Decay Rate Of Solutions Of Magnetohydrodynamics Equations In3-dimensional Half Space

In this paper, we study the L2spatial decay rate of the weak solution (u, b) ofthe incompressible Magnetohydrodynamics equations in3-dimensional half space,When the Magnetohydrodynamics equations satisfying the initial conditions u(x,0)=u0(x), b(x,0)=b0(x),(x223u0, x23b0)∈L(R+3),(x3u0, x3b0)∈L65(R+3),(u0, b0)∈L1(R+3)∩L2(R+3), divu0=divb0=0, but without slip boundary condition u(x1, x2,0, t)=0, b(x1, x2,0, t)=0, we derive that (1+t)58(x3u(t), x3b(t))L2(R+3)is uniformly boundedon t. The decay rate of weak solutions of the incompressible Navier-Stokes equations in3-dimensional half space was obtained in H.J.Choe and B.J.Jin [2], Our paper state thedecay rate of weak solutions of the incompressible Magnetohydrodynamics equationsbased on the method of [2].The contents of this paper is as follows:In the first chapter, we introduce the research significance and the research back-ground, and some research results of this paper.In the second chapter, we give the main preparatory knowledge needed in theprocess of the main result proving, including basic mark, basic lemma involved in thispaper.In the third chapter,we prove the main conclusion, which is under the assumptionof (x223u0, x23b0)∈L(R+3),(x3u0, x3b0)∈L65(R+3),(u0, b0)∈L1(R+3)∩L2(R+3), divu0=divb0=0, we establish the decay rate of the weak solution of incompressible Magne-tohydrodynamics equations:(1+t)58(x3u(t), x3b(t))L2(≤R+3)c.In the fourth chapter,we conclude this paper and explore some open problemsabout the Magnetohydrodynamics equations.