Forward discretely self-similar solutions of the Navier-Stokes equations II

Abstract

For any discretely self-similar, incompressible initial data $v_0$ which satisfies $\Vert v_0{\Vert }_{L_w^3({\mathbb{R}}^3)}\le c_0$ where $c_0$ is allowed to be large, we construct a forward discretely self-similar local Leray solution in the sense of Lemarié-Rieusset to the 3D Navier-Stokes equations in the whole space. No further assumptions are imposed on the initial data; in particular, the data is not required to be continuous or locally bounded on ${\mathbb{R}}^3\setminus \{0\}$. The same method is used to construct self-similar solutions for any $-1$-homogeneous initial data in $L_w^3$.