Classical Incompressible Fluid Dynamics as a Limit of Relativistic Compressible Fluid Dynamics

Abstract

Properly scaled, the relativistic Euler system for an arbitrary isentropic, causally compressible fluid is shown to formally converge, as c → ∞, to the non-relativistic Euler system for the homogeneously incompressible fluid. The limit is particularly interesting in the case of the relativistic stiff fluid, for which all modes are linearly degenerate in the sense of the theory of hyperbolic systems of conservation laws. This case connects the continuation problem for regular solutions to the incompressible version of the classical Euler equations with the old conjecture that for hyperbolic systems linear degeneracy of all modes prevent gradient blowup. One could say that questions in two different areas of the theory of partial differential equations are linked to each other through Einstein’s theory of relativity.