Recently, Associate Professor Wang Yanqing from the School of Mathematics and Information Science published a research paper titled "On the borderline regularity criterion in anisotropic Lebesgue spaces of the Navier-Stokes equations" in the Journal of Differential Equations (Chinese Academy of Sciences Q1 TOP journal),an internationally renowned journal in the field of mathematics. This work, with Zhengzhou University of Light Industry as the first affiliated institution and Wang Yanqing as the first author, was conducted in collaboration with Associate Professor Wei Wei from Northwest University, Associate Professor Wu Gang from the University of Chinese Academy of Sciences, and Associate Professor Zhou Daoguo from Hangzhou Normal University.

The incompressible Navier-Stokes equations are one of the fundamental sets of equations in fluid mechanics. These equations, through coupling with other physical quantities, can be widely applied to fields such as fluid mechanics, aircraft manufacturing and weather forecasting. The question of whether the solutions of the incompressible Navier-Stokes equations in three and higher dimensions experience blow-up in finite time is one of the seven Millennium Prize Problems proposed by the Clay Mathematics Institute. A classical result in the regularity of the Navier-Stokes equations in three and higher dimensions is the Ladyzhenskaya-Prodi-Serrin Criterion: The solutions of the Navier-Stokes equations must be regular when they belong to certain critical spaces. The endpoint case of this criterion remained unresolved until 2003, when Escauriaza, Seregin, and Sverak used techniques such as heat kernel backward uniqueness and Carleman-type estimates to provide a final solution. This paper aims to promote the endpoint regularity criterion. Leveraging the Gagliardo-Nirenberg inequality (Journal d'Analyse Mathématique, 2025) in anisotropic Lebesgue spaces and local suitable weak solutions of Navier-Stokes, which were recently proven by Wang Yanqing, Mei Xue, a master's degree candidate at Zhengzhou University of Light Industry, and Wei Wei, Associate Professor from Northwest University, it extended the Escauriaza-Seregin-Sverak results from classical Lebesgue spaces to anisotropic Lebesgue spaces. Furthermore, a small-energy regularity criterion for the Navier-Stokes equations in higher dimensions was established using the De Giorgi iteration technique, further proving the endpoint regularity criterion for the Navier-Stokes equations in higher dimensions in critical anisotropic Lebesgue spaces. This finding enriches and refines the regularity results for the incompressible Navier-Stokes equations in critical endpoint spaces.

This research was funded and supported by the General Program of the National Natural Science Foundation of China and the Science Fund for Distinguished Young Scholars of Henan Province.

Journal article link: https://www.sciencedirect.com/science/article/pii/S002203962500378X