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Rigorous Numerics for Partial Differential Equations: the Kuramoto-Sivashinsky equation

P. Zgliczynski, K. Mischaikow

TL;DR

The method is based on the concept of the self-consistent a priori bounds, which permit the rigorous justification of the use of Galerkin projections, and obtains a low-dimensional system of ODEs subject to rigorously controlled small perturbation from the neglected modes.

Abstract

We present a new topological method for the study of the dynamics of dissipative PDE's. The method is based on the concept of the self-consistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we obtain a low-dimensional system of ODE's subject to rigorously controlled small perturbation from the neglected modes. To this ODE's we apply the Conley index to obtain information about the dynamics of the PDE under consideration. As an application we present a computer assisted proof of the existence of fixed points for the Kuramoto-Sivashinsky equation.

Rigorous Numerics for Partial Differential Equations: the Kuramoto-Sivashinsky equation

TL;DR

The method is based on the concept of the self-consistent a priori bounds, which permit the rigorous justification of the use of Galerkin projections, and obtains a low-dimensional system of ODEs subject to rigorously controlled small perturbation from the neglected modes.

Abstract

We present a new topological method for the study of the dynamics of dissipative PDE's. The method is based on the concept of the self-consistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we obtain a low-dimensional system of ODE's subject to rigorously controlled small perturbation from the neglected modes. To this ODE's we apply the Conley index to obtain information about the dynamics of the PDE under consideration. As an application we present a computer assisted proof of the existence of fixed points for the Kuramoto-Sivashinsky equation.

Paper Structure

This paper contains 12 sections, 20 theorems, 141 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Method
  3. Self-consistent Bounds
  4. Conley Index
  5. A Singular Perturbation Result
  6. Remarks on Related Work
  7. Estimates for Kuramoto-Sivashinsky equation
  8. $1\leq k\leq M$
  9. $k>M$
  10. Refining the Self-Consistent Bounds
  11. A Typical Proof
  12. Obtaining the Initial Estimates

Key Result

Theorem 1.1

Let $u(x) = \sum_{k=1}^{28} a_k\sin(kx)$ where the $a_k$ are given in Table table:coef. Then, for $\nu=0.1$ there exists an equilibrium $u^*(x)$ for (eq:KS) such that

Figures (2)

  • Figure 1: The equilibrium solution $u^*(x)$ for $\nu=0.1$
  • Figure 2: The computed function $u(x)$ from Theorem \ref{['thm:proof']}

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 19 more