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In this paper we continue to develop an alternative viewpoint on recentstudies of Navier-Stokes regularity in critical spaces, a program which wasstarted in the recent work by C. Kenig and the second author (Ann Inst HPoincar\'e Anal Non Lin\'eaire 28(2):159-187, 2011). Specifically, we provethat strong solutions which remain bounded in the space $L^3(R^3)$ do notbecome singular in finite time, a known result established by Escauriaza,Seregin and Sverak (Uspekhi Mat Nauk 58(2(350)):3-44, 2003) in the context ofsuitable weak solutions. Here, we use the method of "critical elements" whichwas recently developed by Kenig and Merle to treat critical dispersiveequations. Our main tool is a "profile decomposition" for the Navier-Stokesequations in critical Besov spaces which we develop here. As a byproduct ofthis tool, assuming a singularity-producing initial datum for Navier-Stokesexists in a critical Lebesgue or Besov space, we show there is one with minimalnorm, generalizing a result of Rusin and Sverak (J Funct Anal 260(3):879-891,2011).

A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion