In a seminal paper, Escauriaza, Seregin, and Sverak introduced a new approach to Navier-Stokes regularity based on ``zooming in" on a potential Navier-Stokes singularity and characterizing the ``blow-up limit." This is known as a Liouville approach to regularity. To execute this method, it is essential to control the sequence of solutions arising from zooming in on the singularity. For this purpose, we develop a new notion of global weak solution to the Navier-Stokes equations with initial data in critical Besov spaces. Our solutions satisfy the following stability property: weak-* convergence of the initial data implies strong convergence of the corresponding solutions. Furthermore, we consider initial data outside the applicability of standard perturbation techniques, e.g., the setting of large scale-invariant initial data, in which non-uniqueness is conjectured to hold. We provide applications to blow-up criteria and scale-invariant solutions. Finally, we discuss various difficulties concerning the space BMO^-1.

Joint work with Tobias Barker (ENS).