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arXiv · 2101.11100

Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three

Yu Deng·Andrea R. Nahmod·Haitian Yue
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Abstract

In this paper we consider the defocusing Hartree nonlinear Schr\"odinger equations on $\mathbb T^3$ with real valued and even potential $V$ and Fourier multiplier decaying like $|k|^{-\beta}$. By relying on the method of random averaging operators in arXiv:1910.08492, we show that there exists $\frac{1}{2} \ll \beta_0 <1 $ such that for $ \beta > \beta_0 $ 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 [7] which requires $\beta >2$ in this case.

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Yu Deng, Andrea R. Nahmod, Haitian Yue. 2021-01-26. Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three. https://doi.org/10.1063/5.0045062

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