Elie Abdo, Quyuan Lin, and Changhui Tan

Abstract

The primitive equations (PE) are a fundamental model in geophysical fluid dynamics. While the viscous PE are globally well-posed, their inviscid counterparts are known to be ill-posed.

In this paper, we study the two-dimensional incompressible PE with fractional horizontal dissipation. We identify a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent \(\alpha=1\). In the critical regime, this dichotomy exhibits a new phenomenon: the transition depends delicately on the balance between the size of the initial data and the viscosity coefficient. Our results precisely quantify the horizontal dissipation required to transition from inviscid instability to viscous regularity. We also establish a global well-posedness theory to the fractional PE, with sufficient dissipation \(\alpha\geq\frac65\).

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

Kunhui Luan, Changhui Tan, and Qiyu Wu

Abstract

We study the 1D pressureless Euler-Poisson equations with variable background states and nonlocal velocity alignment. Our main focus is the phenomenon of critical thresholds, where subcritical initial data lead to global regularity, while supercritical data result in finite-time singularity formation. The critical threshold behavior of the Euler-Poisson-alignment (EPA) system has previously been investigated under two specific setups: (1) when the background state is constant, phase plane analysis was used in the work of Bhatnagar, Liu and Tan [J. Differ. Equ. 375 (2023) 82-119] to establish critical thresholds; and (2) when the nonlocal alignment is replaced by linear damping, comparison principles based on Lyapunov functions were employed in the work of Choi, Kim, Koo and Tadmor [arXiv:2402.12839].

In this work, we present a comprehensive critical threshold analysis of the general EPA system, incorporating both nonlocal effects. Our framework unifies the techniques developed in the aforementioned studies and recovers their results under the respective limiting assumptions. A key feature of our approach is the oscillatory nature of the solution, which motivates a decomposition of the phase plane into four distinct regions. In each region, we implement tailored comparison principles to construct the critical thresholds piece by piece.

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

I am honored to be named a 2025 McCausland Faculty Fellow.

About the Fellowship

The McCausland Faculty Fellowship is the premier faculty fellowship program in the McCausland College of Arts and Sciences. It supports early-career McCausland College of Arts and Sciences faculty who are committed, creative teachers and rising stars in their academic disciplines.

The college established the program with a $10 million endowment from alumnus Peter McCausland (’71 history) and his wife, Bonnie. Through this fellowship, the McCauslands support innovative research and teaching, enhancing the career of faculty and the experience of students in the University of South Carolina's largest college.

Announcement from College of Arts and Sciences

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

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