Fine properties of self-similar solutions of the Navier-Stokes equations

TL;DR

This paper investigates self-similar solutions to the Navier-Stokes equations in multiple dimensions, providing explicit asymptotic formulas that connect the solutions' profiles to their initial data, enhancing understanding of their long-term behavior.

Contribution

It introduces an explicit asymptotic formula linking the self-similar velocity profile to the initial data for solutions derived from small, homogeneous initial conditions.

Findings

Derived explicit asymptotic relation between initial data and self-similar profile

Extended analysis to solutions in dimensions d ≥ 2

Enhanced understanding of long-term behavior of Navier-Stokes solutions

Abstract

We study the solutions of the nonstationary incompressible Navier--Stokes equations in $\R^d$, $d\ge2$, of self-similar form $u(x,t)=\frac{1}{\sqrt t}U\bigl(\frac{x}{\sqrt t}\bigr)$, obtained from small and homogeneous initial data $a(x)$. We construct an explicit asymptotic formula relating the self-similar profile $U(x)$ of the velocity field to its corresponding initial datum $a(x)$.