Quyuan Lin, Changhui Tan

Abstract

We establish global well-posedness of classical solutions to the two-dimensional primitive equations with fractional horizontal dissipation for arbitrarily large initial data in the full subcritical range \(1<\alpha\leq2\). Together with the known ill-posedness results for \(0\leq\alpha<1\), this establishes the sharp dissipation threshold for large-data global well-posedness in the corresponding solution framework.

The key ingredient is a hydrostatic energy estimate obtained by splitting the nonlinear energy into symmetric and antisymmetric parts and exploiting the commutator structure of the latter. Using anisotropy and incompressibility, we bound the nonlinear energy by the \(L^\infty\) norm of the hydrostatic vorticity times a quadratic velocity norm with only one-half additional horizontal derivative. The vorticity maximum principle then yields enhanced velocity bounds, which close the vorticity estimates and verify the continuation criterion throughout the subcritical regime.

	This work is supported by NSF grants DMS #2238219
	This work is supported by a USC VPR ASPIRE grant

Roman Shvydkoy, Changhui Tan

Abstract

We study the hydrodynamic limit of kinetic flocking models with nonlinear velocity alignment and strong local Fokker--Planck forcing. In the limit of small Knudsen number the kinetic densities converge to a local Maxwellian centered around macroscopic quantities satisfying the compressible Euler system with isothermal pressure \(p = \sigma\rho\), while macroscopic alignment is obtained by convolution of the microscopic velocity law with the standard Gaussian, resulting in a different nonlinearity. Such discrepancy has been observed already in the work of Black and Tan [KRM, 18(4):609–632, 2025].

In this note we rigorously justify the limit for admissible weak kinetic solutions and non-vacuous limiting macroscopic solution. Our analysis provides a quantitative rate of convergence in terms of relative entropy: \(\mathcal H(f_\varepsilon | \mu) \lesssim \varepsilon \bigl(1+|\log\varepsilon|^{p-2}\bigr)\) provided initially \(\mathcal H(f_\varepsilon(0) | \mu(0)) \leq \varepsilon\), where \(p\) is the order of non-linearity in the alignment force. In the linear case \(p=2\) we recover the known result of Karper, Mellet, and Trivisa [M3AS, 25(01):131–163, 2015].

This work is supported by NSF grants DMS #2238219

Changhui Tan, Liutang Xue

Abstract

We prove global well-posedness of the periodic unidirectional Euler-alignment system in every dimension for supercritical dissipation of order \(0<\alpha<1\) and arbitrary non-vacuum initial data in the Sobolev class of the local theory. The key idea is to propagate simultaneously two moduli of continuity at critical scales for the density \(\rho\) and the potential gradient \(\Gamma=\nabla\Lambda^{-\alpha}u\), both of which have scaling-invariant amplitudes. We also establish exponential alignment of the velocity and exponential convergence of the density to a traveling profile.

This work is supported by NSF grants DMS #2238219

 I have been awarded a grant from the Office of the Vice President for Research at the University of South Carolina, on a one-year project: Transition between inviscid and viscous primitive equations.

Project Summary

The primitive equations are the fundamental mathematical models governing large-scale atmospheric and oceanic dynamics. They arise from the Navier-Stokes equations under the hydrostatic approximation and form the core of modern climate and weather models. Despite their central role, the mathematical behavior of the primitive equations depends sensitively on the strength and structure of viscosity, and transitions between well-posed and ill-posed regimes remain poorly understood.

This project investigates the transition between inviscid and viscous primitive equations using fractional horizontal dissipation, which provides a continuous interpolation between these regimes. Recent joint work by the PI established the first rigorous results in two dimensions, identifying a sharp threshold separating local well-posedness from ill-posedness and proving global regularity in certain subcritical regimes. However, the underlying transition mechanisms and their robustness remain largely unexplored.

The proposed research aims to extend this framework in three directions:
(1) resolving the open transition regime between local and global regularity,
(2) understanding the effect of vertical viscosity, and
(3) initiating the study of three-dimensional models, where new mechanisms such as vorticity stretching arise.

The project will develop analytical tools that clarify how viscosity stabilizes geophysical flows, produce publishable results, and lay the groundwork for external funding applications to NSF-DMS and related programs.

USC Office of Research Awards Announcement Page

Trevor M. Leslie, Changhui Tan

Abstract

We study weak solutions of the one-dimensional pressureless Euler-Poisson-alignment system. When smooth solutions develop singularities, distributional weak solutions are not unique. We introduce an entropy-based selection principle via an associated scalar balance law with time-dependent flux and establish global well-posedness for its entropy solutions. The resulting entropic solution yields a uniquely selected weak solution of the Euler-Poisson-alignment system. In the attractive regime, it is compatible with sticky particle dynamics, while in the repulsive regime atomic states may disperse, revealing a fundamental qualitative difference between the two cases.

This work is supported by NSF grants DMS #2238219

Vinh Nguyen, Roman Shvydkoy, Changhui Tan

Advances in Mathematics, Volume 494, 110929, 45pp. (2026)

Abstract

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol \(\phi\) and a non-linear coupling of velocities given by the power law \(A(v) = |v|^{p-2}v\), \(p>2\). The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the \(k\)-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.

	doi:10.1016/j.aim.2026.110929
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	This work is supported by NSF grants DMS #2238219

 Changhui Tan, Zhuan Ye

Abstract

In this paper, we investigate the two-dimensional incompressible primitive equations with fractional horizontal dissipation. Specifically, we establish global well-posedness of strong solutions for arbitrarily large initial data when the dissipation exponent satisfies \(\alpha\geq\alpha_{0}\approx1.1108\). In addition, we prove global well-posedness of strong solutions for small initial data when \(\alpha \in [1, \alpha_0)\). Notably, the smallness assumption is imposed only on the \(L^\infty\) norm of the initial vorticity.

	This work is supported by NSF grants DMS #2238219
	This work is supported by a USC VPR ASPIRE grant

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
	This work is supported by a USC VPR ASPIRE grant

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