Long-time asymptotics of the Navier-Stokes and vorticity equations on R^3

Abstract

We use the vorticity formulation to study the long-time behavior of solutions to the Navier-Stokes equation on R^3. We assume that the initial vorticity is small and decays algebraically at infinity. After introducing self-similar variables, we compute the long-time asymptotics of the rescaled vorticity equation up to second order. Each term in the asymptotics is a self-similar divergence-free vector field with Gaussian decay at infinity, and the coefficients in the expansion can be determined by solving a finite system of ordinary differential equations. As a consequence of our results, we are able to characterize the set of solutions for which the velocity field converges to zero faster than t^(-5/4) in energy norm. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space.

32 pages, 2 figures

References in corpus (3)

• Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R^2
• 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 (17)

• Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R^2
• Global stability of vortex solutions of the two-dimensional Navier-Stokes equation
• On blow-up "twistors" for the Navier--Stokes equations in $R^3$: a view from reaction-diffusion theory
• Viscous evolution of point vortex equilibria: The collinear state
• On the sharp time decay rates for the 2D generalized quasi-geostrophic equation and the Boussinesq system
• On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem
• Boundary Characteristic Point Regularity for Navier-Stokes Equations: Blow-up Scaling and Petrovskii-type Criterion (a Formal Approach)
• Asymptotic profiles for the third grade fluids equations
• 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
• Navier and Stokes meet Poincar\' e and Dulac
• Space-time decay of Navier-Stokes flows invariant under rotations
• Asymptotic approximation of a modified compressible Navier-Stokes system
• Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity
• Sturmian Multiple Zeros for Stokes and Navier--Stokes Equations in $\re^3$ via Solenoidal Hermite Polynomials
• Forced Rapidly Dissipative Navier--Stokes Flows
• Time-Asymptotic Study of a Viscous Axisymmetric Fluid without Swirl