Hydrodynamic Stability, Boundary layers, Free boundaries, and Polymeric Flows

了解流体流动的行为在许多科学和技术领域，包括工程，物理学和生物物理学中具有根本的重要性。在基本层面上，流体流动的动力学行为由欧拉或纳维尔-斯托克斯方程描述。这些复杂的系统的非线性偏微分方程是非常难以研究使用经典的技术和只有少数精确的解决方案是已知的这一天。在许多应用中，不需要精确解，本项目的主要目标是在某些极限情况下给出这些方程的解的定性描述。 实际上，当在某些限制情况下，由于存在小参数或考虑长时间时，这些复杂的方程组可以简化为更简单的模型。 这些更简单的模型能够提供这些解决方案的行为的重要定性描述，而不必显式地计算它们。 这种通用方法在复杂系统的长期行为，奇异性的发展，特殊模式的形成，稳定性和不稳定性之间的过渡和过渡到湍流的第一步的研究中有应用。该项目将涉及研究生和博士后的培训。本计画的主要问题是研究二维Euler方程和二维Navier-Stokes方程的某些切变流的渐近稳定性。在Gevrey正则性的周期性条件下，对Couette流的无粘阻尼的研究取得了一些重要进展。 该项目的主要目标是将该研究扩展到更一般的剪切流情况。 一个新的困难来自于这样一个事实，即线性化的问题更难分析，需要从泛函分析中获得一些深刻的思想来克服缺乏对解决方案的简单明确的描述。第二个项目是在整个空间（即没有周期性的假设）的问题的研究。这里的主要困难来自低频混合的均匀性的缺乏。第三个项目是理解粗糙扰动下的行为，即仅在Sobolev空间中的扰动。 在此基础上，还将发现一些新的非线性级联。第四个项目是研究阻尼不发生时的情况，并尝试识别特殊的解决方案，如猫眼流。 这四个项目也可以制定的Navier-Stokes演化和这里的主要问题是要了解小粘度限制。

Understanding the behavior of fluid flows is of fundamental importance in many scientific and technological fields, including engineering, geophysics, and biophysics. At the basic level, the dynamical behavior of fluid flow is described by the Euler or the Navier-Stokes equations. These complex systems of nonlinear partial differential equations are very difficult to study using classical techniques and only a few exact solutions are known to this day. In many applications, an exact solution is not needed and a main goal of this project is to give a qualitative description of solutions to these equations in some limiting cases. Indeed, when taken in some limiting situations, due to the presence of a small parameter or when considered for large time, these complex sets of equations may be reduced to simpler models. These simpler models are capable of providing important qualitative descriptions of the behavior of these solutions without having to compute them explicitly. This general method has applications in the study of the long-time behavior of complex systems, the development of singularities, the formation of special patterns, the transition between stability and instability and the first steps of transition towards turbulence. This project will involve the training of graduate students and postdocs. The main problem to be addressed in this project is the study of the asymptotic stability of some shear flows for the 2D Euler and the 2D Navier-Stokes equations. Some important progress was made recently in the study of the inviscid damping around Couette flow in a periodic setting for Gevrey regularity. A main goal of this project is to expand this study to the case of more general shear flows. A new difficulty comes from the fact that the linearized problem is more difficult to analyze and some deep ideas from functional analysis will be needed to overcome the lack of a simple explicit description of the solution. The second project is the study of the problem in the whole space (i.e. ,without the assumption of periodicity). The major difficulty here comes from the lack of uniformity of the mixing for low frequencies. The third project is to understand the behavior under rougher perturbations, namely perturbations which are only in Sobolev spaces. Some new nonlinear cascades should be discovered here. The fourth project is to study the case when damping does not occur, and try to identify special solutions such as cat's eyes flows. These four projects can also be formulated for the Navier-Stokes evolution and the major problem here is to understand the small viscosity limit.