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Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation

Francis Filbet, Myeongju Kang

TL;DR

This work analyzes the inertial Kuramoto–Sakaguchi–Fokker–Planck equation with inertia $m$, coupling $\kappa$, and noise $\sigma$, yielding two main results. First, for small coupling relative to noise, solutions converge exponentially to a phase-homogeneous stationary state in a weighted $L^2_\gamma$ framework, achieved via a micro–macro hypocoercivity strategy and a modified energy that couples microscopic dissipation to macroscopic relaxation. Second, in the diffusion regime of identical oscillators ($g(\nu)=\delta_0$), the kinetic density approaches a drift-diffusion limit for the phase density $\rho$, with explicit error estimates in the mass parameter $m$; the density satisfies $\partial_t\rho-\widetilde{\sigma}\partial_\theta^2\rho-\widetilde{\kappa}\partial_\theta((\sin*\rho)\rho)=0$. Compared to prior works, the paper provides simpler proofs, explicit convergence rates, and quantitative diffusion-limit error bounds, relying on a robust weighted $L^2$ hypocoercivity framework and a carefully constructed modified energy. These results have potential implications for the design of structure-preserving numerical schemes and offer refined insights into synchronization dynamics under noise and inertia.

Abstract

In this paper, we study the inertial Kuramoto-Sakaguchi equation for interacting oscillatory systems. On the one hand, we prove the convergence toward corresponding phase-homogeneous stationary states in weighted Lebesgue norm sense when the coupling strength is small enough. In [10], it is proved that when the noise intensity is sufficiently large, equilibrium of the inertial Kuramoto-Sakaguchi equation is asymptotically stable. For generic initial data, every solutions converges to equilibrium in weighted Sobolev norm sense. We improve this previous result by showing the convergence for a larger class of functions and by providing a simpler proof. On the other hand, we investigate the diffusion limit when all oscillators are identical. In [19], authors studied the same problem using an energy estimate on renormalized solutions and a compactness method, through which error estimates could not be discussed. Here we provide error estimates for the diffusion limit with respect to the mass m $\ll$ 1 using a simple proof by imposing slightly more regularity on the solution.

Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation

TL;DR

This work analyzes the inertial Kuramoto–Sakaguchi–Fokker–Planck equation with inertia , coupling , and noise , yielding two main results. First, for small coupling relative to noise, solutions converge exponentially to a phase-homogeneous stationary state in a weighted framework, achieved via a micro–macro hypocoercivity strategy and a modified energy that couples microscopic dissipation to macroscopic relaxation. Second, in the diffusion regime of identical oscillators (), the kinetic density approaches a drift-diffusion limit for the phase density , with explicit error estimates in the mass parameter ; the density satisfies . Compared to prior works, the paper provides simpler proofs, explicit convergence rates, and quantitative diffusion-limit error bounds, relying on a robust weighted hypocoercivity framework and a carefully constructed modified energy. These results have potential implications for the design of structure-preserving numerical schemes and offer refined insights into synchronization dynamics under noise and inertia.

Abstract

In this paper, we study the inertial Kuramoto-Sakaguchi equation for interacting oscillatory systems. On the one hand, we prove the convergence toward corresponding phase-homogeneous stationary states in weighted Lebesgue norm sense when the coupling strength is small enough. In [10], it is proved that when the noise intensity is sufficiently large, equilibrium of the inertial Kuramoto-Sakaguchi equation is asymptotically stable. For generic initial data, every solutions converges to equilibrium in weighted Sobolev norm sense. We improve this previous result by showing the convergence for a larger class of functions and by providing a simpler proof. On the other hand, we investigate the diffusion limit when all oscillators are identical. In [19], authors studied the same problem using an energy estimate on renormalized solutions and a compactness method, through which error estimates could not be discussed. Here we provide error estimates for the diffusion limit with respect to the mass m 1 using a simple proof by imposing slightly more regularity on the solution.
Paper Structure (13 sections, 10 theorems, 186 equations)

This paper contains 13 sections, 10 theorems, 186 equations.

Table of Contents

  1. Introduction and main results
  2. Long time asymptotics
  3. Diffusion limit for identical oscillators
  4. Long time behavior for small coupling strength
  5. Outline of the proof of Theorem \ref{['th:1']}
  6. Basic properties
  7. Toward the modified energy $\mathcal{E}[f]$
  8. Proof of Theorem \ref{['th:1']}
  9. Diffusion limit when $g=\delta_0$
  10. Outline of the proof of Theorem \ref{['th:2']}
  11. Basic estimates
  12. Proof of Theorem \ref{['th:2']}
  13. Conclusion

Key Result

Theorem 1.1

Consider an initial data $f_{\rm in}\geq 0$ such that Then there exists a constant $C_\infty>0$, only depending on $\|g\|_{L^2_{\bar{\gamma}}}$, such that if the coupling strength $\widetilde{\kappa}>0$ and the noise intensity $\widetilde{\sigma}>0$ satisfy hence the solution $f$ to K3-1-K3-2 converges to the phase-homogeneous stationary state fInt denoted by $f_\infty$ exponentially fast where

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • ...and 14 more