On the well-posedness of relativistic viscous fluids

Abstract

Using a simple and well-motivated modification of the stress-energy tensor for a viscous fluid proposed by Lichnerowicz, we prove that Einstein's equations coupled to a relativistic version of the Navier-Stokes equations are well-posed in a suitable Gevrey class if the fluid is incompressible and irrotational. These last two conditions are given an appropriate relativistic interpretation. The solutions enjoy the domain of dependence or finite propagation speed property. We also derive a full set of equations, describing a relativistic fluid that is not necessarily incompressible or irrotational, which is well-suited for comparisons with the system of an inviscid fluid.

Force, inertia, metric in Newtonian relativity and general relativity

“…22). This fact is very clear in numerical relativity (Disconzi 2014(Disconzi p. 1918Wilson et al 2007 § 2.2;Gourgoulhon 2012 § 6.3.2;Baumgarte et al 2010 ch. 5;Rezzolla et al 2013 § 2.3).…”

The formulation of the Navier–Stokes equations on Riemannian manifolds

“…This is a consequence of the lack of a variational formulation for the Navier-Stokes equations, which prevents us from uniquely determining a stress-energy tensor for viscous fluids in the context of relativity 4 . Over the years, different proposals have been put forward to address this issue (see [12] for background and references). Naturally, any suitable candidate must recover the standard, non-relativisitic Navier-Stokes equations in the non-relativistic limit.…”

“…Alternatives to (4.1) have been proposed by Lichnerowicz [28], Choquet-Bruhat [9], and Freistühler and Temple [17]. Each of these has been shown to yield a satisfactory theory of relativistic viscous fluids under different assumptions [9,10,12,17] although these results fall short of covering all situations of physical interest, and the matter of how to correctly formulate relativistic viscous phenomena remains largely open. However, these approaches all give (4.16) in the non-relativistic limit, as we now explain.…”

Bianchi I cosmology in the presence of a causally regularized viscous fluid

“…A key-point consists in the fact that expression (5) reduces to the traditional description provided by Eckart upon setting F = 1. Furthermore, one of the main assumptions of the well-posedness theorems (see [21,22] for details) requires that…”