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