# Displaying items by tag: Euleralignment system

## Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

*Xiang Bai, Changhui Tan and Liutang Xue*

Journal of Differential Equations, Volume 407, pp. 269-310 (2024).

**Abstract**

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter \(\alpha\in(0,1)\). This highlights the distinct feature of the alignment operator.

doi:10.1016/j.jde.2024.06.020 | |

Download the Published Version | |

This work is supported by NSF grants DMS #2108264 and DMS #2238219 |

## Hydrodynamic limit of a kinetic flocking model with nonlinear velocity alignment

*McKenzie Black, and Changhui Tan*

**Abstract**

We investigate a class of Vlasov-type kinetic flocking models featuring nonlinear velocity alignment. Our primary objective is to rigorously derive the hydrodynamic limit leading to the compressible Euler system with nonlinear alignment. This study builds upon the work by Figalli and Kang [Anal. PDE, 12(3), 843-866, 2018], which addressed the scenario of linear velocity alignment using the relative entropy method. The introduction of nonlinearity gives rise to an additional discrepancy in the alignment term during the limiting process. To effectively handle this discrepancy, we employ the monokinetic ansatz in conjunction with the relative entropy approach. Furthermore, our analysis reveals distinct nonlinear alignment behaviors between the kinetic and hydrodynamic systems, particularly evident in the isothermal regime.

This work is supported by NSF grants DMS #2108264 and DMS #2238219 |

## Global well-posedness and refined regularity criterion for the uni-directional Euler-alignment system

*Yatao Li, Qianyun Miao, Changhui Tan and Liutang Xue*

**Abstract**

We investigate global solutions to the Euler-alignment system in d dimensions with unidirectional flows and strongly singular communication protocols \(\phi(x)=|x|^{-d+\alpha}\) for \(\alpha\in(0,2)\). Our paper establishes global regularity results in both the subcritical regime \(1<\alpha<2\) and the critical regime \(\alpha=1\). Notably, when \(\alpha=1\), the system exhibits a critical scaling similar to the critical quasi-geostrophic equation. To achieve global well-posedness, we employ a novel method based on propagating the modulus of continuity. Our approach introduces the concept of simultaneously propagating multiple moduli of continuity, which allows us to effectively handle the system of two equations with critical scaling. Additionally, we improve the regularity criteria for solutions to this system in the supercritical regime \(0<\alpha<1\).

This work is supported by NSF grants DMS #2108264 and DMS #2238219 |

## Finite- and Infinite-Time Cluster Formation for Alignment Dynamics on the Real Line

*Trevor M. Leslie and Changhui Tan*

Journal of Evolution Equations, Volume 24, Article 8, 45pp. (2024).

**Abstract**

We show that the locations where finite- and infinite-time clustering occurs for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated to a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated to a scalar balance law formulation of the system.

doi:10.1007/s00028-023-00939-2 | |

Download the Published Version | |

This work is supported by NSF grants DMS #2108264 and DMS #2238219 |

## Asymptotic behaviors for the compressible Euler system with nonlinear velocity alignment

*McKenzie Black and Changhui Tan*

Journal of Differential Equations, Volume 380, pp. 198-227 (2024)

**Abstract**

We consider the pressureless compressible Euler system with a family of nonlinear velocity alignment. The system is a nonlinear extension of the Euler-alignment system in collective dynamics. We show the asymptotic emergent phenomena of the system: alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are investigated, resulting in a variety of different asymptotic behaviors.

doi:10.1016/j.jde.2023.10.044 | |

Download the Published Version | |

This work is supported by NSF grants DMS #2108264 and DMS 2238219 | |

This work is supported by a UofSC VPR SPARC grant. |

## Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment

*Xiang Bai, Qianyun Miao, Changhui Tan and Liutang Xue*

Nonlinearity, Volume 37, 025007, 46pp. (2024).

**Abstract**

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behavior and optimal decay estimates of the solutions as \(t\to\infty\).

doi:10.1088/1361-6544/ad140b | |

Download the Published Version | |

This work is supported by NSF grant DMS #1853001 and DMS #2108264 |

## Sticky particle Cucker-Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

*Trevor M. Leslie, and Changhui Tan*

Communications in Partial Differential Equations, Volume 48, No. 5, pp. 753-791 (2023)

**Abstract**

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density and bounded velocity. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker-Smale dynamics. Our approach is largely inspired by the work of Brenier and Grenier [SIAM J. Numer. Anal, 35(6):2317-2328, 1998] on the pressureless Euler equations.

doi:10.1080/03605302.2023.2202720 | |

Download the Published Version | |

This work is supported by NSF grant DMS #1853001 and DMS #2108264 |

## Eulerian dynamics in multi-dimensions with radial symmetry

*Changhui Tan*

SIAM Journal on Mathematical Analysis, Volume 53, No 3, pp. 3040–3071 (2021).

**Abstract**

We study the global wellposedness of pressureless Eulerian dynamics in multidimensions, with radially symmetric data. Compared with the one-dimensional system, a major difference in multidimensional Eulerian dynamics is the presence of the spectral gap, which is difficult to control in general. We propose a new pair of scalar quantities that provides significantly better control of the spectral gap. Two applications are presented: (i) the Euler-Poisson equations: we show a sharp threshold condition on initial data that distinguish global regularity and finite time blowup; (ii) the Euler-alignment equations: we show a large subcritical region of initial data that leads to global smooth solutions.

doi:10.1137/20M1358682 | |

Download the Published Version | |

This work is supported by NSF grant DMS #1853001 |

## On the global classical solution to compressible Euler system with singular velocity alignment

*Li Chen, Changhui Tan, and Lining Tong*

Methods and Applications of Analysis, Volume 28, No.2, pp. 155-174 (2021).

Dedicated to Professor Ling Hsiao's 80th birthday.

**Abstract**

We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.

doi:10.4310/MAA.2021.v28.n2.a3 | |

Download the Published Version | |

This work is supported by NSF grant DMS #1853001 |

## Global regularity for a 1D Euler-alignment system with misalignment

*Qianyun Miao, Changhui Tan, and Liutang Xue*

Mathematical Models and Methods in Applied Sciences, Volume 31, No 3, pp. 473-524 (2021).

**Abstract**

We study one-dimensional Eulerian dynamics with nonlocal alignment interactions, fea- turing strong short-range alignment, and long-range misalignment. Compared with the well-studied Euler-alignment system, the presence of the misalignment brings different behaviors of the solutions, including the possible creation of vacuum at infinite time, which destabilizes the solutions. We show that with a strongly singular short-range alignment interaction, the solution is globally regular, despite the effect of misalignment.

doi:10.1142/S021820252150010X | |

Download the Published Version | |

This work is supported by NSF grant DMS #1853001 |