Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation
Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation
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DOI：
10.1016/j.jde.2009.06.009
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发表时间：
2009-09
影响因子：
2.4
通讯作者：
Nobu Kishimoto
中科院分区：
文献类型：
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作者：
Nobu Kishimoto
本文建立了具有二次非线性项c1u2+c2u‘2的一维和二维非线性最小二乘问题的双线性估计，证明了一维情形下S⩾−1和二维情形下S&−1在Hs中柯西问题的局部适定性。这是我们研究的继续[N.Kishimoto，二次非线性薛定谔方程柯西问题的局部适定性。纯苹果。肛门。Kenig，Ponce和Vega，以及Colliander，Delort，Kenig和Staffilani以前的论文分别在一维和二维建立了S和−3/4的局部适定性，当非线性限制到Cu2时，Bejenaru和Tao，Bejenaru和De Silva的论文将这些结果改进到一维的S⩾−1和二维的S>；−1。2D的双线性估计还改进了2D立方NLS的有限能量全局时间解的Soblev范数的增长率。
In this article we establish the bilinear estimates corresponding to the 1D and 2D NLS with a quadratic nonlinearity c1u2+c2u¯2, which imply the local well-posedness of the Cauchy problem in Hsfor 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 u¯2, Commun. Pure Appl. Anal. 7 (2008) 1123–1143] on the 1D NLS with nonlinearity cu¯2. 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.