Leray's backward self-similar solutions to the 3D Navier-Stokes equations in Morrey spaces

Abstract

In this paper, it is shown that there does not exist a non-trivial Leray's backward self-similar solution to the 3D Navier-Stokes equations with profiles in Morrey spaces ${\dot{\mathcal{M}}}^{q,1}({\mathbb{R}}^3)$ provided $3/2<q<6$, or in ${\dot{\mathcal{M}}}^{q,l}({\mathbb{R}}^3)$ provided $6\le q<\infty$ and $2<l\le q$. This generalizes the corresponding results obtained by Nečas-Råužička-Šverák [19, Acta.Math. 176 (1996)] in $L^3({\mathbb{R}}^3)$, Tsai [25, Arch. Ration. Mech. Anal. 143 (1998)] in $L^p({\mathbb{R}}^3)$ with $p\ge 3$,, Chae-Wolf [3, Arch. Ration. Mech. Anal. 225 (2017)] in Lorentz spaces $L^{p,\infty }({\mathbb{R}}^3)$ with $p>3/2$, and Guevara-Phuc [11, SIAM J. Math. Anal. 12 (2018)] in ${\dot{\mathcal{M}}}^{q,\frac{12-2q}{3}}({\mathbb{R}}^3)$ with $12/5\le q<3$ and in $L^{q,\infty }({\mathbb{R}}^3)$ with $12/5\le q<6$.

24 pages, 4 figures