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

Th. Gallay; C. E. Wayne

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arXiv · math/0102198

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

Th. Gallay·C. E. Wayne

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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 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.

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BibTeX RIS

Th. Gallay, C. E. Wayne. 2001-02-26. Long-time asymptotics of the Navier-Stokes and vorticity equations on R^3. https://arxiv.org/abs/math/0102198

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