Regularity of the 3D Navier-Stokes equations in the largest critical space and related problems

三维不可压Navier-Stokes方程（NSE）解的正则性问题一直是一个重要的研究课题。尽管这个问题还远未解决，但自20世纪30年代Leray的工作以来，已经证明了许多正则性准则。该奖项资助的研究将扩展经典的正则性和唯一性结果，特别是最近建立的Escauriaza-Seregin-Sverak准则，以更广泛的Besov空间。所用的方法将结合联合收割机谐波分析工具和Navier-Stokes方程的经典技术。此外，还计划研究3D NSE和相关模型的全局吸引子，例如二元（又名壳）湍流模型。研究整体吸引子的结构对于更好地理解湍流现象是必不可少的。由于缺乏唯一性证明，不知道3D NSE是否具有解算子半群，因此经典的半流理论不能用于该系统。该奖项旨在继续发展进化系统的全局吸引子理论，这是可以应用于这种情况的动力系统的推广。该奖项将支持有关流体运动方程的一些基本开放问题的研究。这些方程是在两个世纪前提出的，但在数学上仍然没有得到很好的理解。尽管这些方程被物理学家和工程师广泛用于实际应用，但（经典）解的存在性和唯一性仍然未知。对这个问题的回答有望揭示与湍流相关的基本问题。 湍流，通常被称为经典物理学中最后一个未解决的问题，是发生在飞机机身，车辆，船舶和涡轮机叶片周围的流体流动中的基本现象。对湍流的更好的数学理解将导致这些物体设计的改进。

The regularity of solutions of the three-dimensional incompressible Navier Stokes equations (NSE) remains a significant problem. Even thought it is far from been solved, numerous regularity criteria have been proved since the work of Leray in the 1930s. The research funded with this award will extend classical regularity and uniqueness results, in particular, the recently established Escauriaza-Seregin-Sverak criterion, to wider classes of Besov spaces. The methods to be used will combine harmonic analysis tools and classical techaniques for the Navier-Stokes equations. In addition, it is planned to study global attractors of the 3D NSE and related models, such as the dyadic (a.k.a. shell) models of turbulence. The study of the structure of the global attractor is essential for a better understanding of turbulent phenomena. Due to the lack of a uniqueness proof, it is not known whether the 3D NSE possesses a semigroup of solution operators, and consequently a classical theory of semiflows cannot be used for this system. It is proposed to continue developing a theory of global attractors for an evolutionary system, which is a generalization of a dynamical system that can be applied in this situation.This award will support research on some fundamental open questions concerning the equations of fluid motion. The equations were introduced almost two centuries ago but are still not well understood mathematically. Even though the equations are broadly used by physicists and engineers for real-life applications, the existence and uniqueness of (classical) solutions is still not known. An answer to this question is expected to shed light on fundamental issues related to turbulence. Turbulence, often referred to as the last unsolved problem in classical physics, is a fundamental phenomenon occurring in fluid flows around airplane bodies, vehicles, ships, and blades of turbines. A better mathematical understanding of turbulence will result in improvements in the design of these objects.