| Fluid dynamic equations are one of the main research fields of partial differential equations and it is the most advanced and crucial content of the whole field of PDE.In particular,fluid dynamic equations which are dominated by the Navier-Stokes equations are one of the main parts of modern PDE research.Meanwhile,models related to Navier-Stokes equations are also the focus of research.The object of this paper is devoted to the study of the well-posedness of the density-dependent incompressible micropolar fluid equation and the density-dependent incompressible magnetohydrodynamics equation.In chapter 1,we mainly introduce the research background,significance,the research status at home and abroad,and existing research problems on the density-dependent incompressible micropolar fluid equation and density-dependent incompressible magnetohydrodynamics equation.In chapter 2,the preliminary knowledge.This paper mainly introduces some basic knowledge,the working spaces,and some commonly used inequalities.In chapter 3,we investigate the solvability of the density-dependent incompressible micropolar fluid equation with only bounded nonnegative initial density andH01（（?））initial velocities.In the 2-D case,using some logarithmic-type inequalities,we construct the global existence of the solution with large initial data.For the 3-D situation,we first construct the local existence of the solution with initial large data,and then imposed some small conditions on the initial date,combined with the continuity method and the compactness method,and construct the global existence of the solution of this model.In order to obtain the uniqueness of the solution,we first developed the Lipschitz estimation of the velocity field by using the time-weighted estimation.Based on this,the uniqueness of the solution is proved by the Lagrangian method.In particular,the initial vacuum is allowed.In chapter 4,we study the solvability of the 3-D density-dependent incompressible magnetohydrodynamics equation with rough initial density,initial vacuum,initial velocity and initial magnetic field inH01（W）frame.Using the energy method,we construct the local existence of the solution with large initial data of this model.For some small initial data conditions,combined with the continuity method and the compactness method,we construct the global existence of the solution.Finally,the uniqueness of the solution is proved by the Lagrangian method.In chapter 5,the main results of the full paper are summarized and we present the relevant questions and future research in the direction. |
