Study of Potential Singularities in the Navier-Stokes Equations

California Institute Of Technology, Pasadena CA

Investigators

Abstract

The investigator studies one of the most important and long-standing open problems in mathematics: whether smooth initial conditions in the three-dimensional (3D) incompressible Navier-Stokes equations can lead to a finite-time singularity. The Navier-Stokes equations describe the motion of fluids and are fundamental to science and engineering, with applications ranging from weather forecasting and ocean modeling to aircraft design and pipe flow. Despite their widespread use, key theoretical questions remain unanswered—most notably, the global regularity and uniqueness of solutions in three space dimensions. This question is one of the seven Clay Millennium Prize Problems. The proposed research aims to develop a new mathematical and computational approach to uncover possible mechanisms that lead to singularity formation in the 3D Navier-Stokes equations. This work is significant because understanding how such singularities form could provide new insight into turbulence, which plays a central role in many physical systems. The broader impacts include the interdisciplinary training of graduate students in mathematical analysis, modeling, and large-scale simulation. This research directly supports NSF’s mission to promote the progress of science and advance national welfare. The investigator develops a novel approach to study the potential finite-time singularity in the 3D incompressible Navier-Stokes equations with smooth initial data of finite energy. The project begins by analyzing a generalized version of the equations in a space of fractional dimension slightly above three. In this setting, the space dimension is treated as a tunable parameter that helps eliminate scaling instability and reveals the formation of an asymptotically self-similar blowup solution. The investigator then considers the vanishing viscosity limit as the space dimension approaches three. The numerical evidence suggests that the potential singularity is not of exactly self-similar, but rather of Type II with logarithmic corrections. Using a novel two-scale dynamic rescaling formulation and a derived leading-order reduced two-dimensional system, the investigator aims to identify and eliminate unstable modes in the solution. This will allow the construction of a stable singularity profile and potentially enable a computer-assisted proof of finite-time blowup in a generalized Navier-Stokes system. The successful execution of this research could provide a pathway toward resolving the Clay Millennium Problem and advance the theoretical understanding of turbulence. Graduate students involved in the project will gain valuable experience in rigorous partial differential equation analysis and high-performance numerical computation. 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.