Regularity Problems in Mathematical Fluid Mechanics

数学流体力学中的正则性问题

• 批准号: RGPIN-2014-06461
• 负责人: Yu, Xinwei
• 金额: $ 1.31万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635162
• 关键词: Regularity; Problems; Mathematical; Fluid; Mechanics

I propose to study the regularity of solutions for several representative partial differential equations in mathematical fluid mechanics.Equations from fluid mechanics have always played an important role in the development of the theory of partial differential equations. Among the many open problems these equations present, one of the most important is their “well-posedness" regarding the existence and uniqueness of the solutions. The key to settling the well-posedness problem is to understand how the solutions can stay regular or form finite-time singularities. The significance of this problem is two-fold. From the mathematical point of view, regularity (or lack thereof) of solutions is the very first and most fundamental issue to be settled in any theory of partial differential equations; from the physical point of view, regularity of solutions directly relates to the validity of the equations as mathematical models for physical phenomena. There are three major obstacles to successful mathematical analysis of equations from mathematical fluid mechanics: nonlinear terms, nonlocal operators, and coupling between unknown quantities. These difficulties have inspired the invention of several new methods and techniques for partial differential equations in recent years. However the progress is still far from satisfactory. I plan to contribute to the theory of partial differential equations through detailed study of four representative systems: the two-dimensional generalized magnetohydrodynamical (GMHD) equations, the Euler-Poincare equations, a one-dimensional nonlinear nonlocal system, and the Onsager model for liquid crystals. These equations are chosen to achieve a balance of difficulty/impact and accessibility. On one hand, all four exhibit most, if not all, of the three difficulties in mathematical fluid mechanics: nonlinearity, nonlocality, and coupling. As a consequence, progress in the study of these equations would shed light on the study of other fluid mechanical equations. Furthermore, as many mathematical models in chemistry, biology, and engineering are derived using ideas from fluid mechanics, the proposed research will also have impact on those fields. On the other hand, there is evidence that the well-posedness problem of these systems, though still open, are among the more tractable ones in the many open problems in mathematical fluid mechanics. Therefore, these equations are ideal for the training of HQP. The outcome of the proposed research will significantly improve our understanding of nonlinearity, nonlocality, and coupling in partial differential equations and will shed light on the study of a wide variety of equations from not only fluid mechanics but also other fields such as mathematical biology. It will also contribute to our understanding of turbulence. Progress in the proposed research will be of interest to both the partial differential equations community and the fluid mechanics community. Part of the proposed research will also draw attention from the community of nonlinear functional analysis. The proposed research will benefit from existing and potential national and international collaborations. Through working on the proposed projects, HQP will receive comprehensive training in partial differential equations, harmonic analysis, nonlinear functional analysis, fluid mechanics, and scientific computing.

我建议研究数学流体力学中几种代表性部分微分方程的解决方案的规律性。流体力学的方程在偏微分方程理论的发展中始终起着重要作用。在这些方程式存在的许多开放问题中，最重要的是它们关于解决方案的存在和独特性的“适合性”。解决良好的问题的关键是了解解决方案如何保持规则或形成有限的奇异点。这个问题的意义是两个倍。从数学的角度来看，解决方案理论中要解决的第一个也是最根本的问题是规律性（或缺乏解决方案）。从物理的角度来看，解决方案的规律性与方程式作为物理现象的数学模型直接相关。从数学流体机制中成功进行数学分析的数学分析存在三个主要障碍：非线性术语，非本地运算符和未知数量之间的耦合。这些困难启发了近年来针对部分微分方程的几种新方法和技术的发明。但是，进度远非满意厂。我计划通过对四个代表性系统的详细研究来促进部分微分方程的理论：二维一般磁性水动力学（GMHD）方程，Euler-PointCare方程，一维非线性非局部性系统，以及液晶的Onsager模型。选择这些方程是为了达到难度/影响和可访问性的平衡。一方面，在数学流体力学中，这三个困难中的三个困难中，这四个方面的表现最多：非线性，非局部性和耦合。结果，对这些方程的研究进展将揭示出其他流体机械方程的研究。此外，由于化学，生物学和工程学中的许多数学模型是使用流体机制的思想得出的，因此拟议的研究也将对这些领域产生影响。另一方面，有证据表明，在数学流体机制中许多开放问题中，这些系统的适应性问题虽然仍然开放，但仍是更容易挑战的问题。因此，这些等式是培训HQP的理想选择。拟议的研究的结果将显着提高我们对部分微分方程中非线性，非局部性和耦合的理解，并将阐明研究不仅是流体力学的各种方程式的研究，而且还阐明了其他领域，例如数学生物学。这也将有助于我们对湍流的理解。拟议的研究的进展将引起部分微分方程社区和流体力学界的兴趣。拟议的研究的一部分还将引起非线性功能分析社区的关注。拟议的研究将受益于现有和潜在的国家和国际合作。通过从事拟议的项目，HQP将通过部分微分方程，谐波分析，非线性功能分析，流体力学和科学计算进行全面的培训。