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 Ṁq,1 (ℝ³) provided 3/2 < q < 6, or in Ṁq,1 (ℝ³) provided 6 ≤ q < ∞ and 2 < l ≤ q. This generalizes the corresponding results obtained by Nečas, Råužička, and Šverák [ Acta. Math., 176 (1996), pp. 283--294] in L³ (ℝ³); Tsai [Arch. Ration. Mech. Anal., 143 (1998), pp. 29-51] in Lp(ℝ³) with p ≥ 3; Chae and Wolf [Arch. Ration. Mech. Anal., 225 (2017), pp. 549-572] in Lorentz spaces Lp,∞ (ℝ³) with p > 3/2; and Guevara and Phuc [SIAM J. Math. Anal., 50 (2018), pp. 541--556] in Ṁq,12-2q/3 (ℝ³) with 12/5 ≤ q < 3 and in Lq,∞ (ℝ³) with 12/5 ≤ q < 6. [ABSTRACT FROM AUTHOR]