Leray's Self-Similar Solutions to the Navier--Stokes Equations with Profiles in Marcinkiewicz and Morrey Spaces

Abstract

We rule out the existence of Leray's backward self-similar solutions to the Navier--Stokes equations with profiles in $L^{12/5}({R}^3)$ or in the Marcinkiewicz space $L^{q,\infty}({R}^3)$ for $q\in(12/5, 6)$. This follows from a more general result formulated in terms of Morrey spaces and the first-order Riesz potential.