Mathematical Structure of Blowup Solutions in Inviscid, Incompressible Flow

无粘、不可压缩流中爆破溶液的数学结构

• 批准号: 0204268
• 负责人: Norman Zabusky
• 金额: $ 12.38万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204268&HistoricalAwards=false
• 关键词: Mathematical; Structure; Blowup; Solutions; Inviscid

This proposal concerns the question of whether some solutions of the equations of incompressible, inviscid fluid flow develop spontaneous singularities in finite time. The objective is to uncover the mathematical structure of possible blowup flows suggested by the PI previous investigations. Two projects are planned. The first will be to use advanced computational techniques to reveal the asymptotic behavior associated with collapse and blowup. Previous work indicates that there exists an inner Leray-type self-similar solution which exhibits a finite-time singularity, a regular outer solution and a matching region of poloidal vorticity. The other project expands upon the observation that all candidate blowup flows found by all investigators have one or more discrete symmetries and can be associated with the finite number of point and space groups. All fluid field variables can be associated with irreducible representations. How the characteristics of the discrete point groups, such as degeneracy, commutivity, associated lattices and space groups effect flow solutions will be explored. The equations that govern the motion of an incompressible fluid, such as water, are routinely used by engineers and scientists for design and research. The most fundamental question about these equations, however, has been open for more than 250 years and reflects directly on the validity of using these equations as a model of fluid flow: whether solutions remain smooth or blowup in finite time. Finding an answer to this question for frictionless flow is addressed in this work. It is a very challenging problem computationally and will be a severe test on modern numerical techniques. Because of its importance, the related viscous problem has been selected as one of seven Clay Millennium prize problems regarded as the Hilbert problems for this century.

这一建议涉及的问题是否不可压缩，无粘流体流动方程的某些解决方案在有限时间内发展自发奇点。其目的是揭示PI以前的调查所建议的可能的爆破流的数学结构。计划了两个项目。第一个将是使用先进的计算技术来揭示与坍缩和爆破相关的渐近行为。前人的工作表明，存在一个具有有限时间奇异性的Leray型自相似内解，一个正则外解和一个极向涡度匹配区域。另一个项目扩展后的观察，所有的候选人爆破流发现的所有调查人员有一个或多个离散的对称性，可以与有限数量的点和空间群。所有流场变量都可以与不可约表示相关联。我们将探讨离散点群的特征，如退化性、交换性、关联格和空间群对流解的影响。控制不可压缩流体（如水）运动的方程通常被工程师和科学家用于设计和研究。然而，关于这些方程的最基本的问题已经公开了250多年，并直接反映了使用这些方程作为流体流动模型的有效性：解在有限时间内是否保持光滑或爆破。在这项工作中解决了这个问题的无摩擦流的答案。这是一个非常具有挑战性的计算问题，也是对现代数值计算技术的严峻考验。由于它的重要性，相关的粘性问题已被选为七个粘土千年奖问题之一，被视为本世纪的希尔伯特问题。