论文标题
某些分散方程的概率点融合问题
Probabilistic pointwise convergence problem of some dispersive equations
论文作者
论文摘要
在本文中，我们分别研究了自由KDV方程，自由波方程，游离椭圆形和非纤维化schrödinger方程的几乎肯定的融合问题。我们首先建立了一些与频率空间的维也纳分解相关的估计值，这些估计只是引理2.1-2.6。其次，通过使用引理2.1-2.6、3.1，我们建立了一些随机序列的概率估计值，这些序列只是引理3.2-3.11。最后，将L $^{2} $中的密度定理与引理3.2-3.11相结合，几乎可以肯定地获得解决方案与相应方程式的，其中具有$ l^{2} $的随机初始数据的相应方程，这比初始数据的规律性要小得多。同时，我们介绍概率密度定理，本文中是引理3.11。
In this paper, we investigate the almost surely pointwise convergence problem of free KdV equation, free wave equation, free elliptic and non-elliptic Schrödinger equation respectively. We firstly establish some estimates related to the Wiener decomposition of frequency spaces which are just Lemmas 2.1-2.6 in this paper. Secondly, by using Lemmas 2.1-2.6, 3.1, we establish the probabilistic estimates of some random series which are just Lemmas 3.2-3.11 in this paper. Finally, combining the density theorem in L$^{2}$ with Lemmas 3.2-3.11, we obtain almost surely pointwise convergence of the solutions to corresponding equations with randomized initial data in $L^{2}$, which require much less regularity of the initial data than the rough data case. At the same time, we present the probabilistic density theorem, which is Lemma 3.11 in this paper.
