Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation
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| Format: | Preprint |
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2023
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| _version_ | 1866914139357577216 |
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| author | Filbet, Francis Kang, Myeongju |
| author_facet | Filbet, Francis Kang, Myeongju |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_19344 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation Filbet, Francis Kang, Myeongju Analysis of PDEs 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. |
| title | Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.19344 |