ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R~2

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on R~2 as follows:{?-△u =-u~3,u(0, x) = u(x), ?(0, x) = u(x),where the initial data(u, u) ∈ H~s(R~2) × H(R~2). It is shown that the IVP is global well-posedness in H~s(R~2) × H(R~2) 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].