Periodic Reporting for period 2 - FluFloRan (Mathematical analysis of fluid flows: the challenge of randomness)
Reporting period: 2022-09-01 to 2024-02-29
The project delves into a comprehensive exploration of the effects of randomness within the realm of fluid dynamics models, with a specific emphasis on the intricate interplay within the Navier-Stokes and Euler equations when subjected to a diverse stochastic perturbations. This undertaking is motivated by countless open questions concerning existence and (non)uniqueness of solutions in various settings. Moreover, our investigation extends to encompass the long-time behavior of these solutions as well as their underlying qualitative properties. One of the enduring, paramount challenges has been to establish whether a unique ergodic invariant measure exists for the stochastic Navier-Stokes equations, even in scenarios where stochastic perturbations may exhibit degeneracy. In addition to these inquiries, we also focus on the intriguing phenomenon of regularization induced by noise, aiming to identify instances where specific stochastic perturbations genuinely yield beneficial effects on the underlying deterministic dynamics. These complex questions not only interest mathematicians but also form part of the foundation of our understanding of fluid dynamics models. Solving them is not just an academic curiosity; it's crucial for pushing the boundaries of mathematical knowledge.
The foremost breakthrough emerged as a decisive response to a long-standing unresolved open problem within the field of stochastic fluid dynamics, specifically, the demonstration of non-uniqueness in law for stochastic Navier-Stokes equations within the project by Hofmanova, Zhu, Zhu. This achievement was promptly succeeded by a pivotal result regarding the ill- and well-posedness of stochastic Euler equations, achieved by the same authors. These accomplishments served as the foundational bedrock for an array of unforeseen and momentous findings, far exceeding the original scope of the project. Notably, we succeeded in establishing the existence and non-uniqueness of globally defined, probabilistically strong and Markov solutions for the stochastic Navier-Stokes equations. Additionally, we unveiled the global existence and non-uniqueness of solutions for Navier-Stokes equations subjected to space-time white noise perturbations. We also delved into the treatment of a class of critical/supercritical equations perturbed by a very singular noise. These achievements paved the way for promising new avenues of research. Prior to the publication of our research findings, there lingered a prevailing optimism within the stochastic fluid dynamics community regarding the potential efficacy of suitably chosen stochastic perturbations in mitigating the issues of ill-posedness inherent in the deterministic framework. Our results have substantially altered this perspective, effectively debunking these hopeful assumptions, at least with regard to specific forms of stochastic perturbations. Simultaneously, we undertook a deeper exploration of the phenomenon of noise-induced regularization, identifying various instances where noise indeed exerts a positive influence.
In the past, there was optimism that introducing an additive stochastic noise, possibly with appropriate low regularity, could potentially lead to the establishment of uniqueness in law for the Navier-Stokes equations. As a result, numerous esteemed mathematicians dedicated their efforts to exploring this avenue, approaching the problem from diverse angles. However, our findings have refuted these assumptions, at least in the context of weak solutions. The scope of possibilities offered by our new methods exceeded our expectations and extended well beyond what was initially envisioned in the project proposal. Additionally, we demonstrated that in certain scenarios, noise indeed plays a helpful role in extending the realm of well-posedness. Achieving a comprehensive understanding of the wide spectrum of stochastic perturbations and their impact on fluid dynamics models remains an important undertaking for the future.