The Attenuation Of The Nature Of The Mhd Equations

We consider the following nonstationary MHD equations in R³ × [0,∞) :The unknown functions u = u(x, t), B = B(x,i), p = p(x, t) are the velocity fields, the magnetic fields and the pressure respectively, u₀ = u₀(x), B₀ = B₀{x) are the initial velocity and the magnetic field respectively.In this paper, we mainly study the spatial decays, temporal and spatial decays for weak solutions of the MHD equations and the temporal and spatial decays for strong solutions. The contents of the paper include the following three parts:First, we construct a sequence of approximate solutions and derive the integral representations of the approximate solutions. To do so, we employ the solutions for the Cauchy problem of the Stokes equations to construct the sequence of approximate solutions for the Cauchy problem of the linearized MHD equations. In order to derive an integral expression for the approximate solutions, we use the fundamental solution of the Stokes equations and the singular integral expression of the Helmholtz projection operator.Second, we consider the spatial decay estimates and the temporal and spatial decay estimates for the weak solutions. Applying the Young inequalities, Holder inequalities, Sobolev inequalities,Gronwall inequalities and the properties of singular integral, we are able to get the spatial decay estimates for the weak solutions. Furthermore, we also get the temporal and spatial decay estimates for the weak solutions.