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arXiv 1608.03136math.APmath.FA
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具有$L^2$亚临界数据的导数非线性薛定谔方程的局部适定性

Local Well-Posedness for the Derivative Nonlinear Schrödinger Equations with $L^2$ Subcritical Data

Shaoming Guo, Xianfeng Ren, Baoxiang Wang

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摘要：

我们将证明它在调制空间$M^{1/2}_{2,q}({\Real})$ $(2\leq q<\infty) $中的局部适定性。众所周知，$H^{1/2}$是DNLS的临界Sobolev空间，所以它在$H^s$中对$s\geq 1/2$是局部适定性，在$H^{s'}$中对$s'<1/2.$是不适定性。注意到$M^{1/2}_{2,q} \subset B^{1/q}_{2,q}$是一个尖锐嵌入，而$L^2 \subset B^0_{2,\infty}$是$M^{1/2}_{2,q}$中的一个次临界数据，它包含了$L^2\setminus H^{1/2}$中的一类函数。

英文摘要：

We will show its local well-posedness in modulation spaces $M^{1/2}_{2,q}({\Real})$ $(2\leq q<\infty) $. It is well-known that $H^{1/2}$ is a critical Sobolev space of DNLS so that it is locally well-posedness in $H^s$ for $s\geq 1/2$ and ill-posed in $H^{s'}$ with $s'<1/2.$ Noticing that that $M^{1/2}_{2,q} \subset B^{1/q}_{2,q}$ is a sharp embedding and $L^2 \subset B^0_{2,\infty}$, our result contains all of the subcritical data in $M^{1/2}_{2,q}$, which contains a class of functions in $L^2\setminus H^{1/2}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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