Long-time asymptotics of the Navier-Stokes and vorticity equations on R(3) |
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Authors: | Gallay Thierry Wayne C Eugene |
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Affiliation: | Université de Grenoble I, Institut Fourier, BP 74, 38402 Saint-Martin d'Hères, France. |
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Abstract: | We use the vorticity formulation to study the long-time behaviour 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 ||u(.,t)||(L(2)) = o(t(-5/4)) as t-->+ infinity. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space. |
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