# Study of Potential Singularities in the Navier-Stokes Equations

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · California Institute of Technology (CA) · $300,000

## 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

## Key facts

- **NSF award ID:** 2508463
- **Awardee organization:** California Institute of Technology (CA)
- **SAM.gov UEI:** U2JMKHNS5TG4
- **PI:** Thomas Y Hou
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** Artificial Intelligence (AI)
- **Estimated total:** $300,000
- **Funds obligated:** $300,000
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2027

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2508463

## Citation

> US National Science Foundation, Award 2508463, Study of Potential Singularities in the Navier-Stokes Equations. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2508463. Licensed CC0.

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