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^3) , 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 et al. (Uspekhi Mat Nauk 58(2(350)):3–44, 2003 ) concerning suitable weak solutions blowing up in L^3(ℝ^3) . Our proof uses profile decompositions and is based on our previous work (Gallagher et al., Math. Ann. 355(4):1527–1559, 2013 ), which provided an alternative proof of the L^3(ℝ^3) 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^3(ℝ^3) setting.