Self Similar Solutions for Navier Stokes Equations in R

Niveau: Supérieur, Licence, Bac+2
Self-Similar Solutions for Navier-Stokes Equations in R 3 M.Cannone ? CEREMADE, Universite Paris IX Dauphine, F-75775 Paris Cedex 16 F.Planchon Centre de Mathematiques, U.R.A. 169 du C.N.R.S., Ecole Polytechnique, F-91 128 Palaiseau Cedex Abstract We construct self-similar solutions for three-dimensional incom- pressible Navier-Stokes equations, providing some examples of func- tional spaces where this can be done. We apply our results to a par- ticular case of L2 initial data. Introduction We are interested in the Navier-Stokes equations for an incompressible vis- cous fluid filling the whole space. We denote the unknown velocity field by u(x, t) = (u1, u2, u3)(x, t), x ? R3 , ? · u = ∂∂x1u1 + ∂ ∂x2u2 + ∂ ∂x3u3; u · ? is the differential operator u1 ∂∂x1 +u2 ∂ ∂x2 +u3 ∂ ∂x3 . Then the Navier-Stokes equations are for t > 0 { ∂u ∂t + (u · ?) u = ?∆u??p ? · u = 0, (1) where the initial data is u(x, 0) = u0(x). For the sake of simplicity, we restrict ourselves to ? = 1, as a rescaling allows us to obtain any value.

• similar solution


• divergence free


• solutions exist


• let u0 ?


• navier stokes equations


• unique self


• self