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Almost global existence for some nonlinear Schr{ö}dinger equations on $\mathbb{T}^d$ in low regularity

We are interested in the long time behavior of solutions of the nonlinear Schr{ö}dinger equation on the $d$-dimensional torus in low regularity, i.e. for small initial data in the Sobolev space $H^{s_0}(\mathbb T^d)$ with $s_0>d/2$. We prove that, even in this context of low regularity, the $H^s$-norms, $s\geq 0$, remain under control during times, $T_\varepsilon= \exp \big(-\frac{|\log\varepsilon|^2}{4\log|\log\varepsilon|} \big)$, exponential with respect to the initial size of the initial datum in $H^{s_0}$, $\|u(0)\|_{H^{s_0}}=\varepsilon$. For this, we add to the linear part of the equation a random Fourier multiplier in $\ell^\infty(\mathbb Z^d)$ and show our stability result for almost any realization of this multiplier. In particular, with such Fourier multipliers, we obtain the almost global well posedness of the nonlinear Schr{ö}dinger equation on $H^{s_0}(\mathbb T^d)$ for any $s_0>d/2$ and any $d\geq1$.