Blow-up of critical Besov norms at a potential Navier-Stokes singularity

We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space Ḃ −1+ 3 p p,q (R ), with 3 < p, q < ∞, gives rise to a strong solution with a singularity at a finite time T > 0, then the norm of the solution in that Besov space becomes unbounded at time T . This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza, Seregin and Šverák (Uspekhi Mat. Nauk 58(2(350)):3-44, 2003) concerning suitable weak solutions blowing up in L(R). Our proof uses profile decompositions and is based on our previous work (Math. Ann. 355(4):1527–1559, 2013) which provided an alternative proof of the L(R) result. For very large values of p, an iterative method, which may be of independent interest, enables us to use some techniques from the L(R) setting.