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

在许多科学和技术领域，包括工程、地球物理和生物物理，了解流体流动的行为是非常重要的。在基本水平上，流体流动的动力学行为可用欧拉方程或Navier-Stokes方程描述。这些复杂的非线性偏微分方程组很难用经典的方法来研究，到目前为止，人们只知道几个精确的解。在许多应用中，不需要精确解，本项目的一个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。事实上，在某些极限情况下，由于存在一个小参数或考虑很长时间，这些复杂的方程组可能会简化为更简单的模型。这些更简单的模型能够为这些解决方案的行为提供重要的定性描述，而不必显式计算它们。这种一般方法在研究复杂系统的长时间行为、奇点的发展、特殊模式的形成、稳定性和不稳定性之间的转变以及向湍流转变的第一步等方面都有应用。该项目将涉及研究生和博士后的培训。本课题的主要研究内容是研究二维欧拉方程和二维N-S方程某些剪切流的渐近稳定性。在Gevrey正则性的周期背景下，Couette流的无粘阻尼流的研究最近取得了一些重要进展。这个项目的一个主要目标是将这项研究扩展到更一般的剪切流的情况。一个新的困难来自这样一个事实，即线性化问题的分析更加困难，需要泛函分析的一些深层思想来克服缺乏对解的简单明确描述的问题。第二个项目是对整个空间问题的研究(即，不考虑周期性的假设)。这里的主要困难来自于低频混频的不均匀。第三个项目是了解在更粗糙的扰动下的行为，也就是只在Sobolev空间中的扰动。在这里应该会发现一些新的非线性级联。第四个项目是研究没有发生阻尼的情况，并试图确定特殊的解决方案，如猫眼流动。这四个方案也可以用来描述Navier-Stokes演化，这里的主要问题是理解小的粘性极限。

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.