The Navier-Stokes equations are a popular model for fluids, used in applications from weather forecasting to modelling airflow over an aerofoil. However, much is left to be desired from the mathematical standpoint, such as the global-in-time existence and regularity of solutions. This is the content of one of the millennium problems.

We consider which initial data classes generate global-in-time weak solutions – in particular, we construct “Calderón solutions” for supercritical Besov-space initial data, and consider an application of the splitting method to quantify the potential singular set under a Type-I blow-up ansatz.