Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R^2

Abstract

We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R^2 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.

46 pages, 3 figures

References in corpus (3)

• Long-time asymptotics of the Navier-Stokes and vorticity equations on R^3
• Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation
• Non-Linear Stability Analysis of Higher Order Dissipative Partial Differential Equations

Cited by in corpus (64)

• Global stability of vortex solutions of the two-dimensional Navier-Stokes equation
• Universal dynamics for the defocusing logarithmic Schrodinger equation
• Long-time asymptotics of the Navier-Stokes and vorticity equations on R^3
• Factorization for non-symmetric operators and exponential H-theorem
• Interaction of vortices in viscous planar flows
• Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces
• Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices
• Pseudospectral and spectral bounds for the Oseen vortices operator
• Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations
• Linear superposition in nonlinear wave dynamics
• Three-Dimensional Stability of Burgers Vortices: the Low Reynolds Number Case
• On blow-up "twistors" for the Navier--Stokes equations in $R^3$: a view from reaction-diffusion theory
• Generalized Helmholtz-Kirchhoff model for two dimensional distributed vortex motion
• Using global invariant manifolds to understand metastability in Burgers equation with small viscosity
• Vanishing viscosity limit for axisymmetric vortex rings
• Viscous evolution of point vortex equilibria: The collinear state
• The linearized Vlasov and Vlasov-Fokker-Planck equations in a uniform magnetic field
• Three-dimensional stability of Burgers vortices
• Arnold's variational principle and its application to the stability of planar vortices
• On the asymptotic stability of steady flows with nonzero flux in two-dimensional exterior domains
• Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity
• On the sharp time decay rates for the 2D generalized quasi-geostrophic equation and the Boussinesq system
• Trapping and extreme clustering of finitely-dense inertial particles near a rotating vortex pair
• Spectral analysis of semigroups and growth-fragmentation equations
• Spectral Analysis and Long-Time Behaviour of a Fokker-Planck Equation with a Non-Local Perturbation
• Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations
• Vortices in stably-stratified rapidly rotating Boussinesq convection
• Space-time decay estimates for the incompressible viscous resistive Hall-MHD equations
• Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity
• Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data
• Fermi-Dirac-Fokker-Planck equation: well-posedness and long-time asymptotics
• Boundary Characteristic Point Regularity for Navier-Stokes Equations: Blow-up Scaling and Petrovskii-type Criterion (a Formal Approach)
• Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation
• On the forced surface quasi-geostrophic equation: Existence of steady states and sharp relaxation rates
• Asymptotic profiles for the second grade fluids equations in R^2
• Asymptotic profiles for the third grade fluids equations
• Pseudospectrum for Oseen vortices operators
• On the Krein-Rutman theorem and beyond
• Asymptotic behavior of scalar convection-diffusion equations
• Hexagonal structures in 2D Navier-Stokes flows
• Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator
• On global solutions and blow-up for Kuramoto-Sivashinsky-type models, and well-posed Burnett equations
• No-slip boundary condition for vorticity equation in 2D exterior domains
• Energy bounds for the two-dimensional Navier-Stokes equations in an infinite cylinder
• On the Decay of Infinite Energy Solutions to the Navier-Stokes Equations in the Plane
• Existence and stability of asymmetric Burgers vortices
• Invariant Measures for a Stochastic Fokker-Planck Equation
• Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models
• Asymptotic approximation of a modified compressible Navier-Stokes system
• The Marr Conjecture and Uniqueness of Wavelet Transforms
• The vorticity equations in a half plane with measures as initial data
• Long-time dynamics of classical Patlak-Keller-Segel equation
• Sturmian Multiple Zeros for Stokes and Navier--Stokes Equations in $\re^3$ via Solenoidal Hermite Polynomials
• Two-dimensional incompressible viscous flow around a small obstacle
• Large Time Behavior and Convergence for the Camassa-Holm Equations with Fractional Laplacian Viscosity
• Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent
• Time-Asymptotic Study of a Viscous Axisymmetric Fluid without Swirl
• Decay Asymptotics of the Viscous Camassa-Holm Equations in the Plane
• Asymptotics for a free-boundary model in price formation
• Long-time asymptotics of $3$-D axisymmetric Navier-Stokes equations in critical spaces
• On stability of blow-up solutions of the Burgers vortex type for the Navier-Stokes equations with a linear strain
• From molecular dynamics to coarse self-similar solutions: A simple example using equation-free computation
• Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity
• Sharp relaxation rates for plane waves of reaction-diffusion systems