## Research

Here are the latest updates for Changhui Tan's research profile.

Here is the Curriculum Vitae and List of Publications.

### Local regularity and finite-time singularity for a class of generalized SQG patches on the half-plane

*Qianyun Miao, Changhui Tan, Liutang Xue, and Zhilong Xue*

**Abstract**

In this paper, we investigate a class of inviscid generalized surface quasi-geostrophic (SQG) equations on the half-plane with a rigid boundary. Compared to the Biot-Savart law in the vorticity form of the 2D Euler equation, the velocity formula here includes an additional Fourier multiplier operator \(m(\Lambda)\). When \(m(\Lambda) = \Lambda^\alpha\), where \(\Lambda = (-\Delta)^{1/2}\) and \(\alpha\in (0,2)\), the equation reduces to the well-known \(\alpha\)-SQG equation. Finite-time singularity formation for patch solutions to the \(\alpha\)-SQG equation was famously discovered by Kiselev, Ryzhik, Yao, and Zlato\v{s} [Ann. Math., 184 (2016), pp. 909-948].

We establish finite-time singularity formation for patch solutions to the generalized SQG equations under the Osgood condition \[ \int_2^\infty \frac{1}{r (\log r) m(r)} dr < \infty \] along with some additional mild conditions. Under these assumptions, we demonstrate that there exist patch-like initial data for which the associated patch solutions on the half-plane are locally well-posed and develop a finite-time singularity. Our result goes beyond the previously known cases. Notably, our result fills the gap between the globally well-posed 2D Euler equation (\(\alpha = 0\)) and the \(\alpha\)-SQG equation (\(\alpha > 0\)). Furthermore, in line with Elgindi's global regularity results for 2D Loglog-Euler type equations [Arch. Rat. Mech. Anal., 211 (2014), pp. 965-990], our findings suggest that the Osgood condition serves as a sharp threshold that distinguishes global regularity and finite-time singularity in these models.

In addition, we generalize the local regularity and finite-time singularity results for patch solutions to the \(\alpha\)-SQG equation, as established by Gancedo and Patel [Ann. PDE, 7 (2021), no. 1, Art. no. 4], extending them to cases where \(m(r)\) behaves like \(r^\alpha\) near infinity but does not have an explicit formulation.

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

### On the Aw-Rascle-Zhang traffic models with nonlocal look-ahead interactions

*Thomas Hamori and Changhui Tan*

**Abstract**

We present a new family of second-order traffic flow models, extending the Aw-Rascle-Zhang (ARZ) model to incorporate nonlocal interactions. Our model includes a specific nonlocal Arrhenius-type look-ahead slowdown factor. We establish both local and global well-posedness theories for these nonlocal ARZ models.

In contrast to the local ARZ model, where generic smooth initial data typically lead to finite-time shock formation, we show that our nonlocal ARZ model exhibits global regularity for a class of smooth subcritical initial data. Our result highlights the potential of nonlocal interactions to mitigate shock formations in second-order traffic flow models.

Our analytical approach relies on investigating phase plane dynamics. We introduce a novel comparison principle based on a mediant inequality to effectively handle the nonlocal information inherent in our model.

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

### 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 |