论文标题
缩放性非线性schrödinger方程的初始数据的病理集合
Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations
论文作者
论文摘要
这项工作的目的是证明一系列初始数据的病理集，在任意短时间内，通过卷积的正则解决方案经历了一种规范通气机制。结果是在Sun和Tzvetkov的构造精神上，该病理套件包含集中在不同点的轮廓的叠加。由于波方程的有限传播速度并在一定时间内，最多有一个轮廓显示出显着的增长。但是，对于Schrödinger-type方程，我们不能排除彼此之间相互作用的轮廓。取而代之的是，我们提出了一种利用近似身份的正规化效果的方法，该方法在给定的尺度上排除了集中在较小尺度上的轮廓的范数通胀。
The purpose of this work is to evidence a pathological set of initial data for which the regularized solutions by convolution experience a norm-inflation mechanism, in arbitrarily short time. The result is in the spirit of the construction from Sun and Tzvetkov, where the pathological set contains superposition of profiles that concentrate at different points. Thanks to finite propagation speed of the wave equation, and given a certain time, at most one profile exhibits significant growth. However, for Schrödinger-type equations, we cannot preclude the profiles from interacting between each other. Instead, we propose a method that exploits the regularizing effect of the approximate identity which, at a given scale, rules out the norm inflation of the profiles that are concentrated at smaller scales.
