1. Recently, the internationally renowned academic journal "Communications in Mathematical Physics" published online a research paper titled “Exponential mixing for the randomly forced NLS equation,” authored by Assistant Professor Zhao Jiacheng from the School of Mathematical Sciences at Shenzhen University in collaboration with Chen Yuxuan, a doctoral student at Peking University, Researcher Xiang Shengquan, and Professor Zhang Zhifei.
This paper is part of the authors’ series of works on the long-term statistical behavior of randomly dispersed equations. It investigates the one-dimensional nonlinear Schrödinger equation under spatial local damping and local random perturbations, focusing on revealing the intrinsic connection between asymptotic compactness in infinite-dimensional dynamical systems, stabilization in control theory, and exponential ergodicity in stochastic dynamical systems. The paper demonstrates that exponential ergodicity is not an isolated probabilistic phenomenon, but rather results from the combined action of multiple stability mechanisms in deterministic dynamics: (1) asymptotic compactness constrains the long-term dynamics of solutions in the vicinity of a high-regular compact set; (2) global exponential stabilization of the equation without external forces (with the local damping term acting as linear state feedback) ensures that trajectories originating from any state have a positive probability of approaching the equilibrium point; (3) Local stability along non-stationary trajectories allows two random trajectories to gradually approach each other via finite-dimensional control. These three points implicitly imply exponential ergodicity through probabilistic coupling methods.
Based on this framework, the authors overcame difficulties such as the lack of parabolic smoothing effects in dispersive equations and the fact that damping and noise act only locally in space. By comprehensively applying methods such as the Carleman estimate, nonlinear smoothing, and frequency analysis, they proved that the equation possesses a unique invariant measure in the space H^s (s ≥ 1), and that the probability distribution of the solutions converges at an exponential rate. This work provides a new paradigm for studying the statistical behavior of stochastic dispersive equations using stabilization and control theory.
“Communications in Mathematical Physics” is an internationally renowned journal in the field of mathematical physics, founded in 1965 by Springer in Germany. It primarily publishes original research driven by modern physics problems and characterized by a high degree of mathematical rigor.
Full text link: https://link.springer.com/article/10.1007/s00220-026-05732-z
2. Recently, the internationally renowned academic journal "Journal de Mathématiques Pures et Appliquées" published a research paper titled “Invariant manifolds of the phase-field system in two and three dimensions,” authored by Assistant Professor Zhao Jiacheng from the School of Mathematical Sciences at Shenzhen University in collaboration with Professor Wang Rongnian from Shanghai Normal University and Professor Wu Jianhong from York University in Canada. The paper was submitted in August 2023 and published online in June 2026.
This paper considers phase-field systems in two- and three-dimensional domains. This system consists of a coupled set of hyperbolic–parabolic equations, which, within the framework of the Landau–Ginzburg theory, is used to describe the static and dynamic behavior at the interface between two phases in specific materials. The authors prove the existence of a finite-dimensional invariant manifold for the phase-field system in the absence of a spectral gap, and demonstrate that this manifold is a global attractor for the solutions. Consequently, under the conditions of the paper’s main theorem, the long-time behavior of this PDE system can be reduced to that of an ordinary differential equation system. In 1988, two mathematicians, John Mallet-Paret and George R. Sell, proved the existence of an inertial manifold for the reaction-diffusion equation in the absence of a spectral gap in the "Journal of the American Mathematical Society", one of the four major mathematics journals. Compared to the reaction-diffusion equation, the core difficulties faced by phase-field systems include their hyperbolic nature and the complex structure of the linear part of the coupled system’s spectrum. To overcome these difficulties, the paper decomposes the parabolic equations in the system into stable and unstable parts based on the point spectrum of their linear sections, while treating the hyperbolic equations (which also possess a dissipative mechanism) as the stable part of the entire system. At the same time, the authors first established the (exponential) attractivity of the manifold on a subspace of higher regularity, and then proved the attractivity of the manifold over the entire phase space by showing the asymptotic compactness of the system.
“Journal de Mathématiques Pures et Appliquées,” founded in 1836 by the renowned French mathematician Joseph Liouville, is a long-standing international journal in the field of mathematics. It primarily publishes groundbreaking and significant results in various areas of pure and applied mathematics and enjoys a high academic reputation.
Full text link: https://www.sciencedirect.com/science/article/pii/S0021782426000966?dgcid=author
Biography: Zhao Jiacheng is an Assistant Professor at the School of Mathematical Sciences, Shenzhen University. He received his Ph.D. from the School of Mathematics and Physics at Shanghai Normal University in June 2023. In December 2023, he joined the School of Mathematical Sciences at Peking University as a full-time postdoctoral researcher (Boya Program). He joined Shenzhen University in September 2025. His primary research interests include the stability and stabilizability of partial differential equations and related control theory. His research has been accepted for publication in journals such as CMP, JMPA, SIMA, Nonlinearity, JDE, and JDDE.
