Time: 2022-10-17

News Source & Photographer: School of Mathematics and Statistics

Editor: Wang Huan Reviewer: Chen Ke

Translator: Long Xiaofei & Qin Yao

Recently, Professor Zhu Rongchan, School of Mathematics and Statistics, BIT, published a research paper entitled “On Ill - and Well-Premises of Dissipative Martingale Solutions to Stochastic 3D Euler Equations” on the top international academic journal Communications on Pure and Applied Mathematics. In this study, for the first time, the concept of dissipative martingale solution is proposed for the stochastic three-dimensional Euler equation, and the existence, weak-strong uniqueness, nonuniqueness in law, existence of strong Markov solution and nonuniqueness of strong Markov solutions of dissipative martingale solution are proved. Furthermore, Professor Zhu Rongchan and her collaborators proved that the Markov solution of the stochastic three-dimensional Navier-Stokes equation is not unique, and constructed a strong solution of the stochastic three-dimensional Navier-Stokes equation in the sense of probability for the first time, solving a problem that has puzzled the field of stochastic partial differential equations for many years. The relevant results were recently received by The Annals of probability, the top journal of probability theory.

Three-dimensional Navier-Stokes equations and three-dimensional Euler equations are important models in fluid equations, which have been widely studied by many mathematicians. The existence of global smooth solutions for three-dimensional Navier-Stokes equations is one of Clay's millennium problems. Stochastic fluid equation is an important model to study turbulence because it is easier to obtain ergodicity. In 1941, Kolmogolov put forword a formal derivation that turbulence meets the statistical law, and its strict mathematical proof is an open problem. Due to convex integration methods, determined fluid equations have made great progress in recent time, including Isett's proof of Onsager conjecture of three-dimensional Euler equations, Buckermaster-Vicol (2019 Clay Joint Research Award)'s proof of the nonunique results of weak solutions of three-dimensional Navier Stokes equations. Professor Zhu Rongchan and her collaborators first applied the convex integral method to the study of stochastic fluid equations, discussed the distribution properties of solutions, studied the distribution properties of solutions of stochastic three-dimensional Euler equations and Navier Stokes equations, and obtained that the distribution of solutions is not unique. Furthermore, the paper proposes a suitable concept of martingale solution for the stochastic three-dimensional Euler Navier Stokes equation, from which the Markov process solution can be constructed. The nonuniqueness of strong Markov solution has also been put forward. The relevant results were evaluated as “Remarkable” by Professor Vicol of Clare Award winner in [Bull. Amer. Math. Soc. 58 (1): 1-44, 2021].

When it comes to the stochastic three-dimensional Euler/Navier Stokes equation, how to construct a strong solution in probabilistic sense is an open problem in the field of stochastic partial differential equations due to the orbital uniqueness. In this paper, a new iterative elimination method for constructing solutions of stochastic partial differential equations is given by introducing the convex integral method. From this method, the probabilistic strong solutions of stochastic three-dimensional Euler/Navier Stokes equations can be constructed for the first time.

The research work was completed by Professor Zhu Rongchan, Professor Hofmanova of Bielefeld University in Germany and researcher Zhu Xiangchan of the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences. Professor Zhu Rongchan is the corresponding author of this article. This work is supported by the National Natural Science Foundation of China.

Paper link address:https://onlinelibrary.wiley.com/doi/10.1002/cpa.22023

Brief introduction of the research team and the person in charge attached:

The probability team of the School of Mathematics and Statistics of BI T has actively carried out substantive international cooperation research and has made a series of important research achievements.

Zhu Rongchan is a professor and doctoral supervisor of BIT. She graduated from Sichuan University with an undergraduate degree. During her doctoral studies, she was jointly trained by the Institute of Mathematics and Systems Science of the Chinese Academy of Sciences and the University of Bielefeld in Germany. She is also a visiting scholar at the University of Bielefeld in Germany. She has been engaged in the research of stochastic partial differential equations, Dirichlet type and stochastic analysis for a long time, and has successively presided over/completed the National Natural Science Foundation for Distinguished Young Scholars, general programs and youth programs. She has published more than 30 SCI papers as the first/corresponding author in Communications on Pure and Applied Mathematics, The Annals of Probability, Probability Theory and Related Fields, Communication in Mathematical Physics, Journal of Functional Analysis and other journals.