Niveau: Supérieur, Doctorat, Bac+8
Long-time asymptotics of the Navier-Stokes and vorticity equations on R3 Thierry Gallay Universite de Grenoble I Institut Fourier BP 74 38402 Saint-Martin d'Heres France C. Eugene Wayne Department of Mathematics and Center for BioDynamics Boston University 111 Cummington Street Boston, MA 02215, USA Abstract We use the vorticity formulation to study the long-time behavior of solutions to the Navier-Stokes equation on R3. 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)?L2 = o(t?5/4) as t ? +∞. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space. 1 Introduction We consider the motion of an incompressible viscous fluid filling the whole space R3. If no external force is applied, the velocity u(x, t) of the fluid satisfies the Navier-Stokes equation ∂tu+ (u · ?)u = ?∆u? 1 ??p , divu = 0 , (1) where ? is the density of the fluid, ? is
- finite dimensional invariant
- vorticity ?
- has global
- navier- stokes equation
- dimensional navier-stokes
- self-similar variables
- vorticity equations