The Israel–Stewart equations are a model for relativistic viscous fluids in which the viscous fluxes are not given by constitutive laws, as in the Navier–Stokes equations, but are independent unknowns governed by additional equations. Away from vacuum, the Israel–Stewart equations can be written as a first-order symmetric hyperbolic system, and local existence and uniqueness follow from the classical theory. In this talk, we consider the Israel–Stewart equations with bulk viscosity as the only source of dissipation, for a gas surrounded by vacuum: the density vanishes on the fluid–vacuum interface, which is advected by the fluid velocity. In this setting, the system degenerates at the vacuum boundary and the classical theory no longer applies.

A natural regime is the one in which the boundary moves with finite, nonzero acceleration. In this case, the density and the bulk viscous pressure are required to decay at specific rates near the boundary. These decay rates are referred to as the physical vacuum boundary condition. We show that solutions satisfying this condition exist locally in time in weighted Sobolev spaces. The main difficulties are the degeneracy at the boundary and a loss of derivatives in the estimates for the linearized equations. They are overcome by a Nash–Moser iteration scheme combining both Lagrangian and Eulerian coordinates.