论文标题
关于二维周期性非线性schrödinger方程的概率良好，具有二次非线性$ | u | u |^2 $
On the probabilistic well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with the quadratic nonlinearity $|u|^2$
论文作者
论文摘要
我们使用二维非线性schrödinger方程（NLS）研究了二次非线性$ | u |^2 $。特别是，我们研究具有高斯自由场的分数衍生物（$α\ geq 0 $）的二次NLS。在删除了零频率下的奇异性之后，我们证明二次NLS几乎肯定在本地适当地符合$α<\ frac {1} {2} $，并且在适当的意义上以$α\ geq \ frac {3} {4} $而言是$α\ geq \ geq \ geq \ geq \ geq \ geq \ geq \ geq \ geq \ geq \ geq \ geq \ frac {4} $。这些结果表明，在粗糙的随机初始数据和二次非线性的情况下，NLS的标准概率良好的拟态度理论在达到临界值$α= 1 $之前，由Deng，Nahmod和Yue（2019）引起的缩放分析所预测的临界值$α= 1 $，因此本文的工作是OH和OHEN的诸如OH和OKAMOTO的启动（202211）（20211）（202211）（2022）对于NLS。
We study the two-dimensional periodic nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $|u|^2$. In particular, we study the quadratic NLS with random initial data distributed according to a fractional derivative (of order $α\geq 0$) of the Gaussian free field. After removing the singularity at the zeroth frequency, we prove that the quadratic NLS is almost surely locally well-posed for $α< \frac{1}{2}$ and is probabilistically ill-posed for $α\geq \frac{3}{4}$ in a suitable sense. These results show that in the case of rough random initial data and a quadratic nonlinearity, the standard probabilistic well-posedness theory for NLS breaks down before reaching the critical value $α= 1$ predicted by the scaling analysis due to Deng, Nahmod, and Yue (2019), and thus this paper is a continuation of the work by Oh and Okamoto (2021) on stochastic nonlinear wave and heat equations by building an analogue for NLS.
