Critical regularity criteria for Navier–Stokes equations in terms of one directional derivative of the velocity

Abstract

In this paper, we consider the 3D Navier–Stokes equations in the whole space. We investigate some new inequalities and a priori estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely, ∂3bold-italicu∈Lpfalse(false(0,Tfalse);Lqfalse(ℝ3false)false), 2p+3q=2, 32<q≤6. Moreover, we extend the range of q while the solution is axisymmetric, that is, the axisymmetric solution u is regular in (0, T], if ∂3u3∈Lpfalse(false(0,Tfalse);Lqfalse(ℝ3false)false), 2p+3q=2, 32<q<∞.

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

“…(a) Let u be a weak solution to the problem ( 1)- (4). Suppose that u ∈ L p (0, T; L r (Ω)) for some p, r such that p > 2, r > 3 and 2∕p + 3∕r = 1.…”

“…There are many follow‐up results in the literature (see, e.g., Chen et al, 4 Guo et al, 5 or Han et al 6 ). In our paper, we are inspired by the Constantin and Fefferman, Beirao da Veiga, and Vasseur's works.…”

“…For simplicity, we suppose the kinematic viscosity 𝜈 to be equal to 1. In the case that Ω is bounded, we add Navier boundary conditions u • n = 0, [((∇u) + (∇u) T )n] 𝜏 + 𝛾u = 𝟎 on 𝜕Ω × (0, T), (4) where 𝛾 > 0 denotes the friction coefficient between the fluid and the fixed wall and n is the outer normal vector on 𝜕Ω, or Navier-type boundary conditions u • n = 0, 𝝎 × n = 𝟎 on 𝜕Ω × (0, T), (5) where 𝝎 = ∇ × u, or Dirichlet boundary conditions u = 𝟎 on 𝜕Ω × (0, T). (6) In this paper, we study the conditional regularity of the weak solutions to the Navier-Stokes equations.…”

A survey of geometric constraints on the blowup of solutions of the Navier--Stokes equation

“…Very recently, Chen, Fang and Zhang [8] and Giang and Khai [15] extended the range of exponents to 3 ≤ q ≤ 6 using different methods, which brings the range of exponents for which there is a scale critical regularity criterion in terms of one directional derivative of velocity to 3 2 < q ≤ 6, with the endpoint case in particular remaining open.…”

Serrin-type regularity criteria for the 3D MHD equations via one velocity component and one magnetic component

“…if the weak solution u satisfies (1.3) u ∈ L p (0, T ; L q (R 3 )), 2 p + 3 q = 1, 3 q ∞, then the weak solution is regular in (0, T ]. There are several notable results [3,4,5,6,11,24] to weaken the above criteria by imposing constraints only on partial components or directional derivatives of velocity field. In particular, D. Chae and J. Wolf [3] made an important progress and obtained the regularity of solution under the condition…”