Existence Of Solutions Of The Mhd Equations And Blasting Guidelines

In this paper we consider the following Cauchy problems for the MHD equations,where u = u(x,t) = (ul(x,t),v(x,t)),B = B(x,t) = (B1(x,t),B2(x,t)), p = p(x, t) denote the unknown velocity field of the fluid,magnetic field and pressure of the fluid, respectively; F= f(x,t) = (F(x, t), f2(x, t)) denotes the given external force, u0=u0(x)=(U (X), u0(X)), B0=B0(x)=(B0(X),B0(X)) denote the given initial data and divito=0, divfio=0. We will discuss the local existence , blow-up criteria and we will also obtain the local existence of the above problem through vanishing the viscous term . Furthermore, we will discuss the corresponding decay rates problem when the viscous term tends to zero.The first part is about the local existence. We will regularize the equations by the standard mollifier to get the approximate solutions . Then by making uniform estimates to the approximate solutions and by passing to the limit, we obtain the local existence for the above problem.Then, based on the local existence of the classical solution , we discuss the blow-up criterion. We will give a blow-up criterion which is only related to magnetic field B. But the global existence of the weak solution is still not obtained.Finally, we discuss local existence and decay rates through vanishing the viscous term. We will again obtain the local existence of the classical solution of the above problems in this part by using viscous approximations. Then we will discuss the decay rates of the viscous approximations when the viscous term tends to zero.