论文标题
2D Hartree方程的站立波和全球适应性和良好的互动
Standing waves and global well-posedness for the 2d Hartree equation with a point interaction
论文作者
论文摘要
我们研究了具有点样的奇异扰动和Hartree非线性的二维非线性Schrödinger方程。游离拉普拉斯的点状奇异扰动引起了研究基态和进化流量所必需的适当扰动的索波列夫空间。我们将质量次临界和批判性哈特里的非线性都包括在内。我们的分析是两个方面：我们建立了基态的存在，对称性和规律性，并且我们证明了在奇异的扰动能量空间中相关的cauchy问题的适当性。与当前工作并行的其他处理不同，第一个目标是通过对修改后的Weinstein功能的Schwartz对称性的标准特性的非平地适应来实现的。这产生了修改的Gagliardo-Nirenberg类型的不平等现象，这些不平等现象可以有效地控制非线性并通过能量方法获得良好的能力。事实证明，进化流程在散落的情况下是全球的，在聚焦和质量亚临界情况下。对于最佳的Gagliardo-Nirenberg常数而言，它在聚焦和质量关键案例中也是全球性的。
We study a class of two-dimensional non-linear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a non-trivial adaptation of the standard properties of Schwartz symmetrisation for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.
