Abstract :
In this article we establish the bilinear estimates corresponding to the 1D and 2D NLS with a quadratic nonlinearity , which imply the local well-posedness of the Cauchy problem in Hs for s −1 in the 1D case and for s>−1 in the 2D case. This is a continuation of our study [N. Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity , Commun. Pure Appl. Anal. 7 (2008) 1123–1143] on the 1D NLS with nonlinearity . Previous papers by Kenig, Ponce and Vega, and Colliander, Delort, Kenig and Staffilani established local well-posedness for s>−3/4 in 1D and in 2D, respectively, and when the nonlinearity is restricted to cu2, papers by Bejenaru and Tao, and Bejenaru and De Silva improved these results to s −1 in 1D and s>−1 in 2D. The bilinear estimate for 2D also yields an improvement on the growth rate of Sobolev norms of finite energy global-in-time solutions to the 2D cubic NLS.
