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

Abstract

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⁎:limt→T−⁎⁡‖u(⋅,t)‖L3(Ω∩B(x⁎,r))=∞∀r>0. Additionally, u generates a non-trivial ‘mild bounded ancient solution’ in R3 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.