Discretely self-similar solutions to the Navier-Stokes equations with Besov space data

We construct self-similar solutions to the three dimensional Navier-Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov space ${\dot{B}}_{p,\mathrm{\infty }}^{-1+3\mathrm{/}p}$ where $3<p<6$. We also construct discretely self-similar solutions for divergence free initial data in ${\dot{B}}_{p,\mathrm{\infty }}^{-1+3\mathrm{/}p}$ for $3<p<6$ that is discretely self-similar for some scaling factor $\lambda >1$. These results extend those of \cite{BT1} which dealt with initial data in $L_w^3$ since $L^3_w\subsetneq \dot B^{-1+3/p}_{p,\infty}$for$p>3$. We also provide several concrete examples of vector fields in the relevant function spaces.