Local well-posedness for quadratic nonlinear Schrödinger equations and the ``good'' Boussinesq equation
Local well-posedness for quadratic nonlinear Schrödinger equations and the ``good'' Boussinesq equation
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DOI：
10.57262/die/1356019307
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发表时间：
2010-05
影响因子：
1.4
通讯作者：
Nobu Kishimoto;K. Tsugawa
中科院分区：
文献类型：
--
作者：
Nobu Kishimoto;K. Tsugawa
． 在（cid:107） f (cid:107) H s,a = (cid:107) (1+ | ξ |) s - a | ξ | a b f (cid:107) l2定义的空间中，研究了具有二次非线性的一维非线性Schr¨odinger方程的Cauchy问题，并在这些空间中得到了包含u¯u项的非线性的明显的局部适定性和病态性结果。特别是，当a = 0时，先前Kenig， Ponce和Vega（1996）给出的H s， s > - 1 / 4的适定性结果改进为s≥- 1 / 4。这也扩展了Otani（2004）在H s，a中的结果。该证明基于类似Kenig， Ponce和Vega的迭代论证，并对傅里叶限制范数的空间进行了修改。我们的结果也适用于“好”的Boussinesq方程，并在H s × H s−2中得到s >−1 / 2的局部适定性，这是Farah（2009）给出的先前结果的改进。
. The Cauchy problem for 1-D nonlinear Schr¨odinger equations with quadratic nonlinearities are considered in the spaces H s,a defined by (cid:107) f (cid:107) H s,a = (cid:107) (1+ | ξ | ) s − a | ξ | a b f (cid:107) L 2 , and sharp local well-posedness and ill-posedness results are obtained in these spaces for nonlinearities including the term u ¯ u . In particular, when a = 0 the previous well-posedness result in H s , s > − 1 / 4, given by Kenig, Ponce and Vega (1996), is improved to s ≥ − 1 / 4. This also extends the result in H s,a by Otani (2004). The proof is based on an iteration argument similar to that of Kenig, Ponce and Vega, with a modification of the spaces of the Fourier restriction norm. Our result is also applied to the “good” Boussinesq equation and yields local well-posedness in H s × H s − 2 with s > − 1 / 2, which is an improvement of the previous result given by Farah (2009).