Regularity Criteria On Partial Components For The 3D Incompressible Magnetohydrodynamics Equations

In this thesis,the conditional regularity of weak solutions for the 3D MHD equations is studied.It is provided if(?)3u3 and the magnetic field b satisfy certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces,then the solutions are smooth.Moreover,we established regularity conditions of axisymmetric weak solutions to the 3D incompressible MHD equations.The main contents of this thesis are as follows:Firstly,we introduce the background and research progresses of MHD equations.Then,some definitions,lemmas and inequalities which will be used in the proof of theorems are listed.The third part is the main part of the thesis.We established regularity criteria in terms of partial derivative of only one component of velocity field and magnetic field with the aid of Troisi inequality,where the difficulty of some a prior estimates for nonlinear terms is overcome.Then the regularity conditions of single component of velocity field and one component of the vorticity are established in a larger space for axisymmetric weak solutions to the 3D MHD equations under the scaling invariant point of view.Finally,we summarize the full thesis.