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Self-consistent bounds method for dissipative PDEs

Daniel Wilczak, Piotr Zgliczyński

TL;DR

The paper develops a rigorous self-consistent bounds framework for dissipative PDEs with periodic boundary conditions, establishing C^0 and C^1 convergence of Galerkin projections and their variational equations under norm- and set-based convergence assumptions. It introduces good sequence spaces (GSS), logarithmic norms, and block decompositions to manage tail behavior and provide uniform bounds essential for rigorous numerics. The method is shown to apply to canonical torus PDEs, including Kuramoto–Sivashinsky and Navier–Stokes, by verifying the isolation property and convergence conditions (S,C,VC) on appropriate polynomial- or exponential-decay sets. The results yield guaranteed bounds on forward trajectories and their variational dynamics, enabling rigorous, computable enclosures for dissipative PDE dynamics and their sensitivity to initial conditions with potential for robust numerical implementations.

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.

Self-consistent bounds method for dissipative PDEs

TL;DR

The paper develops a rigorous self-consistent bounds framework for dissipative PDEs with periodic boundary conditions, establishing C^0 and C^1 convergence of Galerkin projections and their variational equations under norm- and set-based convergence assumptions. It introduces good sequence spaces (GSS), logarithmic norms, and block decompositions to manage tail behavior and provide uniform bounds essential for rigorous numerics. The method is shown to apply to canonical torus PDEs, including Kuramoto–Sivashinsky and Navier–Stokes, by verifying the isolation property and convergence conditions (S,C,VC) on appropriate polynomial- or exponential-decay sets. The results yield guaranteed bounds on forward trajectories and their variational dynamics, enabling rigorous, computable enclosures for dissipative PDE dynamics and their sensitivity to initial conditions with potential for robust numerical implementations.

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.

Paper Structure

This paper contains 29 sections, 30 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Sequence spaces with good norms
  3. Linear forms on GSS
  4. Bounded linear self-maps on GSS
  5. Galerkin projection of linear maps on GSS
  6. Block decomposition
  7. Logaritmic norms and estimates for li-near equa-tions
  8. $C^0$-convergence
  9. $\mathcal{C}^1$-convergence of Galerkin projections
  10. Some remarks about $C^1$ functions on compact subsets of GSS spaces
  11. Variational equations and Galerkin projections
  12. Block decomposition
  13. Isolation property
  14. Dissipative PDEs on the torus
  15. Some examples
  16. ...and 14 more sections

Key Result

Lemma 1

If $(H,\|\cdot\|)$ is GSS, then for any $J \subset \mathbb{Z}_+$ the projection $P_J$ is continuous and for any $w \in H$ there holds

Theorems & Definitions (42)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Definition 2
  • Definition 3
  • ...and 32 more