Weak in Space, Log in Time Improvement of the Lady{\v{z}}enskaja-Prodi-Serrin Criteria

In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak $L^p$ norms in the space variables and log improvement in the time variable.Comment: 14 pages, to appea

Logarithmically Improved Blow up Criterion for Smooths Solution to the 3D Micropolar Fluid Equations

Blow-up criteria of smooth solutions for the 3D micropolar fluid equations are investigated. Logarithmically improved blow-up criteria are established in the Morrey-Campanto space

Logarithmic improvement of regularity criteria for the Navier-Stokes equations in terms of pressure

XY is partially supported by a grant from NSERC.In this article we prove a logarithmic improvement of regularity criteria in the multiplier spaces for the Cauchy problem of the incompressible Navier-Stokes equations in terms of pressure. This improves the main result in [S. Benbernou, A note on the regularity criterion in terms of pressure for the Navier-Stokes equations, Applied Mathematics Letters 22 (2009) 1438–1443].PostprintPeer reviewe

Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations

X.Y. is partially supported by the Discovery Grant No. RES0020476 from NSERC.In this article we prove new regularity criteria of the Prodi-Serrin-Ladyzhenskaya type for the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations. Our results improve the classical Lr(0,T;Ls) regularity criteria for both velocity and pressure by factors of certain nagative powers of the scaling invariant norms ||u||L3 and ||u||H1/2.PostprintPeer reviewe

Regularity results for solutions of micropolar fluid equations in terms of the pressure

This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we prove that the weak solution is regular on $(0, T]$ provided that either the norm $\left\Vert \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R}^{3}))}$ with $\frac{2}{\alpha }+ \frac{3}{\beta } = 2$ and \frac{3}{2} < \beta < \infty or $\left\Vert \nabla \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R} ^{3}))}$ with $\frac{2}{\alpha }+\frac{3}{\beta } = 3$ and 1 < \beta < \infty is sufficiently small

From concentration to quantiative regularity:a short survey of recent developments for the Navier-Stokes equations

In this short survey paper, we focus on some new developments in the study of the regularity or potential singularity formation for solutions of the 3D Navier-Stokes equations. Some of the motivating questions are: Are certain norms accumulating/concentrating on small scales near potential blow-up times? At what speed do certain scale-invariant norms blow-up? Can one prove explicit quantitative regularity estimates? Can one break the criticality barrier, even slightly? We emphasize that these questions are closely linked together. Many recent advances for the Navier-Stokes equations are directly inspired by results and methods from the field of nonlinear dispersive equations