论文标题
Ostrovsky方程的侧面收敛问题，带有粗糙的数据和随机数据
Pointwise convergence problem of Ostrovsky equation with rough data and random data
论文作者
论文摘要
在本文中，我们考虑了带有粗糙数据和随机数据的自由奥斯特罗夫斯基方程的重点收敛问题。首先，我们在$ h^{s}中以$ s \ geq \ geq \ frac {1} {4} $和粗糙的数据显示了几乎所有到处的无处不在的融合。其次，我们提出反描述，表明如果$ s <\ frac {1} {4} $，与免费Ostrovsky方程相关的最大函数估计可能会失败。最后，对于\ mathbb {r} $中的每一个$ x \，我们在$ l^{2}中几乎可以肯定地显示了免费的ostrovsky方程的点，并显示了随机数据。主要工具是密度定理，高低频率思想，Wiener分解和引理2.1-2.6，以及某些随机系列的概率估计，这只是引理3.2-3.4。主要困难是零是自由奥斯特罗夫斯基方程的相位函数的单数点。我们使用高低的频率想法来征服困难。
In this paper, we consider the pointwise convergence problem of free Ostrovsky equation with rough data and random data. Firstly, we show the almost everywhere pointwise convergence of free Ostrovsky equation in $H^{s}(\mathbb{R})$ with $s\geq \frac{1}{4}$ with rough data. Secondly, we present counterexamples showing that the maximal function estimate related to the free Ostrovsky equation can fail if $s<\frac{1}{4}$. Finally, for every $x\in \mathbb{R}$, we show the almost surely pointwise convergence of free Ostrovsky equation in $L^{2}(\mathbb{R})$ with random data. The main tools are the density theorem, high-low frequency idea, Wiener decomposition and Lemmas 2.1-2.6 as well as the probabilistic estimates of some random series which are just Lemmas 3.2-3.4 in this paper. The main difficulty is that zero is the singular point of the phase functions of free Ostrovsky equation. We use high-low frequency idea to conquer the difficulties.
