Study On Regularity Criteria Of The Incompressible Magnetohydrodynamics Equations

In this thesis,We study the conditional regularity of weak solu-tions for the three dimensional incompressible MHD equations.It is proved that the solutions can be smooth when the gradient of velocity component and magnetic field or the gradient of velocity componen-t and partial components of magnetic field satisfy certain space-time integrability conditions.The whole thesis is made up of four chapters:In the first chapter we introduce the research background and re-lated results on MHD equations,and put forward the contents to be studied in this thesis.In the second chapter we present some preliminaries,mainly intro-duce some mathematical symbols,inequalities,definitions and lemmas which will be used in this thesis.In the third chapter,the main research results are shown,that is,some sufficient conditions are given to ensure the regularity of weak solutions.The conditions here mainly involve:(1)(?)u3,b(2)(?)hu3,(?)3u3,bh,b3(3)(?)u3,bh(4)(?)3u3,(?)u3,bFor the regularity condition in(2),only(?)3u3,bh or(?)3u3,b3 are optimal.For(3),the regularity condition of bh is optimal.The regularity condi-tions in(4)are all optimal.The fourth chapter summarizes the full thesis.