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 $L^{12/5}({\mathbb{R}}^3)$ or in the Marcinkiewicz space $L^{q,\infty }({\mathbb{R}}^3)$ for $q\in (12/5,6)$. This follows from a more general result formulated in terms of Morrey spaces and the first order Riesz's potential.

arXiv commentA error is fixed in this version. The structure is also slightly reorganized