Abstract
We study the solutions of the nonstationary incompressible Navier–Stokes equations in \({\mathbb{R}^d}, d\geqq2\), 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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Brandolese, L. Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations. Arch Rational Mech Anal 192, 375–401 (2009). https://doi.org/10.1007/s00205-008-0149-x
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DOI: https://doi.org/10.1007/s00205-008-0149-x