In this paper we provide a complete local well-posedness theory for the free
boundary relativistic Euler equations with a physical vacuum boundary on a
Minkowski background. Specifically, we establish the following results: (i)
local well-posedness in the Hadamard sense, i.e., local existence, uniqueness,
and continuous dependence on the data; (ii) low regularity solutions: our
uniqueness result holds at the level of Lipschitz velocity and density, while
our rough solutions, obtained as unique limits of smooth solutions, have
regularity only a half derivative above scaling; (iii) stability: our
uniqueness in fact follows from a more general result, namely, we show that a
certain nonlinear functional that tracks the distance between two solutions (in
part by measuring the distance between their respective boundaries) is
propagated by the flow; (iv) we establish sharp, essentially scale invariant
energy estimates for solutions; (v) a sharp continuation criterion, at the
level of scaling, showing that solutions can be continued as long as the the
velocity is in $L^1_t Lip$ and a suitable weighted version of the density is at
the same regularity level.
Our entire approach is in Eulerian coordinates and relies on the functional
framework developed in the companion work of the second and third authors
corresponding to the non relativistic problem. All our results are valid for a
general equation of state $p(\varrho)= \varrho^\gamma$, $\gamma > 1$.
