Regularity Problems in Mathematical Fluid Mechanics

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理论的发展中一直起着重要的作用。在这些方程提出的许多开放问题中，最重要的问题之一是它们关于解的存在性和唯一性的“适定性”。解决适定性问题的关键是理解解如何保持规则或形成有限时间奇点。这个问题的重要性是双重的。从数学的角度来看，解的正则性（或不正则性）是任何偏微分方程理论中需要解决的第一个也是最基本的问题；从物理学的观点来看，解的规律性直接关系到方程作为物理现象的数学模型的有效性。**对数学流体力学方程进行成功的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一些新的偏微分方程的方法和技术的发明。然而，进展还远远不能令人满意。**我计划通过详细研究四个代表性系统，即二维广义磁流体动力学（GMHD）方程、欧拉-庞加莱方程、一维非线性非局部系统和液晶的Onsager模型，为偏微分方程理论做出贡献。选择这些方程式是为了达到难度/影响和易用性之间的平衡。一方面，所有这四个问题都表现出数学流体力学中三个困难中的大部分（如果不是全部的话）：非线性、非局部性和耦合。因此，这些方程的研究进展将对其他流体力学方程的研究有所启发。此外，由于化学、生物学和工程学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，这些系统的适定性问题，虽然仍然是开放的，但在数学流体力学的许多开放问题中是比较容易处理的。因此，这些方程对于HQP的训练是理想的。**提出的研究结果将显著提高我们对偏微分方程中的非线性、非局域性和耦合的理解，并将为流体力学以及数学生物学等其他领域的各种方程的研究提供启示。它也将有助于我们对湍流的理解。所提出的研究进展将引起偏微分方程界和流体力学界的兴趣。部分拟进行的研究也将引起非线性泛函分析界的关注。拟议的研究将受益于现有的和潜在的国家和国际合作。通过参与计划项目，HQP将在偏微分方程、谐波分析、非线性泛函分析、流体力学和科学计算等方面得到全面的训练。

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.