Spatial asymptotic expansions in the Navier-Stokes equation

We prove that the Navier-Stokes equation for a viscous incompressible fluid in ${\mathbb{R}}^d$ is locally well-posed in spaces of functions allowing spatial asymptotic expansions with log terms as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ of any a priori given order. The solution depends analytically on the initial data and time so that for any $0<\vartheta <\pi \mathrm{/}2$ it can be holomorphically extended in time to a conic sector in $\mathbb{C}$ with angle $2\vartheta$ at zero. We discuss the approximation of solutions by their asymptotic parts.