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We consider the Cauchy problem for the nonlinear Schrodinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on \begin{document}$ \mathbb{R}^d $\end{document} , \begin{document}$ d = 5,6 $\end{document} , and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (ⅰ) we prove almost sure global well-posedness of the (standard) energy-critical NLS on \begin{document}$ \mathbb{R}^d $\end{document} , \begin{document}$ d = 5, 6 $\end{document} , in the defocusing case, and (ⅱ) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities