Spatial asymptotic expansions in the Navier-Stokes equation

TL;DR

This paper establishes local well-posedness of the Navier-Stokes equations for viscous incompressible fluids in spaces with prescribed spatial asymptotic expansions, including log terms, and explores solution dependence and approximation.

Contribution

It introduces a framework for analyzing Navier-Stokes solutions with prescribed asymptotic behavior at infinity, including log terms, and demonstrates their analytic dependence on initial data and time.

Findings

Solutions depend analytically on initial data and time.

Solutions can be extended holomorphically in a conic sector of the complex plane.

Approximation of solutions by their asymptotic parts is discussed.

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.