2001/02/26 by Th. Gallay, C. Eugene Wayne, Gallay, Th. +1
Mathematics · Engineering · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.math/0102198
We use the vorticity formulation to study the long-time behavior of solutions to the Navier-Stokes equation on R3. 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.