On the very weak solution to the Navier–Stokes equations

Abstract In this paper, we establish a supercritical criterion to distinguish Leray–Hopf weak solutions from very weak solutions. Moreover, we also establish a localized version of Galdi’s theory in Galdi (2019 Proc. Am. Math. Soc. 147 5349–59) and extend the localized Ladyzhenskaya–Prodi–Serrin criterion from weak solutions to very weak solutions.