Weak in Space, Log in Time Improvement of the Ladyženskaja–Prodi–Serrin Criteria

A regularity criterion for the Navier-Stokes equations in terms of the pressure gradient

“…[1,3,6,7,12,13,16,20] and references therein). Here, we are mainly interested in the results of [13], where the following improvement in time of (2) is provided.…”

“…The purpose of this article is to establish a novel regularity criterion in terms of the pressure gradient, valid either when Ω is bounded or Ω = R 3 , and in presence of an external force φ ∈ L 2 (0 T ; L 2 ). This is done in the spirit of Theorem 1.2, yielding an improved (in time) version of (3).…”

“…Let Ω ⊂ R 3 be either the whole space or a bounded domain with smooth boundary ∂Ω. For an arbitrarily fixed T > 0, we consider the dimensionless form of the Navier-Stokes equations in the space-time cylinder Ω T = Ω × (0 T )…”

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

“…Theorem 1.1 in the whole space Ω = R 3 can be found in [1], whereas Theorem 1.2 is proved in [9] for a small constant c > 0, although the value of this constant is not explicit. On the contrary, in our proof the value of c can be in principle explicitly calculated.…”

New Regularity Criteria Based on Pressure or Gradient of Velocity in Lorentz Spaces for the 3D Navier–Stokes Equations

“…Before formulating our results, we mention that, as (1.8), regularity criteria in terms of pressure Π or gradient of pressure ∇Π with only space direction belonging to Lorentz spaces can be found in [7,33]. As for other regularity criteria involving Lorentz spaces, see [5,10,13,21,22,31]. Now our first result is stated as follows.…”