Vinh Nguyen, Roman Shvydkoy, Changhui Tan

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.

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

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