Self-consistent bounds method for dissipative PDEs

TL;DR

This paper introduces a self-consistent bounds method for dissipative PDEs, providing a convergence framework and demonstrating its applicability to classical equations like Kuramoto-Sivashinsky and Navier-Stokes on the torus.

Contribution

It develops a theoretical convergence framework for a new bounds method applicable to various dissipative PDEs, including classical examples.

Findings

Proves convergence theorems for the bounds method.

Shows applicability to Kuramoto-Sivashinsky and Navier-Stokes equations.

Provides a general framework for solution bounds and dependence on initial conditions.

Abstract

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.