Well-posedness for the free boundary hard phase model in general relativity

TL;DR

This paper establishes the mathematical well-posedness of the free boundary hard phase model in general relativity, demonstrating existence, uniqueness, and stability of solutions involving a relativistic fluid coupled with Einstein's equations.

Contribution

It provides the first rigorous proof of a priori estimates and well-posedness for the relativistic hard phase fluid with free boundary in a general relativistic setting.

Findings

Derived a priori estimates for the model.

Proved well-posedness in Sobolev spaces.

Established estimates for curvature using Bianchi equations.

Abstract

The hard phase model describes a relativistic barotropic and irrotational fluid with sound speed equal to the speed of light. In the framework of general relativity, the fluid, as a matter field, affects the geometry of the background spacetime. Therefore the motion of the fluid must be coupled to the Einstein equations which describe the structure of the underlying spacetime. In this work we prove a priori estimates and well-posedness in Sobolev spaces for this model with free boundary. Estimates for the curvature are derived using the Bianchi equations in a frame that is parallel transported by the fluid velocity. The fluid velocity is also decomposed with respect to this parallel frame, and its components are estimated using a coupled interior-boundary system of wave equations.