Toward criticality of the Navier-Stokes regularity problem

纳维-斯托克斯正则性问题的关键性

基本信息

  • 批准号:
    2009607
  • 负责人:
  • 金额:
    $ 24.33万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2020
  • 资助国家:
    美国
  • 起止时间:
    2020-07-01 至 2023-06-30
  • 项目状态:
    已结题

项目摘要

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.
三维Navier-Stokes方程所描述的流动中是否会形成奇点是数学物理中的一个重要的公开问题。它的重要意义源于这样一个事实,即Navier-Stokes方程在科学和工程中已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域)下对该模型进行严格的验证,特别是排除奇异性的形成,是更可靠地模拟湍流流动的关键。自20世纪30年代纳维-斯托克斯正则性问题开始以来,在防止可能形成奇点所需要的东西和从方程中可以严格获得的东西之间一直存在着一道“鸿沟”。根据该奖项将完成的研究的目的是弥合这一差距并达到“临界点”;这不会完全排除奇点的形成,但将极大地限制奇点可能发生的情况。考虑到自然界和工程世界中无处不在的湍流现象,这项研究的影响将超出学科的界限。在指导初级科学家方面,该奖项将资助一名研究生研究助理。该奖项将在Pi和L.Xu最近的工作基础上进行,首次证明了Navier-Stokes正则性问题的渐近临界性质。该方法基于对速度场高阶导数的正、负分量的超水平集的适当定义的稀疏性的研究,并通过调和测量优化原理来利用稀疏性(由于任何光滑流动都是在空间变量中自动解析的,因此分量的正、负部分是次调和的)。超水平集越稀疏,调和测度优化原理在生成超范数上的“自我改进”界时就越有效。在这个框架内,证明了当导数的阶数趋于无穷大时,正则类和相应的先验界之间的“标度差”缩小到零,这是渐近临界性的一个表现。将进行的研究的主要目标是得出临界性的更强和/或更经典的表现形式。有两种自然的途径可以采取,相对于扩散强度的临界性和关于非线性强度的临界性。这正是两个主要的项目,前者将发生在3D,超临界,超耗散的Navier-Stokes方程的背景下,而后者将发生在通过涡度方向的局部一致性来几何耗尽非线性的领域。该奖项反映了NSF的法定使命,并通过使用基金会的智力优势和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
A regularity criterion for 3D NSE in ‘dynamically restricted’ local Morrey spaces
“动态限制”局部 Morrey 空间中 3D NSE 的规律性准则
  • DOI:
    10.1080/00036811.2021.1906418
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    1.1
  • 作者:
    Grujić, Zoran;Xu, Liaosha
  • 通讯作者:
    Xu, Liaosha
Toward criticality of the Navier-Stokes regularity problem
纳维-斯托克斯正则问题的关键性
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Zoran Grujic其他文献

Zoran Grujic的其他文献

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{{ truncateString('Zoran Grujic', 18)}}的其他基金

Three problems in fluid mechanics through the lens of sparseness of the regions of intense fluid activity
从流体活动强烈区域的稀疏性角度来看流体力学中的三个问题
  • 批准号:
    2307657
  • 财政年份:
    2023
  • 资助金额:
    $ 24.33万
  • 项目类别:
    Standard Grant
Collaborative research: Turbulent cascades and dissipation in the 3D Navier-Stokes model
合作研究:3D Navier-Stokes 模型中的湍流级联和耗散
  • 批准号:
    1515805
  • 财政年份:
    2015
  • 资助金额:
    $ 24.33万
  • 项目类别:
    Standard Grant
Collaborative research: Turbulent cascades and regularity theory in physical scales of 3D incompressible fluid flows
合作研究:3D 不可压缩流体物理尺度中的湍流级联和规律性理论
  • 批准号:
    1212023
  • 财政年份:
    2012
  • 资助金额:
    $ 24.33万
  • 项目类别:
    Standard Grant

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