The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

Abstract

We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and transport-div-curl equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type.

The stability of simple plane-symmetric shock formation for 3D compressible Euler flow with vorticity and entropy

“…While our present work treats only the non-relativistic case, it is likely that the relativistic case can also be treated in the same way. This is because the relativistic compressible Euler equations also admit a similar reformulation as we consider here, and likewise the variables in the reformulation also exhibits a very similar null structure [26].…”

“…The framework we introduced in [35,36,50] is useful in other low-regularity settings. See for example results on improved regularity for vorticity/entropy in [26], and results on local existence with rough data in [25,52].…”

On the existence of solutions and causality for relativistic viscous conformal fluids

“…It is useful to compare these characteristics to those of the ideal fluid. In the latter case we have the flow lines and the sound cone (i.e., the characteristics of the sound waves; see [17] for a detailed discussion of the role of the sound cone in the relativistic Euler equations). Here it is as if the the sound cone had "split" into two sound-type characteristics.…”

Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics

“…We have already showed that conditions λ, η, χ 1 > 0 together with (4.2)-(4.6) (which are the assumptions in Theorem 2.1) guarantee that 0 ≤ β a ≤ 1. Therefore, comparing (u α ξ α ) 2 − β a Π αβ ξ α ξ β with the characteristics of an acoustical metric (see, e.g., [34]), we conclude that det(m αβ ξ α ξ β ) is a product of hyperbolic polynomials.…”

“…While there remain questions about the details of how to fully derive relativistic fluid dynamics from an underlying microscopic theory [16,50,85,10,25,27], and rigorous mathematical results in this direction are few [90,37], the overwhelming success of the relativistic fluid dynamics more than justifies the importance of studying its mathematical properties. Furthermore, from a purely mathematical point of view, relativistic fluid dynamics has also been a fertile source of mathematical problems (see, e.g., [21,22,20,9,84,34] and references therein).…”