Changhui Tan, Liutang Xue, and Zhilong Xue

Abstract

In this paper, we revisit the patch solutions for a class of inviscid whole-space active scalar equations that interpolate between the 2D Euler equation and the \(\alpha\)-SQG equation. Compared with the 2D Euler equation in vorticity form, there is an additional Fourier multiplier \(m(\Lambda)\) (\(\Lambda = (-\Delta)^{1/2}\)) in the Biot–Savart law. If the symbol \(m\) satisfies the Osgood-type condition \[\int_2^{+\infty} \frac{1}{r (\log r) m(r)} = +\infty\] and certain mild assumptions, the system is referred to as the 2D Loglog-Euler type equation.

First, we prove a Yudovich-type theorem establishing the existence and uniqueness of a global weak solution for the Loglog-Euler type equation associated with bounded and integrable initial data. This result directly applies to patch solutions, which are weak solutions corresponding to patch initial data given by characteristic functions of disjoint, regular, bounded domains.

Next, we revisit the seminal result by Elgindi [ARMA 211 (2014) 965-990] and provide a different proof under explicit assumptions on \(m\), showing that for the 2D Loglog-Euler type equation with \(C^{1,\mu}\) (\(0<\mu<1\)) single-patch initial data, the evolved patch boundary globally preserves the \(C^{1,\mu-\varepsilon}\) regularity for any \(\varepsilon \in (0,\mu)\). In contrast to the frequency-space argument in [ARMA 211 (2014) 965-990], we develop an entirely physical-space-based approach that avoids the Littlewood–Paley theory and offers advantages for potential extensions to the half-plane or bounded smooth domains.

Furthermore, we investigate the global propagation of higher-order \(C^{n,\mu}\) boundary regularity for patch solutions with any \(n \in \mathbb{N}^\star\), and analyze the evolution of multiple patches.

This work is supported by NSF grants DMS #2238219

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