Interior Regularity for Solutions to the Modified Navier--Stokes Equations (original) (raw)
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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*
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.
A new regularity criterion for weak solutions to the Navier–Stokes equations
Journal de Mathématiques Pures et Appliquées, 2005
In this paper we obtain a new regularity criterion for weak solutions to the 3-D Navier-Stokes equations. We show that if any one component of the velocity field belongs to L α ([0, T); L γ (R 3)) with 2 α + 3 γ ≤ 1 2 , 6 < γ ≤ ∞, then the weak solution actually is regular and unique. Titre. Un nouveau critère de régularité pour les solutions faibles deséquations de Navier-Stokes Resumé. Dans cet article, on obtient un nouveau critère de régularité pour les solutions faibles deséquations de Navier-Stokes en dimension 3. On démontre que si une conposante quelconque du champ de vitesse appartientà L α ([0, T ]; L γ (R 3)) avec 2 α + 3 γ ≤ 1 2 , 6 < γ ≤ ∞, alors la solution faible est régulière et unique.