[Paper Review] A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary

Mahir Hadžić, Steve Shkoller+1 more|arXiv (Cornell University)|Nov 23, 2015

Cosmology and Gravitation Theories31 references18 citations

TL;DR

This paper establishes local-in-time a priori estimates for solutions to the relativistic Euler equations with a moving fluid-vacuum boundary, using Lagrangian coordinates to handle the degeneracy caused by density vanishing at the boundary. The key contribution is a novel energy method that overcomes the failure of standard hyperbolic energy estimates by exploiting the full nonlinear structure of the equations and extracting 3D spatial regularity from 4D spacetime operators.

ABSTRACT

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods.

Motivation & Objective

• To derive local-in-time a priori estimates for solutions to the relativistic Euler equations with a dynamic fluid-vacuum boundary where density vanishes at a specific rate.
• To extend non-relativistic physical vacuum techniques to the relativistic setting, accounting for the non-trivial geometry of proper time hypersurfaces.
• To control vorticity source terms arising from non-zero fluid acceleration in the relativistic case, which are absent in the non-relativistic limit.
• To extract three-dimensional spatial regularity from four-dimensional spacetime operators in energy and elliptic-type estimates.
• To establish a functional framework compatible with the equation of state $ p(\rho) = \rho^\gamma $, $ \gamma > 1 $, particularly near the physical vacuum boundary.

Proposed method

• Formulate the relativistic Euler equations in Lagrangian (co-moving) coordinates, where fluid particles are used as spatial labels.
• Use the four-velocity normalization condition $ g_{\alpha\beta}u^\alpha u^\beta = -1 $ as a constraint to close the system and define the fluid evolution.
• Apply a weighted Sobolev framework with time derivatives and spatial derivatives adapted to the fluid's motion, incorporating the physical vacuum condition $ \mathring{n} \sim d^{\gamma-1} $ near the boundary.
• Develop a nonlinear energy method that accounts for degeneracy in the hyperbolic system by using a modified energy functional involving $ \mathring{F} = \mathring{n}\mathring{v}^0 $ and the Jacobian $ \mathscr{J} $.
• Use the wave-map-like structure of the equations to derive energy estimates via integration by parts and Gronwall-type arguments, controlling error terms via smallness assumptions.
• Apply Sobolev embedding and Young’s inequality to bound nonlinear terms, ensuring the energy functional remains under control over time.

Experimental results

Research questions

• RQ1 How can a priori estimates be established for the relativistic Euler equations when the fluid density vanishes at the boundary, leading to degenerate hyperbolicity?
• RQ2 What modifications are required in the energy method to handle the relativistic setting, particularly due to non-constant proper time hypersurfaces and non-zero vorticity sources?
• RQ3 How does the choice of equation of state $ p(\rho) = \rho^\gamma $ affect the required regularity and the structure of the energy estimates?
• RQ4 Can the full nonlinear structure of the relativistic Euler equations be exploited to control vorticity and other nonlinear terms that prevent standard energy method application?
• RQ5 What is the minimal number of time derivatives required in the energy functional to ensure control of spatial regularity when $ \gamma \to 1^+ $?

Key findings

• The authors derive local-in-time a priori estimates in weighted Sobolev spaces for solutions to the relativistic Euler equations with a physical vacuum boundary, under the assumption of a barotropic equation of state.
• The energy method successfully controls the degeneracy at the fluid-vacuum interface by exploiting the full nonlinear structure of the equations in Lagrangian coordinates.
• The method overcomes the failure of standard symmetric hyperbolic energy estimates by extracting 3D spatial regularity from 4D spacetime operators through careful analysis of the wave-map-like system.
• For the equation of state $ p(\rho) = \rho^\gamma $, $ \gamma > 1 $, the required number of time derivatives in the energy norm increases as $ \gamma \to 1^+ $, reflecting reduced regularity near the vacuum boundary.
• The main estimate is of the form $ \mathscr{S}(\tau) \leq \mathring{M} + \tau P(\mathscr{S}(\tau)) $, which via continuity argument yields a time $ T > 0 $ such that $ \mathscr{S}(\tau) \leq 2\mathring{M} $, ensuring the bootstrap assumptions are improved.
• The result extends prior non-relativistic results to the relativistic case, with key new challenges addressed: non-constant proper time slices, non-vanishing vorticity sources, and 4D-to-3D regularity extraction.