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.Mathematics Subject Classification. 35Q30, 76D03, 76D05.

Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations

“…Note that thanks to (9), (10) is a weaker assumption than (5) as it is implied by the latter. The affirmative answer to this conjecture was established in [2], [7] and later refined in many follow-up papers, including [1], [3], [9], [13], [16], [17], [19], [27], [28], [30]. Roughly speaking, most of the aforementioned improvements of (5) [31] it is shown that u is smooth as long as…”

“…Many generalizations and refinements of (5) have been proved, see e.g. [3], [5], [8], [12], [31], [34]. Mathematically the pressure p serves as the Lagrange multiplier of the incompressibility constraint divu = 0.…”

Logarithmic improvement of regularity criteria for the Navier–Stokes equations in terms of pressure

“…However the achievement of this goal is still out of the question. In particular, there is still no satisfactory answer to the question of well-posedness of the Cauchy problem of (1)- (3).…”

“…On could further show that such a smooth Leray-Hopf solution must be the unique solution to (1)- (3). Therefore the well-posedness problem would be settled if all Leray-Hopf solutions could be shown to be smooth.…”

“…Many generalizations and refinements of (5) have been proved, see e.g. [3], [5], [9], [25], [26], [27]. If we formally take divergence of (1) we obtain the following relation between u and p:…”