Existence and Weak* Stability for the Navier-Stokes System with Initial Values in Critical Besov Spaces

In 2016, Seregin and uSverak, conceived a notion of global in time solution (as well as proving existence of them) to the three dimensional Navier-Stokes equation with $L_3$ solenoidal initial data called 'global $L_3$ solutions'. A key feature of global $L_3$ solutions is continuity with respect to weak convergence of a sequence of solenoidal $L_3$ initial data. The first aim of this paper is to show that a similar notion of ' global $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ solutions' exists for solenoidal initial data in the wider critical space $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ and satisfies certain continuity properties with respect to weak* convergence of a sequence of solenoidal $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ initial data. This is the widest such critical space if one requires the solution to the Navier-Stokes equations minus the caloric extension of the initial data to be in the global energy class. For the case of initial values in the wider class of $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$ initial data ($p>4)$, we prove that for any $0Besov spaces $\dot{B}^{-1+\frac 3 p}_{p,\infty}$. This does not appear to obviously follow from the known real interpolation theory.