Rigorous $C_1$ integration of dissipative PDEs

TL;DR

This paper presents a new $C^1$ algorithm for rigorously integrating dissipative PDEs, enabling computer-assisted proofs of attractors in specific equations.

Contribution

The paper introduces a novel $C^1$ algorithm that rigorously controls solutions and derivatives for dissipative PDEs, facilitating proof of attractors.

Findings

Established existence of locally attracting periodic orbits for Chafee-Infante and Burgers equations.

Developed an algorithm suitable for computer-assisted proofs involving derivatives.

Demonstrated the algorithm's effectiveness on non-autonomous dissipative PDEs.

Abstract

We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.