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 $6\leq q<\infty$ and $2<l\leq q$. This generalizes the corresponding results obtained by Ne\v{c}as-R\r{a}u\v{z}i\v{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 $p\geq3$,, 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 $12/5\leq q<3$ and in $L^{q, \infty}(\mathbb{R}^3)$ with $12/5\leq q<6$.