Toward criticality of the Navier-Stokes regularity problem

在三维Navier-Stokes方程所描述的流动中，奇点能否形成的问题是数学物理中的主要开放性问题之一。它的重要性源于Navier-Stokes方程在科学和工程中被广泛用于模拟三维流体流动，而在强烈流体活动（例如，飞机遇到的湍流区）的情况下对模型进行严格验证，特别是排除奇点形成，是更可靠地模拟湍流的关键。自从20世纪30年代Navier-Stokes正则性问题出现以来，在防止奇点可能形成所需的条件与从方程中严格获得的条件之间存在着“差距”。在该奖项下完成的研究的目的是弥合这一差距并达到“临界”；这不会完全排除奇点的形成，但会极大地限制奇点可能发生的情况。考虑到湍流现象无处不在，无论是在自然界还是在工程世界中，研究的影响将超越学科的界限。在指导初级科学家方面，该奖项将支持一名研究生研究助理。在该奖项下进行的研究建立在PI和L. Xu最近的工作基础上，首次证明了Navier-Stokes正则问题的渐近临界性质。该方法基于对速度场高阶导数分量的正、负分量的超水平集的稀疏性尺度的研究，并通过谐波测度最大化原理（分量的正、负部分是次谐波，因为任何平滑的流动在空间变量中都是自动解析的）来利用稀疏性。超水平集越稀疏，调和测度多数化原理在超范数上生成“自完善”界的效率越高。在这个框架内，证明了正则类和相应的先验界之间的“尺度差距”随着导数的阶数趋于无穷而缩小到零，这是渐近临界性的一个表现。要进行的研究的主要目标是达到更强和/或更经典的临界表现。有两种自然的途径，一种是关于扩散强度的临界，另一种是关于非线性强度的临界。这正是两个主要项目，前者将在三维，超临界，超耗散的Navier-Stokes方程的设置中进行，后者将在通过涡度方向的局部相干性的非线性几何耗尽领域中进行。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The problem of whether a singularity can form in a flow described by the 3D Navier-Stokes equations is one of the major open problems in mathematical physics. Its significance stems from the fact that the Navier-Stokes equations have been widely used in science and engineering to model 3D fluid flows, and a rigorous validation of the model in the regime of the intense fluid activity (e.g., the pockets of turbulence encountered by the airplanes), and in particular ruling out the singularity formation, is the key to more reliable modeling of turbulent flows. Since the inception of the Navier-Stokes regularity problem in the 1930s, there has been a 'gap' between what is needed to prevent the possible formation of a singularity and what could be rigorously obtained from the equations. The aim of the research to be accomplished under the award is to bridge this gap and arrive at 'criticality'; this will not completely rule out the formation of singularities, but will drastically restrict the possible scenarios at which it might occur. Given the omnipresence of the turbulent phenomena, both in nature and in the engineered world, the impact of the research will extend beyond the boundaries of the discipline. In the domain of mentoring junior scientists, the award will support a graduate research assistant.The research to be carried out under the award builds on a very recent work by the PI and L. Xu demonstrating--for the first time--asymptotically critical nature of the Navier-Stokes regularity problem. The methodology is based on the study of a suitably defined scale of sparseness of the super-level sets of the positive and the negative parts of the components of the higher-order derivatives of the velocity field, and the sparseness is utilized via the harmonic measure majorization principle (the positive and the negative parts of the components are subharmonic since any smooth flow is automatically analytic in the spatial variables). The sparser the super-level sets are, the more efficient the harmonic measure majorization principle is in generating the 'self-improving' bounds on the sup-norm. Within this framework, it was demonstrated that the 'scaling gap' between the regularity class and the corresponding a priori bound shrinks to zero as the order of the derivative goes to infinity, a manifestation of the asymptotic criticality. The main goal of the research to be performed is to arrive at stronger and/or more classical manifestations of the criticality. There are two natural avenues to take, toward criticality with respect to the strength of the diffusion and toward criticality with respect to the strength of the nonlinearity. These are exactly the two main projects, the former will take place in the setting of the 3D,super-critical, hyper-dissipative Navier-Stokes equations, and the latter in the realm of the geometric depletion of the nonlinearity via the local coherence of the vorticity direction.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.