Global Well-posedness For Incompressible Magnetohydrodynamic Equations In Critical Besov Spaces

The fluid dynamic equations as the basic macro model to describe fluid flow, is an important nonlinear partial differential equations to understand the natural phenomena. They are in the core of the research areas of mathematics and physics. For instance, the MHD system describes the macroscopic behavior of electrically conducting fluids in a magnetic field. They have important applications in astrophysics, geophysics, aero-dynamics, or the universe of plasma physics. In this dissertation, we mainly study the wellposedness of the incompressible MHD equations in the critical Besov spaces. The article is divided into five parts. In the first part, we establish the global well-posedness and asymptotic behavior of incompressible MHD equations in critical Besov spaces. In the second part, we continue to study the homogeneous incompressible MHD equations and obtain the global existence of solutions with initial data under a nonlinear smallness hypothesis. In the third part, we are concerned with the inviscid limit for the MHD equations in the Lp-type space. In the fourth part, we study the Cauchy problem for the MHD system with variable density, variable viscosity and variable conductivity. We ob-tain the global existence of a unique solution for this system without any small conditions imposed on the third component of the initial velocity field and magnetic field when the initial density near to some positive constant in the critical Besov spaces. In the fifth part, we mainly establish the local existence and uniqueness and global existence with small initial data under the initial density away from vacuum but not a near equilibrium state.The detailed results in this dissertation are as following. In Chapters Two and Three, we study the Cauchy problem of the following MHD equations. In Chapter Two, we proved that there exists a global solution to above equations if the initial data satisfy We use the fact that the equations on the vertical component are linear equations with coefficients depending on the horizontal components. Therefore, the equation on the vertical component does not demand any smallness condition. We first get the pressure estimates by using weighted Chemin-Lerner spaces, and get the energy estimates on the horizontal components and vertical component respectively. Then, we choose the proper weighted function to close our energy estimates which implies the result of our main theorem. Finally, we analyze the long behavior of the solutions and get some decay estimates which imply that the solutions are infinite smooth.In Chapter Three, we are concerned with the wellposedness of the n-dimensional in-compressible MHD equations in the critical Besov spaces. We obtain the global existence of solutions with initial data under some nonlinear smallness hypothesis. We also exhibit an initial data satisfying that nonlinear smallness assumption, despite each component of the initial data could be arbitrarily large.In Chapter Four, we continue to study the incompressible MHD equations. We prove that as the viscosity and resistivity go to zero, the solutions of the Cauchy problem for the homogeneous incompressible MHD system converges to the solution of the ideal MHD system in the Lp type space. The convergence rate is also obtained simultaneously.In Chapter Five, we study the following MHD equations with variable density, vari-able viscosity and variable conductivity By the classical Gagliardo-Nirenberg inequality, anisotropic Besov space and Bony’s para-product decomposition, we obtain the global existence of a unique solution for this prob-lem without any small conditions imposed on the third component of the initial velocity field and magnetic field when the initial density near to some positive constant in the critical Besov spaces.In Chapter Six, we are concerned with the nonhomogeneous incompressible MHD equations of the following form:We get the local existence and uniqueness and obtain the global wellposedness with small-ness initial data when the initial density away from vacuum but without assumptions of small density variation. We begin by solving an approximate problem and prove that the solutions are uniformly bounded. Then we study the convergence to the solution of the original equations by a compactness argument. At the same time, we also give a blowup criteria which similar to BKM’s for the strong solution. In the process of extending the local solution to be a global one, we take out a homogeneous incompressible MHD equa-tions to undertake the initial data of the velocity and magnetic field. Moreover, we get higher regularity for the solutions. For the reminder of nonhomogeneous incompressible MHD equations, we get first the L2 energy estimates and then H1 estimates and H2 esti-mates. Then, we use the momentum equation again to get the estimate ‖▽u‖L1((0,∞),L∞) which combining the blowup criteria, implies the global wellposedness.