On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component (original) (raw)
Related papers
Journal of Mathematical Fluid Mechanics, 2019
In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, ∂ 3 u 3. We show that if ∂ 3 u 3 satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor ∂ 1 u 3 are also established, which covers some previous literature.
On the regularity of the solutions of the Navier–Stokes equations via one velocity component
Nonlinearity, 2010
We consider the regularity criteria for the incompressible Navier-Stokes equations connected with one velocity component. Based on the method from [4] we prove that the weak solution is regular, provided u 3 ∈ L t (0, T ; L s (R 3)), 2 t + 3 s ≤ 3 4 + 1 2s , s > 10 3 or provided ∇u 3 ∈ L t (0, T ; L s (R 3)), 2 t + 3 s ≤ 19 12 + 1 2s if s ∈ (30 19 , 3] or 2 t + 3 s ≤ 3 2 + 3 4s if s ∈ (3, ∞]. As a corollary, we also improve the regularity criteria expressed by the regularity of ∂p ∂x 3 or ∂u 3 ∂x 3 .
Regularity criteria for the three-dimensional Navier-Stokes equations
Indiana University Mathematics Journal, 2008
In this paper we consider the three-dimensional Navier-Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one component of the velocity field, or alternatively in terms of one component of the pressure gradient, for the regularity of strong solutions to the three-dimensional Navier-Stokes equations.
Some New Regularity Criteria for the Navier-Stokes Equations Containing Gradient of the Velocity
Applications of Mathematics, 2000
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz 49 (2004) APPLICATIONS OF MATHEMATICS No. 5, 483-493 SOME NEW REGULARITY CRITERIA FOR THE NAVIER-STOKES EQUATIONS CONTAINING GRADIENT OF THE VELOCITY*
Regularity Criteria in Weak L3 for 3D Incompressible Navier-Stokes Equations
Funkcialaj Ekvacioj, 2015
We study the regularity of a distributional solution (u, p) of the 3D incompressible evolution Navier-Stokes equations. Let B r denote concentric balls in R 3 with radius r. We will show that if p ∈ L m (0, 1; L 1 (B 2)), m > 2, and if u is sufficiently small in L ∞ (0, 1; L 3,∞ (B 2)), without any assumption on its gradient, then u is bounded in B 1 × (1 10 , 1). It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.