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The global attractor of the Toner-Tu-Swift-Hohenberg equations of active turbulence and its properties

Daniel W. Boutros, Kolluru Venkata Kiran, John D. Gibbon, Rahul Pandit

TL;DR

This work analyzes the Toner-Tu-Swift-Hohenberg (TTSH) equations on the unit periodic domain $\mathbb{T}^d$ ($d=2,3$) as models of active turbulence. Using Galerkin approximations, absorbing-ball estimates, Lieb–Thirring inequalities and a Lyapunov-exponent framework, the authors prove the existence of a finite-dimensional global attractor $\mathcal{A}$ and show $\mathcal{A}_{L^2}=\mathcal{A}_{H^1}$. They derive explicit upper bounds on the Lyapunov dimension $d_L(\mathcal{A})$ that scale with the Swift–Hohenberg length $\eta_{\text{SH}}=\Gamma_{2}/\Gamma_{0}$, yielding leading-order growth $d_L(\mathcal{A}) \sim (\Gamma_{0}^2/\Gamma_{2}^2 - \alpha \Gamma_{2}^{-1})^{1/2}$ in 2D and $d_L(\mathcal{A}) \sim (\Gamma_{0}^2/\Gamma_{2}^2 - \alpha \Gamma_{2}^{-1})^{3/4}$ in 3D. The authors also perform pseudospectral DNS in 2D to compute Lyapunov spectra and verify the bounds, observing agreement between numerics and theory and highlighting the SH-scale’s dominance in the long-time dynamics. These results provide a rigorous mathematical justification for the SH-scale phenomena observed in bacterial turbulence and offer a principled finite-dimensional description of active-turbulence dynamics.

Abstract

The Toner-Tu-Swift-Hohenberg (TTSH) equations are one of the basic equations that are used to model turbulent behaviour in active matter, specifically the swarming of bacteria in suspension. They combine features of the incompressible Navier-Stokes, the Toner-Tu and Swift-Hohenberg equations, together with the important properties that they are linearly driven, and that the Laplacian diffusion is taken to be negative in combination with hyper-dissipation. We prove that the TTSH equations possess a finite-dimensional compact global attractor on the periodic domain $\mathbb{T}^d$ ($d=2,3$) and we establish explicit estimates for its Lyapunov dimension which agree with the heuristic prediction based on the Swift-Hohenberg length scale. The predominance of this length scale (as a vortex length scale) has been observed in both numerical and experimental studies of bacterial turbulence, so our methods and results provide a rigorous theoretical foundation for this phenomenon. We also carry out pseudospectral direct numerical simulations of these PDEs in dimension $d=2$ through which we obtain Lyapunov spectra for representative parameter values. We show that our numerical results are consistent with the analytically derived rigorous bounds.

The global attractor of the Toner-Tu-Swift-Hohenberg equations of active turbulence and its properties

TL;DR

This work analyzes the Toner-Tu-Swift-Hohenberg (TTSH) equations on the unit periodic domain () as models of active turbulence. Using Galerkin approximations, absorbing-ball estimates, Lieb–Thirring inequalities and a Lyapunov-exponent framework, the authors prove the existence of a finite-dimensional global attractor and show . They derive explicit upper bounds on the Lyapunov dimension that scale with the Swift–Hohenberg length , yielding leading-order growth in 2D and in 3D. The authors also perform pseudospectral DNS in 2D to compute Lyapunov spectra and verify the bounds, observing agreement between numerics and theory and highlighting the SH-scale’s dominance in the long-time dynamics. These results provide a rigorous mathematical justification for the SH-scale phenomena observed in bacterial turbulence and offer a principled finite-dimensional description of active-turbulence dynamics.

Abstract

The Toner-Tu-Swift-Hohenberg (TTSH) equations are one of the basic equations that are used to model turbulent behaviour in active matter, specifically the swarming of bacteria in suspension. They combine features of the incompressible Navier-Stokes, the Toner-Tu and Swift-Hohenberg equations, together with the important properties that they are linearly driven, and that the Laplacian diffusion is taken to be negative in combination with hyper-dissipation. We prove that the TTSH equations possess a finite-dimensional compact global attractor on the periodic domain () and we establish explicit estimates for its Lyapunov dimension which agree with the heuristic prediction based on the Swift-Hohenberg length scale. The predominance of this length scale (as a vortex length scale) has been observed in both numerical and experimental studies of bacterial turbulence, so our methods and results provide a rigorous theoretical foundation for this phenomenon. We also carry out pseudospectral direct numerical simulations of these PDEs in dimension through which we obtain Lyapunov spectra for representative parameter values. We show that our numerical results are consistent with the analytically derived rigorous bounds.
Paper Structure (9 sections, 8 theorems, 115 equations, 2 figures, 1 table)

This paper contains 9 sections, 8 theorems, 115 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof of global well-posedness and the existence of a global attractor
  3. Estimates for the dimension of the global attractor $\mathcal{A}$
  4. Trace formulae and Lieb-Thirring inequalities
  5. Attractor dimension estimate in the two-dimensional case
  6. Attractor dimension estimate in the three-dimensional case
  7. Numerical evaluation of Lyapunov exponents and $d_L$
  8. Conclusions
  9. Acknowledgements

Key Result

Theorem 1

Let $\hbox{\boldmath$u$}_0 \in L^2 (\mathbb{T}^d)$, $T > 0$ and $d=2,3$. Then there exists a unique global weak solution $\hbox{\boldmath$u$} \in C([0,T];L^2(\mathbb{T}^d)) \cap L^2 ((0,T); H^2 (\mathbb{T}^d)) \cap L^4 ((0,T); L^4 (\mathbb{T}^d))$ of the TTSH equations.

Figures (2)

  • Figure 1: The Lyapunov spectrum for the run 7, where the black dots are for the unordered spectrum and the red curve gives the ordered spectrum, with shaded region giving the error bars. The inset gives the ordered Lyapunov spectrum for the run 2.
  • Figure 2: Log-log plot of $d_{L}$ versus $L^2(\Gamma_{0}/\Gamma_{2})$ for different values of $\alpha$. The solid black line corresponds to $10^{-1}\times L^2(\Gamma_{0}/\Gamma_{2})$.

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Lemma 1
  • ...and 8 more