论文标题
几乎可以确保$ \ mathbb {r}^3 $的单位球上的能量超临界NLS的全球能量良好性
Almost sure global well-posedness for the energy supercritical NLS on the unit ball of $\mathbb{R}^3$
论文作者
论文摘要
在本文中，我们使用两种不同的方法在$ \ bbb r^3 $的单位球上提供了两个几乎确定的全球适合度（GWP）结果，以使用两种不同的方法。首先，对于具有亚临界初始数据的代数非线性的NLS，我们显示了几乎确定的全球良好性和基本措施的不变性，并建立了解决方案Sobolev规范增长的控制。该全球结果基于确定性的本地理论和概率全球化。其次，对于具有关键和超临界初始条件的通用功率非线性的NLS，我们证明了几乎确定的全球辅助性，以及在解决方案流下的度量的不变性。该全局结果建立在紧凑的论点和Skorokhod代表定理的基础上。
In this paper, we present two almost sure global well-posedness (GWP) results for the energy supercritical nonlinear Schrödinger equations (NLS) on the unit ball of $\Bbb R^3$ using two different approaches. First, for the NLS with algebraic nonlinearities with the subcritical initial data, we show the almost sure global well-posedness and the invariance of the underlying measures, and establish controls on the growth of Sobolev norms of the solutions. This global result is based on a deterministic local theory and a probabilistic globalization. Second, for the NLS with generic power nonlinearities with critical and supercritical initial conditions, we prove the almost sure global well-posedness, and the invariance of the measure under the solution flows. This global result is built on a compactness argument and the Skorokhod representation theorem.
