01-76 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) Feb 24, 01
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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.

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