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

Abstract

We study the solutions of the nonstationary incompressible Navier--Stokes equations in ${\mathbb{R}}^d$, $d\ge 2$, of self-similar form $u(x,t)=\frac{1}{\sqrt{t}}U\left(\frac{x}{\sqrt{t}}\right)$, 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 $a(x)$.

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• Hexagonal structures in 2D Navier-Stokes flows
• Stability of singular solutions to the Navier-Stokes system
• Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions