On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \mathbb{R}^{N}

In this paper we establish a Serrin’s type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equations in It is proved that if the gradient of pressure belongs to L^{α, γ} with then the weak solution actually is regular and unique. Received: May 4, 2004     

A regularity criterion for the solutions of 3D Navier-Stokes equations

In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier-Stokes equation in R³. More precisely, let u=(u¹,u²,u³) be a weak solution in (0,T)×R³. Then u becomes a classical solution if any two functions of _∂1u¹, _∂2u² and _∂3u³ belong to Lθ(0,T;Lr(R³)) provided with , .     

On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure

We study the Cauchy problem for the n-dimensional Navier-Stokes equations (n?3), and prove some regularity criteria involving the integrability of the pressure or the pressure gradient for weak solutions in the Morrey, Besov and multiplier spaces.     

A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure

In this paper, we study the regularity criterion of weak solutions of the three dimensional micropolar fluid flows. It is proved that if the pressure satisfies where denotes the critical Besov space, then the weak solution (u,w) becomes a regular solution on (0,T]. This regularity criterion can be regarded as log in time improvements of the standard Serrin's criteria established before. Copyright © 2011 John Wiley & Sons, Ltd.     

Limiting case for the regularity criterion of the Navier-Stokes equations and the magnetohydrodynamic equations

Any weak solution u to the Navier-Stokes equations is showed to be regular under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small, which is a limiting case of the regularity criteria derived by Kim and Kozono. Our result gives a positive answer to the question proposed by Kim and Kozono. For the incompressible magnetohydrodynamic equations, we also show the regularity of weak solution only under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small.     

A new scaling invariant regularity criterion for the 3D MHD equations in terms of horizontal gradient of horizontal components

We prove a new scaling invariant regularity criterion for the 3D MHD equations via horizontal gradient of horizontal components of weak solutions. This result improves a recent work by Ni et al. (2012), in the sense that the assumption on the horizontal gradient of the vertical components is removed. As a byproduct, a scaling invariant regularity criterion involving vertical components of vorticity and current density is also obtained.     

Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component

We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of u₃ and ω₃, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M.Pokorný (2004).     

A regularity class for the Navier-Stokes equations in Lorentz spaces

We extend Serrin's regularity class for weak solutions of the Navier-Stokes equations to a larger class replacing the Lebesgue spaces by Lorentz spaces. Received November 30, 2000; accepted January 16, 2001.     

Partial regularity for the stochastic Navier-Stokes equations

The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary solutions it is proved that at any time the set of singular points is empty. The same result holds true for every martingale solution starting from -a.e. initial condition , where is the law at time zero of a stationary solution. Finally, the previous result is non-trivial when the noise is sufficiently non-degenerate, since for any stationary solution, the measure is supported on the whole space of initial conditions.

     

A note on the regularity criteria for the Navier-Stokes equations

We consider the regularity problem for 3D Navier-Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient ∇p/(1+|u|^δ₁+|∇u|^δ₂) for the Navier-Stokes equations is established.     

A note on regularity criteria in terms of pressure for the 3D viscous MHD equations

The notions of almost ω-categoricity and 1-local ω-categoricity are studied. In particular, necessary and sufficient conditions for their equivalence under additional assumptions are found. It is proved that 1-local ω-categorical theories on dense linear orders are Ehrenfeucht and that Ehrenfeucht quite o-minimal binary theories are almost ω-categorical.     

Sufficient condition of local regularity for the Navier-Stokes equations

We prove a local regularity result for the non-stationary three-dimensional Navier-Stokes equations. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2005, pp. 46–54.     

On regularity criteria in terms of pressure for the 3D viscous MHD equations

In this article, we consider regularity of solutions to the 3D viscous MHD equations. Regularity criteria are established in terms of the pressure or the gradient of pressure, which improve the results in Y. Zhou [Regularity criteria for the 3D MHD equations in terms of the pressure, Int. J. Non-Linear Mech. 41(10) (2006), pp. 1174–1180] where additional conditions on the magnetic field are also needed.