Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

Abstract

We deal with the conditional regularity of the weak solutions to the Navier–Stokes equations. We discuss a famous criterion by Vasseur in terms of divfalse(bold-italicufalse/false|bold-italicufalse|false) and extend this criterion to bounded domains with Navier and Navier‐type boundary conditions. Inspired by the equality bold-italicu·∇false|bold-italicufalse|λ=−λfalse|bold-italicufalse|λ+1divfalse(bold-italicufalse/false|bold-italicufalse|false),0.1emλ≥1, we further prove an optimal regularity criterion in terms of bold-italicu·∇false|bold-italicufalse|λ both for the whole three‐dimensional space and bounded domains with Navier's, Navier‐type, and Dirichlet boundary conditions. It specially means for λ=2 that the control of the energy flow in the critical norms LtpLxr provides the regularity of solutions. This criterion is proved by two different techniques.

Navier–Stokes equations: regularity criteria in terms of the derivatives of several fundamental quantities along the streamlines—the case of a bounded domain

“…Since also Ψ p r ∈ L 1 (T 1 , χ), it follows from (23) and by the use of Gronwall's lemma that u ∈ L ∞ (T 1 , χ; L λ+2 (Ω)) ∩ L λ+2 (T 1 , χ; L 3λ+6 (Ω)). Using now (11) and lemmas 11 and 12 u can be regularly prolonged beyond χ and it contradicts the definition of χ.…”

“…and by the use of theorem 1, theorem 3 from [11] can be extended to λ ∈ (0, 1) and r ∈ (max((3λ + 9)/(3λ + 8), 3/(λ + 2)), ∞).…”