Yu Pin Collaborates to Achieve Breakthrough in Research on High-Dimensional Compressible Euler Equations

Yu Pin from Tsinghua University and Luo Tianwen from South China Normal University have made significant progress in the mathematical theory of high-dimensional gas dynamics. Their series of papers titled "Stability of High-Dimensional Rarefaction Waves" was officially published in the Annals of Mathematics on September 26, 2025. In this issue, only three articles were updated and published online, two of which are the series of papers completed by Professor Yu Pin from the Department of Mathematical Sciences in collaboration with Luo Tianwen from South China Normal University. This research is the first to conduct a systematic and rigorous mathematical analysis of rarefaction waves in high-dimensional compressible Euler equations, proving the nonlinear structural stability of classical Riemann problems in the bi-family rarefaction wave region, thereby filling a theoretical gap in this field.

The three articles published in this issue of Annals of Mathematics

A rarefaction wave is an expansion process that occurs in fluids or gases, opposite to the common shock wave (which represents a compression process). Its formation mechanism can be understood through a classic model: Assume a long tube is filled with stationary gas, with uniform pressure and density inside. When the piston at the left end is suddenly and rapidly pulled to the left, the nearby gas expands first, causing a local decrease in pressure and density; this expanded gas layer further pushes adjacent gas layers to expand sequentially, causing the low-pressure area to propagate like a wave. In reality, rarefaction waves widely exist in various physical processes. For example, during an explosion, a shock wave first forms at the explosion center and propagates outward, followed by the recovery process of the low-pressure area formed in the central region, which is dominated by rarefaction waves. Another example is in supersonic flight, where the aircraft's front generates a shock wave, while its tail is often accompanied by the appearance of rarefaction waves.

Abstract of the First Paper

Abstract of the Second Paper

In terms of research methodology, Yu Pin and Luo Tianwen innovatively employed differential geometric tools to characterize the geometric structure of sound propagation in gases. The first paper in the series focuses on the a priori estimates for central rarefaction waves. By analyzing the fine geometric structure of characteristic surfaces near singularities, it proposes an energy method without derivative loss, providing key a priori control for the stability proof. Building on this, the second paper delves into the canonical foliation structure at the singularities, revealing the nonlinear stability mechanism in high-dimensional Riemann problems. This work is an important manifestation of the cross-integration of analysis and geometry in the study of partial differential equations. The developed ideas and methods provide new perspectives and tools for a deeper understanding of high-dimensional compressible Euler equations.

Professor Yu Pin

Yu Pin is currently a professor in the Department of Mathematical Sciences at Tsinghua University. His research focuses on partial differential equations, particularly wave phenomena in fluid mechanics and general relativity. His research work has received support from the National Natural Science Foundation of China, the "New Cornerstone Science Foundation," and the "Xiaomi Chair Professorship" project. While dedicated to cutting-edge scientific research, Professor Yu Pin has long emphasized talent cultivation and teaching practice. He has offered multiple foundational courses such as "Mathematical Analysis" and "Abstract Algebra" for undergraduates and graduate students, which are highly popular among students. Luo Tianwen is currently a professor at the School of Mathematical Sciences, South China Normal University. He previously served as an assistant professor at the Yau Mathematical Sciences Center of Tsinghua University.

Paper Link:

https://annals.math.princeton.edu/2025/202-2/p02

https://annals.math.princeton.edu/2025/202-2/p03