Lerays backward self-similar solutions to the 3D Navier-Stokes equations in Morrey spaces

In this paper, it is shown that there does not exist a non-trivial Lerays 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 $6leq q<infty$ and $2<lleq q$. This generalizes the corresponding results obtained by Nev{c}as-Rr{a}uv{z}iv{c}ka-v{S}ver{a}k [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 $pgeq3$,, 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 $12/5leq q<3$ and in $L^{q, infty}(mathbb{R}^3)$ with $12/5leq q<6$.