Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ℝ2

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on ℝ2 as follows: {∂ttu-Δu=-u3,u(0,x)=u0(x),∂tu(0,x)=u1(x),where the initial data (u0, u1) ɛ Hs(ℝ2) × Hs−1(ℝ2). It is shown that the IVP is global well-posedness in Hs(ℝ2) × Hs−1(ℝ2) for any 1 > s > 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].