Leray's self-similar solutions to the Navier--Stokes equations with profiles in Marcinkiewicz and Morrey spaces

TL;DR

This paper proves the non-existence of certain self-similar solutions to the Navier--Stokes equations with profiles in specific Lebesgue and Morrey spaces, advancing understanding of solution regularity and space constraints.

Contribution

It establishes new non-existence results for Leray's self-similar solutions with profiles in Marcinkiewicz and Morrey spaces, extending previous regularity criteria.

Findings

No Leray's backward self-similar solutions in L^{12/5}( ^3) or Marcinkiewicz spaces for q in (12/5,6)

Results derived using Morrey spaces and Riesz potential analysis

Advances understanding of solution regularity constraints in Navier--Stokes equations

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's potential.