Spatial Asymptotic Expansions in the Navier–Stokes Equation

Abstract

Abstract 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 $|x|\to \infty $ of any a priori given order. The solution depends analytically on the initial data and time so that for any $0<\vartheta <\pi /2$ 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.