Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation |
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Authors: | Nobu Kishimoto |
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Institution: | Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan |
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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. |
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Keywords: | Quadratic Schrö dinger equation Cauchy problem Well-posedness Iteration method Bilinear estimate Fourier restriction norm |
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