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Nonlinear Stability of First-Order Relativistic Viscous Hydrodynamics

Heinrich Freistuhler, Matthias Sroczinski

TL;DR

This work proves nonlinear stability of homogeneous rest states for a broad class of second-order hyperbolic models of dissipative relativistic fluids by verifying a dissipativity criterion (D) together with hyperbolicity (H_B). The authors recast the dynamics in terms of $\psi^\alpha=\theta^{-1}u^\alpha$ and perform a rest-frame, parallel/perpendicular block analysis to reduce the problem to checking algebraic (D1)-(D3) and (H_B) conditions; using a Routh–Hurwitz analysis of the dispersion relation, they show that small perturbations decay in Sobolev spaces at rate $t^{-3/4}$. The results cover two parameter regimes, $(C1)$ and $(C2)$, and unify prior models, including Hughes–Kato–Marsden-type behavior in the positive case. The work also clarifies causality constraints and their compatibility with dissipativity, providing a rigorous foundation for nonlinear stability of relativistic viscous hydrodynamics in a broad, Lorentz-invariant setting.

Abstract

This paper shows nonlinear stability of homogeneous states in second-order hyperbolic systems of partial differential equations that model the dynamics of dissipative relativistic fluids, by checking a dissipativity criterion formulated earlier by the authors and invoking a recent general result by the second author on long-time existence and time-asymptotic stability of small-data solutions to nonlinear hyperbolic systems. Version 3 differs from version 2 by a trivial correction (minus signs in front of six coefficients).

Nonlinear Stability of First-Order Relativistic Viscous Hydrodynamics

TL;DR

This work proves nonlinear stability of homogeneous rest states for a broad class of second-order hyperbolic models of dissipative relativistic fluids by verifying a dissipativity criterion (D) together with hyperbolicity (H_B). The authors recast the dynamics in terms of and perform a rest-frame, parallel/perpendicular block analysis to reduce the problem to checking algebraic (D1)-(D3) and (H_B) conditions; using a Routh–Hurwitz analysis of the dispersion relation, they show that small perturbations decay in Sobolev spaces at rate . The results cover two parameter regimes, and , and unify prior models, including Hughes–Kato–Marsden-type behavior in the positive case. The work also clarifies causality constraints and their compatibility with dissipativity, providing a rigorous foundation for nonlinear stability of relativistic viscous hydrodynamics in a broad, Lorentz-invariant setting.

Abstract

This paper shows nonlinear stability of homogeneous states in second-order hyperbolic systems of partial differential equations that model the dynamics of dissipative relativistic fluids, by checking a dissipativity criterion formulated earlier by the authors and invoking a recent general result by the second author on long-time existence and time-asymptotic stability of small-data solutions to nonlinear hyperbolic systems. Version 3 differs from version 2 by a trivial correction (minus signs in front of six coefficients).
Paper Structure (4 sections, 6 theorems, 65 equations)

This paper contains 4 sections, 6 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Hyperbolicity
  3. Dissipativity
  4. Causality

Key Result

Theorem 1

Fix a number $s>5/2$ and assume that normalized versions $(\kappa,\omega,\nu)=\theta^{-2}(\tilde{\kappa},\tilde{\omega},\tilde{\nu})$, $(\tau,\chi,\mu,\eta)=\theta^{-1}(\tilde{\tau},\tilde{\chi},\tilde{\mu},\tilde{\eta})$ of the dissipation coefficients satisfy and either condition (C1), i.e, or condition (C2), i.e., where $c_s=\sqrt{p'(\theta)/\theta p"(\theta)}>0$ is the speed of sound. Then

Theorems & Definitions (15)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 1
  • Lemma 4
  • ...and 5 more