In this note, a logarithmic improved regularity criteria for the micropolar fluid equations are established in terms of the velocity field or the pressure in the homogeneous Besov space.
In the study of the regularity criteria for Leray weak solutions to threedimensional Navier-Stokes equations, two sufficient conditions such that the horizontal velocity u satisfies u∈L2（0,T;BMO（R3）） or u∈L^2/1＋r（0,T;B∞,∞（R3）） for 0 〈 r 〈 1 are considered.
Regularity criteria of Leray-Hopf weak solutions to the three-dimensional Navier-Stokes equations in some critical spaces such as Lorentz space, Morrey space and multiplier space are derived in terms of two partial derivatives, θ1u1, θ2u2, of velocity fields.
In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian（Φp（u000））0（t）＋a（t）f（t, u（t？τ）, u0（t））=0, 0〈t〈1;u000 （0）=u00 （0）=0, u0 （1）=αu0 （η）;u（t）=0, ？τ ≤t≤0. By using Schauder fixed-point theorem, some su？cient conditions are obtained which guar-antee the fourth-order delay differential equation of boundary value problem with p-Laplacian has at least one positive solution. Some corresponding examples are presented to illustrate the application of our main results.
By using cone expansion-compression theorem in this paper, we study boundary value problems for a coupled system of nonlinear third-order differential equation. Some sufficient conditions are obtained which guarantee the boundary value problems for a coupled system of nonlinear third-order differential equation has at least one positive solution. Some examples are given to verify our results.
