Lagrangian formulation and Eulerian closure in alignment dynamics
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arXiv
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| Natura: | Preprint |
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2026
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| author | Carrillo, José A. Choi, Young-Pil Tadmor, Eitan |
| author_facet | Carrillo, José A. Choi, Young-Pil Tadmor, Eitan |
| contents | We investigate a continuum Lagrangian $p$-alignment system given by a nonlocal mean-field system of ordinary differential equations for interacting agents with weak initial data. We first establish global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We next construct Eulerian variables from the possibly non-injective Lagrangian flow via pushforward and disintegration, which leads to an Euler--Reynolds--alignment system featuring a nonnegative Reynolds stress and, for $p>2$, a nonlinear defect force induced by microscopic velocity fluctuations. Assuming only heavy-tailed interaction, we then show that these defect terms vanish asymptotically, leading to asymptotic mono-kinetic closure in the long-time limit. In the linear case $p=2$, we further obtain global weak solutions to the Euler--alignment system, including a sharp one-dimensional critical-threshold characterization and a global result in higher dimensions under a large-coupling condition. Finally, we establish a uniform-in-time mean-field stability estimate for the particle Cucker--Smale system in the linear regime and deduce uniform-in-time convergence toward the mono-kinetic Eulerian limit; for general $p\ge2$, we also obtain a finite-time mean-field convergence result toward the associated kinetic/Lagrangian alignment dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lagrangian formulation and Eulerian closure in alignment dynamics Carrillo, José A. Choi, Young-Pil Tadmor, Eitan Analysis of PDEs Classical Analysis and ODEs Adaptation and Self-Organizing Systems 35Q35, 76N10, 92D25, 35Q83 We investigate a continuum Lagrangian $p$-alignment system given by a nonlocal mean-field system of ordinary differential equations for interacting agents with weak initial data. We first establish global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We next construct Eulerian variables from the possibly non-injective Lagrangian flow via pushforward and disintegration, which leads to an Euler--Reynolds--alignment system featuring a nonnegative Reynolds stress and, for $p>2$, a nonlinear defect force induced by microscopic velocity fluctuations. Assuming only heavy-tailed interaction, we then show that these defect terms vanish asymptotically, leading to asymptotic mono-kinetic closure in the long-time limit. In the linear case $p=2$, we further obtain global weak solutions to the Euler--alignment system, including a sharp one-dimensional critical-threshold characterization and a global result in higher dimensions under a large-coupling condition. Finally, we establish a uniform-in-time mean-field stability estimate for the particle Cucker--Smale system in the linear regime and deduce uniform-in-time convergence toward the mono-kinetic Eulerian limit; for general $p\ge2$, we also obtain a finite-time mean-field convergence result toward the associated kinetic/Lagrangian alignment dynamics. |
| title | Lagrangian formulation and Eulerian closure in alignment dynamics |
| topic | Analysis of PDEs Classical Analysis and ODEs Adaptation and Self-Organizing Systems 35Q35, 76N10, 92D25, 35Q83 |
| url | https://arxiv.org/abs/2604.10253 |