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

In 2016, Seregin and \u{S}ver\'ak, 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 $0<T<\infty$ there exists a solution to the Navier-Stokes system on $\mathbb{R}^3 \times ]0,T[$ with this initial data. We discuss how properties of these solutions imply a new regularity criteria for 3D weak Leray-Hopf solutions in terms of the norm $\|v(\cdot,t)\|_{\dot{B}^{-1+\frac{3}{p}}_{p,\infty}}$ (as well as certain additional assumptions). The main new observation of this paper, that enables these results, regards the decomposition of homogeneous Besov spaces $\dot{B}^{-1+\frac 3 p}_{p,\infty}$. This does not appear to obviously follow from the known real interpolation theory.