The linearized Israel-Stewart Equations with a Physical Vacuum Boundary

Abstract

The Israel-Stewart theory describes relativistic viscous fluids and provides a robust model for simulating important physical phenomena in astrophysics, high-energy physics, and cosmology. However, it has been studied far less extensively than relativistic ideal fluids governed by the relativistic Euler equations. In this dissertation, we investigate an equation of the Israel-Stewart type with bulk viscosity in the presence of a vacuum. By allowing for a vacuum, we introduces degeneracy in the governing equations near the boundary. It turns out that the decay rates of the fluid’s density and bulk viscosity play a crucial role in addressing this degeneracy. These decay rates also ensure that the boundary can have a finite, nonzero acceleration, allowing us to model some physical scenarios such as star rotation. Based on these decay rates, we define the physical vacuum boundary condition. Our primary goal in the dissertation is to establish the local well-posedness of the linearized system with a physical vacuum boundary. Since the nonlinear system is degenerate near the boundary, the linearized equations are also degenerate near the free boundary. In this case, the local well-posedness of the linearized equations cannot be derived from the standard techniques. To handle these challenges, we incorporate a weight into our energy estimates, which naturally leads to a functional framework based on weighted Sobolev spaces. The core strategy for proving local well-posedness relies on energy estimates in these weighted Sobolev spaces, combined with a duality argument. Within this process, weighted elliptic operators tied to the second-order evolution of the system play a crucial role. We show that these weighted elliptic operators satisfy some specific weighted elliptic estimates and connect the weighted Sobolev spaces at different regularities. Finally, we discuss how the local wellposedness of the linearized equations contribute to constructing solutions for the fully nonlinear system, which will be presented in an upcoming work.