Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

We generalize two results in the Navier-Stokes regularity theory whose proofs rely on 'zooming in' on a presumed singularity to the local setting near a curved portion Γ ⊂ ∂Ω of the boundary.Suppose that u is a boundary suitable weak solution with singularity z * = (x * , T * ), where x * ∈ Ω ∪ Γ.Then, under weak background assumptions, the L3 norm of u tends to infinity in every ball centered at x * :Additionally, u generates a non-trivial 'mild bounded ancient solution' in R 3 or R 3 + through a rescaling procedure that 'zooms in' on the singularity.Our proofs rely on a truncation procedure for boundary suitable weak solutions.The former result is based on energy estimates for L3 initial data and a Liouville theorem.For the latter result, we apply perturbation theory for L∞ initial data based on linear estimates due to K. Abe and Y. Giga.