论文标题
$ \ mathbb {t}^d $在低规律性的$ \ mathbb {t}^d $上的某些非线性schr {Ö}的几乎全局存在
Almost global existence for some nonlinear Schr{ö}dinger equations on $\mathbb{T}^d$ in low regularity
论文作者
论文摘要
我们对$ d $二维的圆环的非线性schr {Ö} dinger方程的长期行为感兴趣，即，对于Sobolev space中的少量初始数据$ h^{s_0}（\ s_0}（\ mathbb t^d），$ s_0> d/2 $ d/2 $。我们证明，即使在这种情况下，即使在这种情况下，$ h^s $ norms，$ s \ geq 0 $，在时间内仍处于控制之下，$ t_ \ varepsilon = \ exp \ big big（ - \ frac {| \ log \ log \ log \ log \ varepsilon |^2} $ h^{s_0} $，$ \ | u（0）\ | _ {h^{s_0}} = \ varepsilon $中的初始基准。为此，我们添加到方程的线性部分中，以$ \ ell^\ infty（\ mathbb z^d）$中的随机傅立叶乘数添加，并显示了几乎任何实现此乘数的稳定性结果。特别是，使用这样的傅立叶乘数，我们在$ h^{s_0}上获得了非线性schr {Ö} dinger方程的几乎全局良好，任何$ s_0> d/2 $和任何$ d $ d \ geq1 $。
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 $d\geq1$.
