Leray's 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 nontrivial Leray backward self-similar solution to the three-dimensional (3D) Navier-Stokes equations with profiles in Morrey spaces (M) over dot(q,1)(R-3) provided 3/2 < q < 6, or in (M) over dot(q,l)(R-3) provided 6 <= q < infinity and 2 < l <= q. This generalizes the corresponding results obtained by Necas, Rauzicka, and Sverak [Acta. Math., 176 (1996), pp. 283-294] in L-3(R-3); Tsai [Arch. Ration. Mech. Anal., 143 (1998), pp. 29-51] in L-p(R-3) with p >= 3; Chae and Wolf [Arch. Ration. Mech. Anal., 225 (2017), pp. 549-572] in Lorentz spaces L-p,L-infinity(R-3) with p > 3/2; and Guevara and Phuc [SIAM J. Math. Anal., 50 (2018), pp. 541-556] in (M) over dot(q,12-2q/3) (R-3) with 12/5 <= q < 3 and in L-q,L-infinity(R-3) with 12/5 <= q < 6.