Journal of Statistical Physics

, Volume 121, Issue 5, pp 779–803

Power Series for Solutions of the 3 \({\mathcal D}\)-Navier-Stokes System on R3

Article

DOI: 10.1007/s10955-005-8670-x

Cite this article as:
Sinai, Y. J Stat Phys (2005) 121: 779. doi:10.1007/s10955-005-8670-x

Abstract

In this paper we study the Fourier transform of the \(3 \mathcal{D}\)-Navier-Stokes System without external forcing on the whole space R3. The properties of solutions depend very much on the space in which the system is considered. In this paper we deal with the space \(\Phi (\alpha , \alpha )\) of functions \(v(k ) \, = \, \frac{c(k)}{|k|^\alpha}\) where \(\alpha = 2 + \epsilon , \, \epsilon > 0\) and c (k) is bounded, \(\sup_{k \in R^3 \, \smallsetminus \, 0} \; | c ( k ) | < \infty\). We construct the power series which converges for small t and gives solutions of the system for bounded intervals of time. These solutions can be estimated at infinity (in k-space) by \(\exp \{ - {\rm const} \, \sqrt{t} | k |\}\).

Keywords

Navier-Stokes SystemFourier transformpower series

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Landau Institute of Theoretical PhysicsMoscowRussia