Niveau: Supérieur, Doctorat, Bac+8
Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R2 Thierry Gallay Universite de Paris-Sud Mathematiques Batiment 425 F-91405 Orsay France C. Eugene Wayne Department of Mathematics and Center for BioDynamics Boston University 111 Cummington Street Boston, MA 02215, USA December 4, 2001 Abstract We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows one to extend those results in a number of ways. 1 Introduction In the last decade and a half, starting with the work of T. Kato, K. Masuda, M. Schon- bek, and M. Wiegner, the long-time behavior of solutions of the Navier-Stokes equation (and the related vorticity equation) on unbounded spatial domains has been extensively studied. (See [20], [22], [26], [19], [31], [1] [15], [27] [5], [6], [11], [28], [24] and [23] for a small sampling of this literature.) This prior work used a variety of techniques including energy estimates, the Fourier splitting method, and a detailed analysis of the semigroup of the linear part of the equation.
- finite dimensional invariant
- hardy-littlewood- sobolev inequality
- navier- stokes equation
- invariant manifold
- wiegner's result
- construct invariant
- method