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Constructive existence proofs and stability of stationary solutions to parabolic PDEs using Gegenbauer polynomials

Maxime Breden, Matthieu Cadiot, Antoine Zurek

Abstract

In this paper, we present a computer-assisted framework for constructive proofs of existence for stationary solutions to one-dimensional parabolic PDEs and the rigorous determination of their linear stability. By expanding solutions in Gegenbauer polynomials, we first develop a general approach for boundary value problems (BVPs), corresponding to the stationary part of the PDE. This yields a computationally efficient sparse structure for both differential and multiplication operators. By deriving sharp, explicit and quantitative estimates for the inverse of differential operators, we implement a Newton-Kantorovich approach. Specifically, given a numerical approximation $\bar{u}$, we prove the existence of a true stationary solution $\tilde{u}$ within a small, rigorously quantified neighborhood of $\bar{u}$. A key advantage of this approach is that the sharp control over the defect $\tilde{u}-\bar{u}$, integrated with the spectral properties of the Gegenbauer basis, enables an accurate enclosure of the linearization's spectrum around $\tilde{u}$. This allows for a definitive conclusion regarding the (in)stability of the verified solution, which is the main contribution of the paper. We demonstrate the efficacy of this method through several applications, capturing both stable and unstable equilibrium states.

Constructive existence proofs and stability of stationary solutions to parabolic PDEs using Gegenbauer polynomials

Abstract

In this paper, we present a computer-assisted framework for constructive proofs of existence for stationary solutions to one-dimensional parabolic PDEs and the rigorous determination of their linear stability. By expanding solutions in Gegenbauer polynomials, we first develop a general approach for boundary value problems (BVPs), corresponding to the stationary part of the PDE. This yields a computationally efficient sparse structure for both differential and multiplication operators. By deriving sharp, explicit and quantitative estimates for the inverse of differential operators, we implement a Newton-Kantorovich approach. Specifically, given a numerical approximation , we prove the existence of a true stationary solution within a small, rigorously quantified neighborhood of . A key advantage of this approach is that the sharp control over the defect , integrated with the spectral properties of the Gegenbauer basis, enables an accurate enclosure of the linearization's spectrum around . This allows for a definitive conclusion regarding the (in)stability of the verified solution, which is the main contribution of the paper. We demonstrate the efficacy of this method through several applications, capturing both stable and unstable equilibrium states.

Paper Structure

This paper contains 17 sections, 21 theorems, 249 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formulation of the problem using Gegenbauer polynomials
  3. Introduction to the Gegenbauer polynomials
  4. Banach algebra and different representations
  5. Differential and boundary operators in coefficient spaces
  6. Linear Problem
  7. Definition of the zero finding problem
  8. Compactness estimates for DK(U̅)
  9. A Newton-Kantorovich approach for studying steady states
  10. A general framework
  11. Computable estimates
  12. Stability
  13. Enclosure of the spectrum using Gershgorin disks
  14. Enclosure of the spectrum directly based on the generalized eigenproblem
  15. Summary
  16. ...and 2 more sections

Key Result

Lemma 2.1

Given $n \in \mathbb{N}_0$, we have the following properties: and with the convention that $G^{(k)}_{-1} = G^{(k)}_{-2} =0$ for all $k \in \mathbb{N}_0.$

Figures (1)

  • Figure :

Theorems & Definitions (47)

  • Lemma 2.1: Evaluation, derivatives and change of basis
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 37 more