The Israel-Stewart theory models relativistic viscous fluids, with important applications in astrophysics and cosmology. In this talk, I will present recent progress on an Israel-Stewart type system with bulk viscosity in the presence of vacuum. By allowing vacuum, we introduce degeneracy near the boundary. In this case, the decay rates of fluid variables play a crucial role in solving the problem.

We focus on decay rates that ensure the boundary maintains a finite, nonzero acceleration-a condition we refer to as the physical vacuum boundary condition. This allows us to model physical phenomena such as star rotation. The core of the talk is on establishing the local well-posedness of the linearized system under these vacuum conditions. Classical hyperbolic theory fails due to the degeneracy near the free boundary, so we incorporate weights in our functional framework and derive weighted energy estimates to construct solutions.