Nonexistence Of Self-Similar Singularities To A Class Of Magnetohydrodynamic Equations

Magnetohydrodynamics has very close relations to industries,such as electronic,mechanical, metallurgical, chemical, medical, astrophysics, controlled thermonuclearetc. With the development of science and technology, magnetohydrodynamics hasirreplaceable superiority when it is applied in these areas. Hence the research onmagnetohydrodynamics is very important and significant. Here we are concerned onthe possibility of finite time singularity in a partially viscous magnetohydrodynamicequations inR3. Under certain theory conditions, we obtain the nonexistence of finitetime singularity in the form of self similarity.In chapter1, some background knowledge and the research present situationsthat are relevant to this survey paper are introduced; Meanwhile the main problemsand conclusions in the paper also are introduced.In chapter2, first of all, by using the modern partial differential equations theoryalong with some relevant skills and methods, the3D incompressible MHD equationsis simplified to the3D incompressible Navier Stokes equations under certainassumptions. Thereupon, we can imply that self similar solutions to the equation areequal to zero almost everywhere inR3. Second, when its magnetic Reynolds numberis negative,3D viscous incompressible MHD equations is a pure mathematicalproblem. Under some suitable conditions on the magnetic field and velocity field, bythe modern partial differential equation theory, the same conclusions could beobtained on the self similar solutions.In chapter3, mainly studying some possible asymptotically self similar singularityto the MHD equation and its existence. And nonexistence of the asymptoticallyself similar singularity is certified under some given hypotheses.