Analysis of Singularity Formation in Three-Dimensional Euler Equations and Search for Potential Singularities in Navier-Stokes Equations

Navier-Stokes equations have been around for more than 150-years. Physicists use them to model ocean currents, weather patterns and turbulent flows behind a commercial jet or ship. Most physicists and engineers believe that the smooth solutions of the Navier-Stokes equations will not break down without external forcing.

On the other hand, a recent study by the PI indicates that the Navier-Stokes equations could develop a catastrophic behavior if one starts with a highly symmetric but perfectly smooth flow. Such scenario corresponds to a "perfect storm" in which all things that could potentially go wrong indeed go wrong. Potentially singular behavior of the Navier-Stokes equations could post tremendous damage to our environment, affect the safety of our planes and ships, and our ability to do accurate weather forecasting.

The purpose of this project is to investigate under what conditions the Euler and the Navier-Stokes equations may develop singular behavior. The understanding of this question would enable us to avoid catastrophic behavior of the fluid flows in nature. The ultimate goal of this research is to develop effective analytical and computational tools that would enhance our ability to model and predict various complex phenomena in nature so that we can have more confidence in the safety of commercial jets and ships, and weather forecasting.

Additional impact of this project will be the involvement of graduate students. The interdisciplinary training they receive in this project will be very important for their future careers in mathematics and science.

Whether the 3D incompressible Euler equations develop finite time singularities from smooth initial data has been a longstanding open question. Built upon the results obtained from the prior NSF support, this project aims at providing a rigorous proof of the potential finite time singularity in the 3D Euler equations with smooth initial data and boundary.

A major approach of the research is to prove the existence and stability of an approximate self-similar profile with a small residual error for the 3D axisymmetric Euler equations. Numerical computations will first be conducted to construct an approximate self-similar profile with a very small residual error. A crucial step in the analysis is to obtain linear stability for the approximate self-similar profile through a dynamic rescaling formulation.

Linear stability will be established by obtaining sharp functional estimates with appropriately chosen singular weights and using space-time estimates with computer assistance. The new techniques and tools developed during this project are likely to have an impact in the neighboring areas of mathematics. Another proposed project is to look for potential singular solutions of the 3D Navier-Stokes equations using specially designed initial data with periodic boundary conditions.

The approach relies on using the dynamic rescaling formulation to solve the axisymmetric Navier-Stokes equations and to avoid the potential numerical instability induced by the frequent changes of the adaptive meshes in recent computations. The successful execution of this research would provide valuable insight to the Clay Millennium Problem on the 3D Navier-Stokes equations and become an important step towards its ultimate resolution.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.