Article

Journal of Statistical Physics

, Volume 121, Issue 5, pp 779-803

First online:

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

  • Yakov SinaiAffiliated withDepartment of Mathematics, Princeton UniversityLandau Institute of Theoretical Physics Email author 

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Abstract

In this paper we study the Fourier transform of the \(3 \mathcal{D}\)-Navier-Stokes System without external forcing on the whole space R 3. 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 System Fourier transform power series