Generic alignment conjecture for systems of Cucker-Smale type
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918119609466880 |
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| author | Shvydkoy, Roman |
| author_facet | Shvydkoy, Roman |
| contents | The generic alignment conjecture states that for almost every initial data on the torus solutions to the Cucker-Smale system with a strictly local communication align to the common mean velocity. In this note we present a partial resolution of this conjecture using a statistical mechanics approach. First, the conjecture holds in full for the sticky particle model representing, formally, infinitely strong local communication. In the classical case the conjecture is proved when $N$, the number of agents, is equal to $2$. It follows from a more general result stating that for a system of any size for almost every data at least two agents align.
The analysis is extended to the open space $\R^n$ in the presence of confinement and potential interaction forces. In particular, it is shown that almost every non-oscillatory pair of solutions aligns and aggregates in the potential well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07860 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generic alignment conjecture for systems of Cucker-Smale type Shvydkoy, Roman Dynamical Systems Analysis of PDEs The generic alignment conjecture states that for almost every initial data on the torus solutions to the Cucker-Smale system with a strictly local communication align to the common mean velocity. In this note we present a partial resolution of this conjecture using a statistical mechanics approach. First, the conjecture holds in full for the sticky particle model representing, formally, infinitely strong local communication. In the classical case the conjecture is proved when $N$, the number of agents, is equal to $2$. It follows from a more general result stating that for a system of any size for almost every data at least two agents align. The analysis is extended to the open space $\R^n$ in the presence of confinement and potential interaction forces. In particular, it is shown that almost every non-oscillatory pair of solutions aligns and aggregates in the potential well. |
| title | Generic alignment conjecture for systems of Cucker-Smale type |
| topic | Dynamical Systems Analysis of PDEs |
| url | https://arxiv.org/abs/2304.07860 |