Forward Discretely Self-Similar Solutions of the Navier-Stokes Equations II

For any discretely self-similar, incompressible initial data which are arbitrarily large in weak , we construct a forward discretely self-similar solution to the 3D Navier-Stokes equations in the whole space. This also gives a third construction of self-similar solutions for any -homogeneous initial data in weak , improving those in JiaSverak and verak (Invent Math 196(1):233-265, 2014) and Korobkov and Tsai (Forward self-similar solutions of the Navier-Stokes equations in the half space, 2016) for Holder continuous data. Our method is based on a new, explicit a priori bound for the Leray equations.