Long-time asymptotics of the Navier-Stokes and vorticity equations on R(3).
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.
Gallay
Thierry
T
Université de Grenoble I, Institut Fourier, BP 74, 38402 Saint-Martin d'Hères, France.
Wayne
C Eugene
CE
D008962:Models, Theoretical; D009038:Motion; D017711:Nonlinear Dynamics; D012212:Rheology / Q000379:methods*; D013997:Time Factors; D014783:Viscosity; D014872:Water Movements