论文标题
部分可观测时空混沌系统的无模型预测
Global existence and blow-up for a stochastic transport equation with non-local velocity
论文作者
论文摘要
在本文中，我们研究了实际线上随机扰动下的非线性和非本地一维运输方程。我们首先建立了一个本地理论，即Sobolev Spaces中的路径解决方案的存在，独特性和爆炸标准$ h^{s} $，$ s> 3 $。此后，我们根据我们考虑的噪声类型给出了解决方案的长时间行为。一方面，我们确定了一个噪音家族，可以以$ 1 $的概率来防止爆炸，从而确保了几乎肯定的全球解决方案的存在和独特性。另一方面，在特定的线性噪声案例中，我们表明奇异性在有限的时间内以正概率发生，并且我们得出了这些概率的下限。总而言之，我们介绍了退出时间的稳定性概念，并表明人们无法改善退出时间的稳定性，并同时提高了对初始数据的依赖性的连续性。
In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces $H^{s}$ with $s>3$. Thereafter, we give a complete picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability $1$, guaranteeing the existence and uniqueness of global solutions almost surely. On the other hand, in the particular linear noise case, we show that singularities occur in finite time with positive probability, and we derive lower bounds of these probabilities. To conclude, we introduce the notion of stability of exiting times and show that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data.
