Professor Qin Yuming from the School of Mathematics and Statistics at Donghua University has recently achieved a significant breakthrough in the theoretical study of the 3D Prandtlequations. The research findings have been published in the renowned journal Journal de Mathématiques Pures et Appliquées, with Donghua University listed as the first institution and Professor Qin as the first author. The paper is titled “Local existence of solutions to 3D Prandtl equations with a special structure”.
The well-posed problem for the 3D Prandtl boundary layer equations, a core mathematical model in fluid mechanics, was highlighted by the internationally renowned mathematician Professor Oleinik in herclassic monograph Mathematical Models in Boundary Layer Theory as a major open challenge in the 21st century. Traditional research, restricted by the Crocco transformation and the assumption of outer flow U ≠ constant, had long been limited to the two dimensional case. The 3D problem remained unresolved due to inherent difficulties such as secondary flows and the loss of xy-derivatives.
To address this challenge, Professor Qin innovatively introduced a polynomially weighted Sobolev space and auxiliary functions with favorable properties. While maintaining Oleinik’s monotonicity assumption, he developed new nonlinear energy estimate methods and a mechanism to prevent the loss of xy-derivatives. With this approach, he provided a rigorous proof of the local existence and uniqueness of solutions for the 3D boundary layer equations in a periodic domain. This breakthrough avoids reliance on the classical Crocco transformation and successfully extends the two dimensional theoretical framework to three dimensions, representing a substantial advancement in the study of solution existence for the 3D Prandtl equations. It also offers a vital mathematical tool for subsequent research into complex flow problems.
This research was supported by grants from the National Natural Science Foundation of China.
Link to the paper: https://www.sciencedirect.com/science/article/pii/S0021782425000145
