# Hydrodynamic Stability and Regularity for the Incompressible Euler and Navier Stokes Equations

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of Minnesota-Twin Cities (MN) · $304,779

## Abstract

This project is focused on the study of incompressible Navier-Stokes and Euler equations, which are the fundamental equations modeling the flow of air (at subsonic speeds) and water, among other fluids. Due to the inherent complexity of fluid dynamics, many questions about these equations are still open. For example, whether the fluid velocity at a small volume could become uncontrollably large, even when the average fluid motion is mild, remains a central open question. In nature, we often observe coherent structures in fluid flows such as vortices (eddies) generated behind an obstacle in a stream of fluid flow, and vortex rings (such as smoke rings) and vortex columns trailing airplanes. Mathematically, these coherent structures represent large (approximate) solutions to the Euler and Navier Stokes equations. A natural question is how stable these solutions are, and what are the precise dynamics of nearby solutions. The main goal of the project is to address these important questions. Theoretical understanding of the Navier Stokes and Euler equations are useful in precise computations of the solutions and in the design of efficient numerical algorithms, which are essential for many scientific and engineering applications. The project provides training opportunities for graduate students, who will learn to use tools from spectral analysis, Fourier analysis, dynamical systems, nonlinear partial differential equations, and numerical simulations in the study of physically signi

## Key facts

- **NSF award ID:** 2453270
- **Awardee organization:** University of Minnesota-Twin Cities (MN)
- **SAM.gov UEI:** KABJZBBJ4B54
- **PI:** Hao Jia
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $304,779
- **Funds obligated:** $304,779
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2453270

## Citation

> US National Science Foundation, Award 2453270, Hydrodynamic Stability and Regularity for the Incompressible Euler and Navier Stokes Equations. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2453270. Licensed CC0.

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