Physical Space Approach to Wave Equation Bilinear Estimates Revisit

Abstract

In the paper by Klainerman, Rodnianski and Tao [7], they give a physical space proof to a classical result of Klainerman and Machedon [3] for the bilinear space-time estimates of null forms. In this paper, we shall give an alternative and very simple physical space proof of the same bilinear estimates by applying div-curl type lemma of Zhou [14] and Wang and Zhou [12, 13]. We have only attained the limited goal of proving the bilinear estimates for the dyadic piece of the solution. Summing up the dyadic parts leads to the bilinear estimates with a Besov loss. As far as we know, the later development of wave maps [1, 2, 8,9,10,11], and the proof of bounded curvature theorem [5, 6] rely on basic ideas of Klainerman and Machedon [3] as well as Klainerman, Rodnianski and Tao [7].