Mathematical Structure of Blowup Solutions in Inviscid, Incompressible Flow

该提议涉及不可压缩、无粘性流体流动方程的某些解是否会在有限时间内产生自发奇点的问题。目的是揭示 PI 之前的研究表明的可能爆破流的数学结构。计划有两个项目。第一个是使用先进的计算技术来揭示与崩溃和爆炸相关的渐近行为。先前的工作表明，存在一个Leray型内自相似解，该解具有有限时间奇点、正则外解和极向涡度匹配区域。另一个项目扩展了所有研究人员发现的所有候选爆炸流都具有一个或多个离散对称性并且可以与有限数量的点和空间群相关联的观察。所有流体场变量都可以与不可约表示相关联。将探讨离散点群的特征（例如简并性、交换性、关联晶格和空间群）如何影响流动解决方案。工程师和科学家通常使用控制不可压缩流体（例如水）运动的方程进行设计和研究。然而，关于这些方程的最基本问题已经开放了 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平滑或爆炸。这项工作解决了寻找无摩擦流动问题的答案。这是一个非常具有挑战性的计算问题，也是对现代数值技术的严峻考验。由于其重要性，相关的粘性问题被选为与本世纪希尔伯特问题并列的七个克莱千禧奖问题之一。

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.