Analytical Study Of Certain Magnetohydrodynamic-α Models With A Slip Boundary Condition

In this thesis, we study the Magnetohydrodynamic-αmodels in a bounded smoothdomain of R3,with slip boundary condition by Galerkin method and Hodge Decomposi-tion theory. We obtain the global weak solution of H1 solution for arbitrary initial dataand the regularity of the weak solution. Finally we discuss that as the length scaleαtendsto zero, a subsequence of solutions of the MHD-αequations converges to a certain so-lution of the three-dimensional MHD equations. In the light of contents, this thesis isdivided into four chapters.The first chapter is to introduce the main problem that we are concerned and thedevelopment of the problem in the in domestic and foreign.In the second chapter, We introduce notations, some important theorems and severalclassical results that will be used in the following proofs.In the third chapter, We consider the MHD-αequations. In§3.1 We show thenonlinearity in MHD-αequations to match with the boundary condition smoothly;In§3.2 we prove some priori estimates; In§3.3 we discuss the existence of H1 solution forarbitrary initial data; In§3.4 we discuss the regularity of the above weak solution; In§3.5 we discuss that as the length scaleαtends to zero, a subsequence of solutions ofthe MHD-αequations converges to a certain solution of the three-dimensional MHDequations.The fourth chapter summaries the work done by us and points out the continue workwe may study.