A Weak-L p Prodi–Serrin Type Regularity Criterion for the Navier–Stokes Equations

Abstract

We give simple proofs that a weak solution u of the Navier–Stokes equations with H 1 initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L s (0, T;L r,∞(Ω)) or if its L s,∞(0, T;L r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1.