Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R 2

2001/02/26 by Thierry Gallay, Th. Gallay, C. Eugene Wayne +1 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Complex system #Compressibility #Fluid Dynamics and Turbulent Flows #Geometry #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Navier-Stokes equation solutions #Navier–Stokes equations #Physics #Ricci-flat manifold #Spacetime #Vortex #Vorticity #math.AP

paper · pdf · doi:10.1007/s002050200200

46 pages, 3 figures

arxiv created 2001/02/26 · openalex publication_date 2002/06/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R² and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows one to extend those results in a number of ways.

Citations

• Non-Linear Stability Analysis of Higher Order Dissipative Partial Differential Equations
• StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions
• Long-time asymptotics of the Navier-Stokes and vorticity equations on R³

Cited by

• On Geometric Evolution and Microlocal Regularity of the Navier-Stokes Equations
• On the Forced Surface Quasi-Geostrophic Equation: Existence of Steady States and Sharp Relaxation Rates