论文标题
衍生物非线性schrödinger方程的全球适应性
Global well-posedness for the derivative nonlinear Schrödinger equation
论文作者
论文摘要
本文致力于研究真实线上的衍生化非线性schrödinger方程。自几十年以来，索博莱夫空间中这个方程式的本地供应良好，而全球良好的良好性并未完全解决。对于后一个问题，最新的结果最新问题要么$ h^{\ frac12} $中的cauchy数据严格少于$4π$，要么在加权sobolev space $ h^{2，2} $中的一般初始条件。在本文中，我们证明了衍生的非线性schrödinger方程在$ h^{\ frac12} $中的cauchy数据方面均得到很好的范围，并且该解决方案的$ h^{\ frac12} $ norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm norm in Globall Global in Alll Global范围内。人们应该回想起，对于$ h^s $，$ s <1/2 $，相关的cauchy问题在某种意义上是不合时宜的，因为相对于初始数据均匀的连续性失败了。因此，我们的结果在Sobolev Spaces $ H^S $的设置中结束了讨论。通过将曲线分解技术与方程的集成性结构相结合来实现证明。
This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in $H^{\frac12}$ with mass strictly less than $4π$ or general initial conditions in the weighted Sobolev space $H^{2, 2}$. In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in $H^{\frac12}$ and that furthermore the $H^{\frac12}$ norm of the solutions remains globally bounded in time. One should recall that for $H^s$, with $s < 1 / 2 $, the associated Cauchy problem is ill-posed in the sense that uniform continuity with respect to the initial data fails. Thus, our result closes the discussion in the setting of the Sobolev spaces $H^s$. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.
