On the existence of solutions to the relativistic Euler equations in 2 spacetime dimensions with a vacuum boundary

Abstract

We prove the existence of a wide class of solutions to the isentropic relativistic Euler equations in 2 spacetime dimensions with an equation of state of the form $p=K{\rho }^2$ that have a fluid vacuum boundary. Near the fluid vacuum boundary, the sound speed for these solutions are monotonically decreasing, approaching zero where the density vanishes. Moreover, the fluid acceleration is finite and bounded away from zero as the fluid vacuum boundary is approached. The existence results of this article also generalize in a straightforward manner to equations of state of the form $p=K{\rho }^{\frac{\gamma +1}{\gamma }}$ with $\gamma >0$.

A major revision of the second half of the paper

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Cited by in corpus (7)

• Conservation laws and evolution schemes in geodesic, hydrodynamic and magnetohydrodynamic flows
• Dynamical compact elastic bodies in general relativity
• Recent developments in mathematical aspects of relativistic fluids
• The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation criterion
• Well-posedness of the free boundary hard phase fluids in Minkowski background and its Newtonian limit
• Dynamical relativistic liquid bodies I: constraint propagation
• Development of singularities in the relativistic Euler equations