论文标题
精致的概率全球适合弱色散的NLS
Refined probabilistic global well-posedness for the weakly dispersive NLS
论文作者
论文摘要
我们继续研究具有非常弱的分散$α> 1 $的立方分数NLS，并根据Gibbs度量分配的数据。我们以$α>α_0= \ frac {31- \ sqrt {233}}} {14} \约1.124 $构建天然强解决方案，该解决方案严格小于$ \ frac {8} {7} {7} $，超出了第一个非整体上的boteritive for sobolev for sobolev sebolev festition for the terecity的阈值。这也改善了我们以前的结果，以sun-tzvetkov \ cite {sun-tz2}。我们依靠Bringmann \ Cite {Bringmann}和Deng-nahmod-yue \ cite {deng2}的最新想法。特别是，我们适应了\ cite {deng2}中新的分辨率ansatz，该分辨率{deng2}捕获了$ x^{s，b} $ type Space中最单一的频率相互作用零件。为了克服弱分散效应造成的困难，我们的具体策略是从这些奇异部分的“几乎”运输效应中受益，并利用其$ l^{\ infty} $以及傅立叶 - lebesgue属性，以从高频部分的线性进化中继承随机特征。
We continue our study of the cubic fractional NLS with very weak dispersion $α>1$ and data distributed according to the Gibbs measure. We construct the natural strong solutions for $α>α_0=\frac{31-\sqrt{233}}{14}\approx 1.124$ which is strictly smaller than $\frac{8}{7}$, the threshold beyond which the first nontrivial Picard iteration has no longer the Sobolev regularity needed for the deterministic well-posedness theory. This also improves our previous result in Sun-Tzvetkov \cite{Sun-Tz2}. We rely on recent ideas of Bringmann \cite{Bringmann} and Deng-Nahmod-Yue \cite{Deng2}. In particular we adapt to our situation the new resolution ansatz in \cite{Deng2} which captures the most singular frequency interaction parts in the $X^{s,b}$ type space. To overcome the difficulties caused by the weakly dispersive effect, our specific strategy is to benefit from the "almost" transport effect of these singular parts and to exploit their $L^{\infty}$ as well as the Fourier-Lebesgue property in order to inherit the random feature from the linear evolution of high frequency portions.
