Study On Properties Of The Solutions To The 3D Axisymmetric Incompressible MHD Equations

Magnetohydrodynamics(magneto fluid dynamics or hydromagnetics or MHD in short) is a physics branch which studies the interaction of plasma and magnetic field. The set of equations that describe MHD are a combination of the NavierStokes equations of fluid dynamics and Maxwell’s equations of electromagnetism.Inspired by the works of 3D incompressible axisymmetric Navier-Stokes equations by predecessor Hou Thomas Y., Lei Zhen, Li Congming, Chae Dongho and Lee Jihoon et al., we apply the energy estimate, Marcinkiewicz multiplier theorem, Fourier transform, the weighted Calderon-Zygmund estimation, truncation function method, embedding theorem and Serrin’s criterion. We also need some important inequalities, such as the H¨older’s inequality, the Calderon-Zygmund inequality, the Sobolev interpolation inequality, the Poincare inequality and the Young’s inequality, etc. We study existence, stability and regularity criterion of the solutions to the 3D incompressible axisymmetric MHD equations.In Chapter 1, we introduce the basic concept of magnetohydrodynamic model,research progress, main research contents and research conclusions of this dissertation.In Chapter 2, firstly, we derive 3D incompressible axisymmetric MHD equations. We apply the coordinate transformation to the 3D incompressible MHD equations in Cartesian coordinate. We obtain three kinds of MHD equations which are equivalent to each other. Secondly, we put forward three types of particular solutions to MHD equations and their corresponding equations.In Chapter 3, we consider the particular solution uθ= Br= Bz= 0. Firstly,we obtain regularity results for all components of velocity field and magnetic field such that the radial component of velocity field u has a higher regularity(i.e.satisfies weighted Serrin-Prodi type condition). Secondly, the question of whether the solutions to the 3D MHD equations can develop finite time singularity from a smooth initial condition with finite energy remains a open problem. We prove the global regularity of the 3D MHD equations for a family of large anisotropic data.We obtain a global bound for the solutions to the MHD equations in terms of its initial data in some Lpnorm. Our results reveal dynamic growth behavior of the solution due to the interaction between the angular magnetic field and the angular velocity field. Finally, we introduce a Banach space, establish the estimation of Calderon- Zygmund of the velocity and vorticity and apply the standard cut-off function. Ifur rsatisfy Serrin’s condition, the solutions is smooth.In Chapter 4, we consider the particular solution Br= Bz= 0. Firstly, we get the regularity of the solution to MHD equations such that the radial component and its negative part of the velocity field u satisfy a generalized weighted SerrinProdi type condition which make this two components have a higher regularity.Secondly, we prove the global regularity of the 3D MHD equations for a family of small anisotropic data. We also obtain a regularity criterion for a family of large anisotropic data. Finally, we apply a smooth cut-off function, the convolution type of weighed singular integral operator, the H¨older’s inequality, the Young’s inequality, the Gagliardo-Nirenberg inequality and other methods. If ursatisfy ordinary Serrin condition, then the solutions is smooth.In Chapter 5, we consider the particular solution Bθ= 0. Firstly, we take the curl operation to the velocity field equation and magnetic field equation and obtain a new set equations for the vorticity and current density. We introduce the standard mollifier in R3. The solutions to MHD equations are smooth by using the energy estimate, the truncation function method, the Serrin’s criterion, the Sobolev interpolation inequality, integration by parts and the angular component of vorticity and current density satisfying suitable condition.In Chapter 6, we consider the situation of the general solution. We introduce a new set of two-dimensional model. A family of three-dimensional exact solutions can be constructed by this two-dimensional model. We also obtain the global regularity from this two dimensional model.