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Stability of fluids in spacetimes with decelerated expansion

David Fajman, Maximilian Ofner, Todd Oliynyk, Zoe Wyatt

TL;DR

This work analyzes the nonlinear stability of homogeneous barotropic relativistic fluids on fixed FLRW spacetimes with decelerated expansion. By reformulating the relativistic Euler equations in expansion-normalized variables and designing a universal corrected $L^2$-energy, the authors establish global future stability under the condition $K<1-\frac{2}{3\alpha}$, with explicit decay rates for the perturbations and a clear mechanism balancing expansion damping against the speed of sound. The approach also yields a streamlined proof of stabilization for linearly expanding cosmologies and clarifies the sharp boundary between stable and unstable regimes, including a detailed shock-formation analysis for dust and radiation in the decelerated setting. The results have implications for cosmological fluid dynamics, highlighting a phase transition in fluid behavior driven by the interplay between expansion rate and sound speed, and are complemented by constructive blow-up scenarios via characteristics in the unstable regimes.

Abstract

We prove the nonlinear stability of homogeneous barotropic perfect fluid solutions in fixed cosmological spacetimes undergoing decelerated expansion. The results hold provided a specific inequality between the speed of sound of the fluid and the expansion rate of spacetime is valid. Numerical studies in our earlier complementary paper provide strong evidence that the aforementioned condition is sharp, i.e. that instabilities occur when the inequality is violated. In this regard, our present result covers the regime of slowest possible expansion which allows for fluids to stabilize, depending on their speed of sound. Our proof relies on an energy functional which is universal in the sense that it also applies to the case of linear expansion and enables a significantly simplified proof of bounds for fluids on linearly expanding spacetimes. Finally, we consider the special cases of dust and radiation fluids in the decelerated regime and prove shock formation for arbitrarily small perturbations of homogeneous solutions.

Stability of fluids in spacetimes with decelerated expansion

TL;DR

This work analyzes the nonlinear stability of homogeneous barotropic relativistic fluids on fixed FLRW spacetimes with decelerated expansion. By reformulating the relativistic Euler equations in expansion-normalized variables and designing a universal corrected -energy, the authors establish global future stability under the condition , with explicit decay rates for the perturbations and a clear mechanism balancing expansion damping against the speed of sound. The approach also yields a streamlined proof of stabilization for linearly expanding cosmologies and clarifies the sharp boundary between stable and unstable regimes, including a detailed shock-formation analysis for dust and radiation in the decelerated setting. The results have implications for cosmological fluid dynamics, highlighting a phase transition in fluid behavior driven by the interplay between expansion rate and sound speed, and are complemented by constructive blow-up scenarios via characteristics in the unstable regimes.

Abstract

We prove the nonlinear stability of homogeneous barotropic perfect fluid solutions in fixed cosmological spacetimes undergoing decelerated expansion. The results hold provided a specific inequality between the speed of sound of the fluid and the expansion rate of spacetime is valid. Numerical studies in our earlier complementary paper provide strong evidence that the aforementioned condition is sharp, i.e. that instabilities occur when the inequality is violated. In this regard, our present result covers the regime of slowest possible expansion which allows for fluids to stabilize, depending on their speed of sound. Our proof relies on an energy functional which is universal in the sense that it also applies to the case of linear expansion and enables a significantly simplified proof of bounds for fluids on linearly expanding spacetimes. Finally, we consider the special cases of dust and radiation fluids in the decelerated regime and prove shock formation for arbitrarily small perturbations of homogeneous solutions.
Paper Structure (37 sections, 16 theorems, 152 equations)

This paper contains 37 sections, 16 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. The relativistic Euler equations
  3. Fluid stabilization
  4. The decelerated regime
  5. Main Result
  6. Fluids on linearly expansion spacetimes
  7. Some results on shock formation
  8. Structure of the paper
  9. Notation, coordinates and definitions
  10. Preliminaries
  11. Energy estimates and the correction mechanism
  12. Evolution of the mean velocity
  13. $L^2$-energy
  14. Corrected $L^2$-energy
  15. Higher order energies
  16. ...and 22 more sections

Key Result

Lemma 2.3

Let $(\mathcal{M},\gamma)$ be a smooth, compact Riemannian manifold. Then for all $f\in H^{1}(\mathcal{M})$, where $\bar{f}=\frac{1}{\text{vol}(\mathcal{M},\gamma)}\int f d\mu(\gamma)$.

Theorems & Definitions (38)

  • Definition 2.1: Spatial derivatives
  • Definition 2.2: Norms
  • Lemma 2.3: Poincaré inequality
  • Lemma 4.1
  • proof
  • Definition 4.2: $L^2$-energy
  • Definition 4.3: Perturbative term
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • ...and 28 more