Global well-posedness in energy space of small amplitude solutions for klein-gordon-zakharov equation in three space dimension

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space {utt−Δu+u=−nu, (x,t)∈ℝ3×ℝ+,ntt−Δu=Δ|u|2, (x,t)∈ℝ3×ℝ+,u(x,0)=u0(x), ∂tu(x,0)=u1(x), n(x,0)=n0(x), ∂tn(x,0)=n1(x),is considered. It is shown that it is globally well-posed in energy space H1×L2×L2×H−1H1×L2×L2×H−1 if small initial data (u0(x),u1(x),n0(x),n1(x))∈(H1×L2×L2×H−1)(u0(x),u1(x),n0(x),n1(x))∈(H1×L2×L2×H−1). It answers an open problem: Is it globally well-posed in energy space H1×L2×L2×H−1H1×L2×L2×H−1 for 3D Klein-Gordon-Zakharov equation with small initial data [1,2]? The method in this article combines the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequalities). We mainly extend the spaces Fs and Ns in one dimension [3] to higher dimension.