Research On The Existence Of Self-similar Leray Solutions Of MHD Equations

Magnetohydrodynamics?MHD?is a branch of physics that studies the interaction between plasma?ideal conductive fluid?and magnetic field.MHD equations follow the conservation of mass,energy and momentum,and are obtained by considering the relationship among fluid velocity,magnetic field,pressure,temperature,etc.MHD equations describe the interaction between viscous or inviscid,compressible or incompressible conductive fluids and magnetic fields.It has a wide range of applications in geophysics,astronomy,engineering and other fields.This paper deals with the existence of self-similar Leray weak solutions in C^???³?for the initial value problems of three-dimensional incompressible MHD equations.Firstly,using the isomorphism of L^3,?space and C^?space,the?-regularity criterion of the suitable weak solution?u,b,p?for the initial value problem of MHD equations is deduced.In proving the?-regularity criterion of?u,b,p?,using the semigroup method,it is proved that there are suitable weak solutions to the initial value problem of the generalized MHD equations.Then,decomposing the initial value appropriately:u₀ u=₀₁+u₀₂,b₀ b=₀₁+b₀₂.By?-regularity criterion of?u,b,p?,it is proved that if?u₁,b₁?is the local mild solution of the MHD equations with the initial value(u₀₁,b₀₁),then the solution?u-u₁,b-b₁?to the initial value problem of the MHD equations has?-H o?5??5?lder continuity.Finally,by using the local regularity of the Leray weak solution of the MHD equations the initial value is scaled.By using the Leray-Schauder fixed point theorem,it is proved that there exist self-similar solutions of the generalized elliptic equations.By the isomorphic relationship between C^?space and X space,the existence of the self-similar solution to the MHD equations in C^???³?is achieved.