The Study Of Global Well-posedness Of Inho-Mogeneous Incompressible Navier-stokes And MHD Equations

In this paper, we mainly study two kinds of important equations of fluid mechanics: in-homogeneous incompressible Navier-Stokes equations and inhomogeneous incompressible mag-netohydrodynamics (MHD) equations. Navier-Stokes equations is a class of motion equations which describe the viscous incompressible fluid's conservation of momentum. And magnetohy-drodynamics (MHD) equations mainly describe the influence between the magnetic fields and conductive fluids (plasma,the liquid metal,etc.).The main structure and contents of this paper are as follows:In chapter 1, we mainly introduce the background and the research situation of Navier-Stokes equations and MHD system, and describe briefly the main works of this article.In chapter 2, we list some preliminary knowledge which will be needed in the later chapters,including some important inequalities and Littlewood-Paley decomposition theory, etc.In chapter 3, we study the global well-posedness of the three dimensional inhomogeneous incompressible Navier-Stokes equations. Specially to say, by using the weighted Chemin-Lerner type Besov spaces norm, the algebraical structure of equations and Gagliardo-Nirenberg inequal-ity, with the polynomial form initial smallness condition, we obtain the global well-posedness result of Navier-Stokes equations by energy estimates on the horizontal components and the ver-tical component of the velocity field respectively. And this substantially improves the result of Paicu and Zhang (J. Funct. Anal. 2012).In chapter 4, we give the same result of three dimensional inhomogeneous incompressible MHD equations as that of Navier-Stokes equations in the chapter 3, and we can shows that any directions of the initial velocity field and magnetic field allow to be large.In chapter 5, without the smallness condition of initial density, we prove the local well-posedness and global well-posedness results of three dimensional inhomogeneous incompressible MHD equations with highly oscillatory initial velocity and magnetic field.When considering the case of background magnetic dependent on the time, in chapter 6, we provide the global stability of the two dimensional MHD equations near the special solution with the linearly growing velocity. And this is also the first attempt to consider the non-equilibrium magnetic background.