Release Date: 2021-5-12 News Source: School of Mathematics and Statistics

Editor: News Agency of BIT

Beijing Institute of Technology, May 12th, 2021: In the Stokes fluid equation, there is a long-term unsolved mathematical problem ---the Kac problem of fluids: can the volume and surface area of this watershed be judged by measuring the frequency emitted by the vibratory Stokes fluid? In layman's terms: aiming at judging the size of this watershed by measuring the "wave tones" of it. A few days ago, Professor Liu Genqian, who is from the School of Mathematics and Statistics of Beijing Institute of Technology(BIT), published an online paper "The geometric invariants for the spectrum of the Stokes operator" in the international authoritative mathematics journal "Mathematische Annalen" ,and gave an affirmative answer to this question. Thus, the famous Kac problem of fluids is completely solved.

Professor Liu Genqian creatively gives the following important spectral asymptotic formulas, using the methods of partial differential equations, differential geometry, pseudo-ditterential operator, singular Green operator and spectral geometry theory.

In this formula, μ is the viscosity constant of Stokes flow, λ_kis the Dirichlet eigenvalue of Stokes operator, |Ω| is the volume of Stokes flow, |∂Ω| is the surface area of the Stokes flow. And u_kis the feature vector corresponding to λ_k,that is

This formula shows that: knowing all the vibration frequencies of the Stokes flow, you can calculate the volume and surface area of the fluid through it. Thus, the Kac problem of fluids has been completely solved. This formula also establishes a close connection between (physical) spectral quantities and (mathematical) geometric quantities. On the other hand, this formula has important practical value. For example, this formula can be used to measure the size of watershed in the fields of navigation, oil and water resources exploration, and military.

The paper is 48 pages long. It took four years from submission and expert review to acceptance by the journal. The two reviewers spoke highly of Professor Liu Genqian’s paper and agreed that the paper was "very interesting and actual."

About the Author：

Liu Genqian, professor and doctoral supervisor of the School of Mathematics and Statistics of BIT, has long been engaged in the research of partial differential equations, geometric analysis, spectral geometry and inverse problems. He has published a series of important papers in "Advances in Mathematics" and other international mathematics authoritative journals, solved a number of long unsolved open problems, which includes the Weyl law of biharmonic Steklov eigenvalues, high-order Sobolev inequality on hyperbolic spaces, Avramidi of elastic eigenvalues, etc.

Paperlink:

https://link.springer.com/article/10.1007/s00208-021-02167-worhttps://doi.org/10.1007/s00208-021-02167-wor as a PDF here

https://link.springer.com/content/pdf/10.1007/s00208-021-02167-w.pdf.