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

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

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

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

Conference Venue

• The conference is held at Petigru College, University of South Carolina (map). 
• All talks will be delivered at Room 108. Poster Session will be held at Room 101 and 102.

Travel Directions

• If you travel by air, University of South Carolina is located close to the Columbia Metropolitan Airport (CAE). The easiest way to travel from CAE airport to campus is by Uber or Lyft. The ride will take approximately 15-20 minutes.

Accommodation

• There are many hotels near the University of South Carolina and the Downtown Columbia area.
• We have reserved a block of rooms at Courtyard by Marriott Columbia Downtown at USC, 630 Assembly St, Columbia, SC 29201, with a group rate of $129 per night. Reservations can be made online from THIS LINK, or by phone at 803-799-7800, mentioning USC Math Conference. The deadline for reservations at this rate is April 29, 2024.

Conference Dinner

• The dinner banquet is held on Tuesday May 28th evening from 6:30 PM to 10:00 PM at the Carolina Room of Capstone Residential Hall (map).

Campus Dining Map

Schedule

• The schedule of the conference can be found HERE.
• The abstract book can be found HERE.

List of Participants

1. Jing An, Duke University.
2. George Androulakis, University of South Carolina. 
3. Peter Binev, University of South Carolina.
4. Amimikh Biswas, University of Maryland, Baltimore County. [Speaker]
5. McKenzie Black, University of South Carolina. [Poster]
6. Victoria Chebotaeva, University of South Carolina. 
7. Dongwei Chen, Clemson University. [Poster]
8. Geng Chen, University of Kansas. [Speaker]
9. Ming Chen, University of Pittsburgh. [Speaker]
10. Peiyi Chen, University of Wisconsin, Madison. [Poster]
11. Alina Chertock, North Carolina State University. [Speaker]
12. Wolfgang Dahmen, University of South Carolina. [Speaker]
13. Ronald DeVore, Texas A&M University. [Keynote Speaker]
14. Di Fang, Duke University. [Speaker]
15. Guosheng Fu, University of Notre Dame.
16. Yuan Gao, Purdue University. [Speaker]
17. Maria Girardi, University of South Carolina. 
18. Anderson Greene, University of South Carolina. 
19. Ziheng Guo, Illinois Institute of Technology. [Poster]
20. Siming He, University of South Carolina. [Organizer]
21. Jianguo Hou, University of South Carolina. 
22. Zhongtian Hu, Duke University.
23. Gautam Iyer, Carnegie Mellon University. [Speaker]
24. Pierre-Emmanuel Jabin, Pennsylvania State University. [Speaker]
25. Ruhui Jin, University of Wisconsin, Madison. [Poster]
26. Yannis Kevrekidis, Johns Hopkins University. [Keynote Speaker]
27. SeHwan Kim, University of South Carolina. 
28. Trevor Leslie, Illinois Institute of Technology.
29. Qin Li, University of Wisconsin, Madison. [Speaker]
30. Wuchen Li, University of South Carolina. 
31. Quyuan Lin, Clemson University. [Speaker]
32. Hailiang Liu, Iowa State University. [Speaker]
33. Jian-Guo Liu, Duke University. [Speaker]
34. Jingcheng Lu, University of Minnesota, Twin Cities. [Poster]
35. Kunhui Luan, University of South Carolina. 
36. Mauro Maggioni, Johns Hopkins University. [Speaker]
37. Anna Mazuccato, Pennsylvania State University. [Speaker]
38. Lorenzo Micalizzi, North Carolina State University.
39. Sebastien Motsch, Arizona State University. [Speaker]
40. Ronghua Pan, Georgia Institute of Technology. [Speaker]
41. Keith Promislow, Michigan State University. [Speaker]
42. Ruiwen Shu, University of Georgia. [Speaker]
43. Roman Shvydkoy, University of Illinois, Chicago. [Speaker]
44. Henry Simmons, University of South Carolina. 
45. Seungjae Son, Carnegie Mellon University. [Poster]
46. Weiran Sun, Simon Fraser University. [Speaker]
47. Yi Sun, University of South Carolina. 
48. Eitan Tadmor, University of Maryland.
49. Changhui Tan, University of South Carolina. [Organizer]
50. Wei-Lun Tsai, University of South Carolina.
51. Wendy Garcia Umbarita, Arizona State University.
52. Li Wang, University of Minnesota. [Speaker]
53. Zhu Wang, University of South Carolina. 
54. Zhaoqing Xu, University of South Carolina. 
55. Xukai Yan, Oklahoma State University. [Speaker]
56. Cheng Yu, University of Florida.
57. Yue Yu, Lehigh University. [Speaker]
58. Qingtian Zhang, West Virginia University. [Speaker]
59. Ming Zhong, Illinois Institute of Technology. [Organizer]
60. Yuhua Zhu, University of California, San Diego. [Speaker]

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

Confirmed Speakers

• Amimikh Biswas, University of Maryland, Baltimore County.
• Alina Chertock, North Carolina State University.
• Wolfgang Dahmen, University of South Carolina.
• Ronald DeVore, Texas A&M University.
• Geng Chen, University of Kansas.
• Ming Chen, University of Pittsburgh.
• Di Fang, Duke University.
• Yuan Gao, Purdue University.
• Gautam Iyer, Carnegie Mellon University.
• Pierre-Emmanuel Jabin, Pennsylvania State University.
• Yannis Kevrekidis, Johns Hopkins University.
• Qin Li, University of Wisconsin, Madison.
• Quyuan Lin, Clemson University.
• Hailiang Liu, Iowa State University.
• Jian-Guo Liu, Duke University.
• Mauro Maggioni, Johns Hopkins University.
• Anna Mazuccato, Pennsylvania State University.
• Sebastien Motsch, Arizona State University.
• Ronghua Pan, Georgia Institute of Technology.
• Keith Promislow, Michigan State University.
• Ruiwen Shu, University of Georgia.
• Roman Shvydkoy, University of Illinois, Chicago.
• Weiran Sun, Simon Fraser University.
• Li Wang, University of Minnesota.
• Xukai Yan, Oklahoma State University.
• Yue Yu, Lehigh University.
• Qingtian Zhang, West Virginia University.
• Yuhua Zhu, University of California, San Diego.

Registration

• REGISTER HERE by May 17th.
• A limited amount of travel and local lodging support is available for researchers in the early stages of their careers who want to attend the full program, especially for graduate students and post-doctoral fellows. Apply by April 28th.

Organizing Committee

• Changhui Tan, University of South Carolina. Email: This email address is being protected from spambots. You need JavaScript enabled to view it.
• Siming He, University of South Carolina.
• Ming Zhong, Illinois Institute of Technology.

Acknowledgment

• Funding provided by NSF Grant DMS-2238219.
• We acknowledge partial support from University of South Carolina: College of Arts and Sciences, Department of Mathematics, and DASIV Smart State Center.

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