The Regularities Of The Solutions To Some Equations Related To Magnetic Fluid

Along with the development of modern science and technology, all kinds ofproblems about the partial differential equations have drawn people’s widely attention.The application of the theory of the partial differential equations appearing in variousfields such as physics, chemistry, and economics,etc has become an important aspect.In this paper, the regularities of the solutions to some equations related to magneticfluid are discussed: One is regularity of the weak solution u to the boundary valueproblem for second-order elliptic equations under some conditions assumptions. Oneis regularity of the weak solution u to the initial/boundary value problem forsecond-order parabolic equations. The other one are two regularity criteria for the3Dincompressible MHD equations.Chapter1is the introduction, recalling the background of the related work andsummarizing the main work of this paper.Chapter2deals with regularity of the weak solution u to the boundary valueproblem for second-order elliptic equations Lu=funder some conditionsassumptions. Via some techniques and methods, we show the interior regularity of theweak solution. These techniques and methods include cut off function, flatten out theboundary, finitely covered theorem, Sobolev-inequalities, interpolation-inequalities,etc.Chapter3deals with the initial/boundary value problem for second-order parabolic equations. Through Galerkin-approximation, the weak convergence of the solutions, Sobolev-inequalities, interpolation-inequalities, etc to studying the regularity ofthe weak solution.Chapter4concerns the two regularity criteria for the3D incompressible MHDequations. The one is in terms of the derivative of the velocity field in one directionwhile another one requires the boundedness of the derivative of the pressure in onedirection.