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.
