UM team published research effort in a top international mathematics journal
澳大數學系團隊在國際顶尖數學期刊發表研究成果

Professor Changfeng GUI and postdoctoral fellow Hao LIU from the Department of Mathematics, Faculty of Science and Technology at the University of Macau, together with their collaborators Dr. Jeaheang BANG from Westlake University, Professor Yun WANG from Soochow University, and Professor Chunjing XIE from Shanghai Jiao Tong University, have recently published a research article entitled “Rigidity of steady solutions to the Navier–Stokes equations in high dimensions and its applications” in the top international mathematics journal “Journal of the European Mathematical Society”. The paper establishes an important rigidity theorem for the Navier–Stokes equations, namely that in spatial dimensions greater than 3, any steady solution satisfying  must be a trivial solution.

The Navier–Stokes (NS) equations are the fundamental equations describing the motion of viscous fluids. The well-posedness theory for these equations has long been one of the central topics in the study of fluid mechanics and partial differential equations, and it is also one of the seven famous Millennium Problems. In the regularity theory for the NS equations, the question of whether one can rule out singularities with scale-invariant bounds and understand the properties of solutions possessing scale-invariant behavior has attracted extensive attention from many internationally renowned mathematicians, including Šverák, Gang TIAN, Zhouping XIN, Hongze YAO, among others. Earlier, Šverák, a member of the American Academy of Arts and Sciences, and his collaborators proved that in dimensions greater than 3, self-similar singularities of steady Navier–Stokes equations can be excluded. Šverák further posed an open problem asking whether more general steady solutions that merely satisfy a scale-invariant bound must also be trivial when the spatial dimension is greater than 3.

This paper provides a positive solution to Šverák’s open problem by proving that, in dimensions greater than 3, any steady solution satisfying  must be trivial without assuming any smallness or self-similarity, but also, by combining blow-up analysis, the authors show that singularities with scale-invariant bounds in general bounded domains can be excluded. This significantly extends prior results of Kozono, former president of the Mathematical Society of Japan, and his collaborators. In addition, by applying the established rigidity theorem, the authors obtain the precise asymptotic profile at infinity for solutions with critical decay rates, showing that the leading term is given by the fundamental solution of the linear Stokes equations.

The core methodology of the paper is to introduce a scale-invariant weighted energy and, using the nonnegativity of the total head pressure, derive weighted energy estimates. These estimates break the scale-invariant bound and ultimately lead to the conclusion that the solution must vanish identically. This approach is also expected to provide useful insights for the study of rigidity of solutions to other fluid equations and related problems.