论文标题
改进的多线性估计和一般非线性波方程的全球规律性在$（1+3）$ dimensions中
Improved multilinear estimates and global regularity for general nonlinear wave equations in $(1+3)$ dimensions
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
This paper is devoted to the investigation of long-time behaviour of solutions to wave equations with quadratic nonlinearity and cubic Dirac equations with Hartree-type nonlinearity. We consider the nonlinearity here with enough simplicity so that we can treat it as a toy model and simultaneously with enough generality so that we can apply our result to wave and Dirac equations with various nonlinearities. The challenging point is that nonlinearity possesses singularity near the origin. Our strategy is to relax such a singularity by exploiting fully an angular momentum operator. In this manner we establish scattering for the critical Sobolev data.
