Navier-Stokes equations are used to model ocean currents, weather patterns, and turbulent flows behind a plane or ship. Mathematicians and physicists believe that an explanation for and the prediction of both breezes and turbulence can be found through an understanding of solutions to Navier-Stokes equations. Though most physicists and engineers believe that the smooth solutions of the Navier-Stokes equations cannot break down without external forcing, currently there is no theoretical guarantee that this is indeed the case. The investigator's recent study with collaborators indicates that the Euler equations, which correspond to the inviscid limit of the Navier-Stokes equations, could develop a catastrophic behavior if one starts with a highly symmetric but perfectly smooth flow. Such a scenario is like a perfect storm in which all things that could potentially go wrong indeed go wrong. A potentially singular behavior of the fluid flows described by the Navier-Stokes equations could negate the ability to forecast the behavior of fluid systems accurately. The investigator studies conditions under which the Euler or Navier-Stokes equations may develop a potentially singular behavior. The ultimate goal of the project is to develop effective analytical and computational tools that enhance our ability to model and predict various complex fluid flows, such as those arising in engineering, oceanography, and weather forecasting. Graduate students and postdoctoral scholars are included in the work of the project. The interdisciplinary training they receive is important for their future careers in mathematics and science.
The project seeks to understand whether the incompressible 3D Euler and Navier-Stokes equations could develop a finite-time singularity from a smooth initial condition with finite energy. A major approach of the project is to study the spatial profiles in potential self-similar singularities of the solutions, which can be obtained by solving a nonlinear eigenvalue problem. A notable aspect of the project is the combination of highly resolved numerical simulations and rigorous mathematical analysis. Numerical computations are first conducted to detect potential finite-time singularity scenarios and gain primary understanding about the singularity formation. Then the evolution equations of spatial profiles in the solutions are studied using a dynamic rescaling formulation both analytically and numerically. The theoretical framework developed in this project introduces an appropriate notion of stability for the self-similar profiles through the dynamic rescaling formulation. Stability of the numerically constructed self-similar profile is a crucial step in constructing a finite-time singularity of the Euler equations. Another interesting aspect of the project is that the dynamic rescaling formulation provides a natural framework to investigate whether the finite-time singularity in the Euler equations may lead to a potential finite-time singularity of the Navier-Stokes equations for certain type of singularities. The stability of self-similar profiles of the Euler equations again plays a crucial role in this study.
