Th. Gallay (Paris XI) and C.E. Wayne (Boston University)
Long-time asymptotics of the Navier-Stokes and
vorticity equations on $R^3$
(217K, Postscript)
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 satisfies
$\|(\cdot,t)\|_{L^2} = o(t^{-5/4})$ as $t \to +\infty$.
In particular, we show that these solutions lie on a smooth
invariant submanifold of codimension $11$ in our function space.