A regularity criterion for the Navier–Stokes equations via one diagonal entry of the velocity gradient (original) (raw)
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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 .
Journal of Mathematical Physics, 2009
We improve the regularity criterion for the incompressible Navier-Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi ͓see "Regularity criteria for the three dimensional Navier-Stokes equations," Indiana Univ. Math. J. 57, 2643 ͑2008͔͒ and Kukavica and Ziane ͓see "Navier-Stokes equations with regularity in one direction," J. Math. Phys. 48, 065203 ͑2007͔͒. In particular, for s ͓2,3͔, we get that the solution is regular if ٌu 3 L t ͑0,T ; L s ͑R 3 ͒͒, 2/ t +3/ s Յ 23 12 .
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*
Energy-Based Regularity Criteria for the Navier–Stokes Equations
Journal of Mathematical Fluid Mechanics, 2009
We present several new regularity criteria for weak solutions u of the instationary Navier-Stokes system which additionally satisfy the strong energy inequality. (i) If the kinetic energy 1 2 u(t) 2 2 is Hölder continuous as a function of time t with Hölder exponent α ∈ ( 1 2 , 1), then u is regular. (ii) If for some α ∈ ( 1 2 , 1) the dissipation energy satisfies the left-side condition lim inf δ→0 1 δ α t t−δ ∇u 2 2 dτ < ∞ for all t of the given time interval, then u is regular. The proofs use local regularity results which are based on the theory of very weak solutions, see [1], , and on uniqueness arguments for weak solutions. Finally, in the last section we mention a local space-time regularity condition. . Primary 35Q30, 76D05, 35B65.
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.