Weak Solution Of The Incompressible Mhd Equations In The Half Space L ~ 2 Decay

We consider the following incompressible MHD equations in R₊ⁿ × (0,∞)where n is the space dimension, u = u(x, t) = (u¹(x, t), ... uⁿ(x, t)), B = B(x, t) = (B¹(x,t), ... Bⁿ(x,t)) and p(x, t) denote the unknown velocity fields, magnetic fields and the scalar function of pressure respectively. u₀(x), B₀(x) denote the given initial velocity and initial magnetic fields.In this paper, we mainly study the decay rate for the weak solutions of the incompressible MHD equations in half spaces. The contents of the paper include the following three parts:1. First, we prove that if u₀(x) ∈ L_σ²∩L^r, B₀(x) ∈L_σ²∩L^r(1 ≤ r < 2), the L² decay rate of u and B is t^{-n/2(1/r-1/2)}.2. Second, based on above results, we show that the L2 decay rate can be improved to t^{-n/2(1/r-1/2)-1/2} under assumptions that u₀(x),B₀(x) ∈ L_σ²∩L^r (1 ≤ r < 2), and ∫_R+ⁿ| y_nu₀(y)|^r dy < ∞, ∫_R+ⁿ| y_nB₀(y)|^r dy < ∞.3. Finally, in view of having the decay rate for the heat equations, Stokes equations and the MHD equations, we estimate the decay rate of difference of the Stokes flow, and MHD velocity flow, and difference between the Heat equation and the MHD magnetic flow.