For any discretely self-similar, incompressible initial data $v_0$ which satisfies $\|v_0 \|_{ L^3_w(\mathbb R^3)}\leq c_0$ where $c_0$ is allowed to be large, we construct a forward discretely self-similar local Leray solution in the sense of Lemari\'e-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 $L^3_w$.
Categories: math.AP