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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612237
• 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.

本文拟研究数学流体力学中几个具有代表性的偏微分方程解的正则性。 流体力学方程在偏微分方程组理论的发展中一直扮演着重要的角色。在这些方程存在的许多公开问题中，最重要的是它们关于解的存在和唯一性的适定性。解决适定性问题的关键是理解解如何保持正则或形成有限时间奇性。这个问题的意义是双重的。从数学的角度来看，解的正则性(或正则性缺乏)是任何偏微分方程组理论中要解决的第一个也是最基本的问题；从物理学的角度来看，解的正则性直接关系到方程作为物理现象的数学模型的有效性。 数学流体力学方程的成功数学分析有三个主要障碍：非线性项、非局部算子和未知量之间的耦合。这些困难激发了近年来偏微分方程组的几种新方法和新技术的发明。然而，进展仍然远远不能令人满意。 我打算通过详细研究四个具有代表性的系统来为偏微分方程组理论做出贡献：二维广义磁流体动力学(GMHD)方程、欧拉-庞加莱方程、一维非线性非局域系统和液晶的Onsager模型。选择这些公式是为了在难度/影响和可获得性之间取得平衡。一方面，这四本书展示了数学流体力学中的三个困难中的大多数，如果不是全部的话：非线性、非局域性和耦合。因此，这些方程的研究进展将为其他流体力学方程的研究提供借鉴。此外，由于化学、生物和工程中的许多数学模型都是利用流体力学的思想推导出来的，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，尽管这些系统的适定性问题仍然是开放的，但在数学流体力学的许多开放问题中，它是更容易处理的问题之一。因此，这些方程是HQP训练的理想选择。 这项研究的结果将极大地提高我们对偏微分方程中的非线性、非局部性和耦合的理解，并将不仅对流体力学而且对数学生物学等其他领域的各种方程的研究有所启示。这也将有助于我们对湍流的理解。偏微分方程界和流体力学界都对拟议研究的进展感兴趣。部分拟议的研究也将引起非线性泛函分析社区的关注。拟议的研究将受益于现有的和潜在的国家和国际合作。通过开展拟议的项目，HQP将接受偏微分方程、调和分析、非线性泛函分析、流体力学和科学计算方面的全面培训。