Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three
Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three
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DOI：
10.1063/5.0045062
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发表时间：
2021-01
影响因子：
1.3
通讯作者：
Yu Deng;A. Nahmod;H. Yue
中科院分区：
文献类型：
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作者：
Yu Deng;A. Nahmod;H. Yue
在本文中，我们考虑 T3 上的散焦 Hartree 非线性薛定谔方程，具有实值甚至势 V 和傅里叶乘数衰减，例如 |k|−β。通过随机平均算子的方法[Deng et al., arXiv:1910.08492 (2019)]，我们证明存在β0，它小于但接近1，这样当β>β0时，我们具有相关吉布斯测度的不变性，并且在其统计集合中全局存在强解。通过这种方式，我们扩展了布尔根的开创性成果[J.布尔根，J.马斯。纯应用程序。 76, 649–702 (1997)]，在这种情况下要求 β > 2。
In this paper, we consider the defocusing Hartree nonlinear Schrodinger equations on T3 with real-valued and even potential V and Fourier multiplier decaying such as |k|−β. By relying on the method of random averaging operators [Deng et al., arXiv:1910.08492 (2019)], we show that there exists β0, which is less than but close to 1, such that for β > β0, we have invariance of the associated Gibbs measure and global existence of strong solutions in its statistical ensemble. In this way, we extend Bourgain’s seminal result [J. Bourgain, J. Math. Pures Appl. 76, 649–702 (1997)], which requires β > 2 in this case.