the density vanishes. Moreover, the fluid acceleration is finite and bounded away from zero as the fluid vacuum boundary is approached. The existence results of this article also generalize in a straightforward manner to equations of state of the form $p=K{\rho }^{\frac{\gamma +1}{\gamma }}$ with $\gamma >0$.