Regularity Of Solutions To Three Dimensional Liquid Crystal Equations And Magnetohydrodynamics Equations

This thesis is divided into two parts.In the first part,we study the blow up criteria of the weak solutions to three-dimensional liquid crystal equations.In the second part,the global wellness of small initial values for three dimensional magnetohydrodynamics equations with mixed dissipation is discussed.In chapter 1,we firstly introduce the development of the liquid crystal equation and the incompressible magnetohydrodynamics equations,and get the conclusions of this article.In chapter 2,we prove that the weak solution to three-dimensional liquid crystal equations is regular if the low-frequency part of(u,?d)is controlled by the scaling invariant of(u0,?d0),and prove that the regularity for 3D nematic liquid crystal flows based on energy distribution at wavenumber bands.In chapter 3,we demonstrate the small initial data global well-posedness of solutions to three dimensional incompressible magnetohydrodynamics equations with mixed dissipation.