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.
