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In this paper, a logarithmically improved regularity criterion for the incompressible magnetohydrodynamics equation is established in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure satisfies the logarithmical Serrin-type conditionthen the solution (u, b) remains smooth on . Compared to the Navier–Stokes result, there is a logarithmic correction involving b in the denominator. This is an extension of earlier regularity results in the Serrin’s type space with

In this note, we consider sufficient conditions for the regularity of Leray Hoph Hopf solutions of the 3D incompressible magnetohydrodynamic equations via the velocity and magnetic fields in terms of B ∞ , ∞ - 1 spaces. We prove that if ( ∇ × u , ∇ × B ) belongs to the space L 2 0 , T ; B ∞ , ∞ - 1 R 3 , then the solution ( u , B ) is regular. This extends recent results contained in Gala (2011) [3].

In this note, we examine small self-similar solutions of the 3D incompressible magnetohydrodynamic equations in homogenous function spaces and show the uniqueness of self-similar solutions when the velocity field is small in some homogeneous function spaces. This extend the recent results given by C. He and Z. Xin. “On the self-similar solutions of the magneto-hydro-dynamic equations. Acta Mathematica Scientia, serie B English Edition 2009; 29 (3): 583–598”. It is a natural way to extend the space widely and improve the previous results. Copyright © 2013 John Wiley & Sons, Ltd.

In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier–Stokes equation in R d . Here we call u a Leray weak solution if u is a weak solution of finite energy, i.e. u ∈ L ∞ ( ( 0 , T ) ; L 2 ) ∩ L 2 ( ( 0 , T ) ; H . 1 ) . It is known that if a Leray weak solution u belongs to (0.1) L 2 1 − r ( ( 0 , T ) ; L d r ) for some 0 ≤ r ≤ 1 , then u is regular (see [J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9 (1962) 187–195]). We succeed in proving the regularity of the Leray weak solution u in terms of pressure under the condition (0.2) p ∈ L 2 2 − r ( ( 0 , T ) ; X . r ( R d ) d ) , where X . r ( R d ) is the multiplier space (a definition is given in the text) for 0 ≤ r ≤ 1 . Since this space X . r is wider than L d r , the above regularity criterion (0.2) is an improvement on Zhou’s result [Y. Zhou, On regularity criteria in terms of pressure for the Navier–Stokes equations in R 3 , Proc. Amer. Math. Soc. 134 (2006) 149–156].

In this paper, we consider sufficient conditions for the regularity of Leray–Hopf solutions of the 3D incompressible magnetohydrodynamic equations via two components of the velocity and magnetic fields in terms of BMO spaces. We prove that if belongs to the space , then the solution (u,b) is regular. This extends recent results contained by Gala, Ji E and Lee J. Copyright © 2013 John Wiley & Sons, Ltd.

Consider the Navier–Stokes equations with the initial data a ∈