Periodic nonlinear Schrödinger equation in critical $H^s(\T^n)$ spaces

In this paper we prove some multi-linear Strichartz estimates for solutions to the linear Schrödinger equations on torus $\T^n$. Then we apply it to get some local well-posed results for nonlinear Schrödinger equation in critical $H^{s}(\T^n)$ spaces. As by-products, the energy critical global well-posed results and energy subcritical global well-posed results with small initial data are also obtained.

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