We discuss the method of self-consistent bounds for dissipative PDEs with periodic boundary conditions. We prove convergence theorems for a class of dissipative PDEs, which constitute a theoretical basis of a general framework for construction of an algorithm that computes bounds for the solutions of the underlying PDE and its dependence on initial conditions. We also show, that the classical examples of parabolic PDEs including Kuramoto-Sivashinsky equation and the Navier-Stokes on the torus fit into this framework.

翻译：本文讨论了自洽界方法在具有周期性边界条件的耗散偏微分方程中的应用。我们证明了一类耗散偏微分方程的收敛定理，这为构建一个通用算法框架奠定了理论基础，该算法能够计算底层偏微分方程解的界及其对初始条件的依赖关系。我们还证明了经典的抛物型偏微分方程实例，包括Kuramoto-Sivashinsky方程和环面上的Navier-Stokes方程，均适用于此框架。