An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr\"odinger equation
An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr\"odinger equation
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三次四阶非线性Schr"odinger方程准不变高斯测度的最优正则结果
DOI：
10.5802/jep.83
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
N. Tzvetkov
中科院分区：
文献类型：
--
作者：
Tadahiro Oh;Philippe Sosoe;N. Tzvetkov
我们研究了圆上三次四阶非线性薛定格方程动力学下索博列夫空间上高斯测度的输运性质。特别是，我们建立了索博列夫空间上均值零高斯测度准不变性的最优正则结果。主要的新成分是通过对能量进行范式约简的无限迭代而建立的改进的能量估计。 功能性的。此外，我们通过在无色散模型的动力学下证明高斯测度的非准不变性，表明色散对于这种准不变性结果至关重要。
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures under the dynamics of the dispersionless model.