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Persistence of invariant manifolds for nonlinear PDEs

Don A. Jones, Steve Shkoller

TL;DR

This work extends Fenichel-type persistence theory to infinite-dimensional Hilbert manifolds, showing that overflowing invariant submanifolds persist under $C^1$ perturbations of PDE semigroups when normal hyperbolicity and normal-attraction conditions hold. The authors develop a tubular-neighborhood framework, graph transforms, and contraction arguments to obtain a $C^1$ perturbed manifold $\bar{M}_{\nu}$ that converges to the original as $\nu\to0$, and they extend the theory to global unstable manifolds, maximum-perturbation analyses, and inertial manifolds. They apply the abstract results to the two-dimensional Navier–Stokes equations with fully discrete approximations, proving persistence of unstable manifolds under numerical schemes and establishing lower semi-continuity of the global unstable manifold and attractors in gradient-like settings. The paper also treats inertial-manifold persistence under a gap condition by introducing a dissipative truncation $\tilde{G}_{h}$ and proving convergence to the unperturbed inertial manifold. Collectively, these results connect long-time PDE dynamics with their numerical approximations, providing rigorous assurance that key invariant structures survive discretization and perturbation in this infinite-dimensional context.

Abstract

We prove that under certain stability and smoothing properties of the semi-groups generated by the partial differential equations that we consider, manifolds left invariant by these flows persist under $C^1$ perturbation. In particular, we extend well known finite-dimensional results to the setting of an infinite-dimensional Hilbert manifold with a semi-group that leaves a submanifold invariant. We then study the persistence of global unstable manifolds of hyperbolic fixed-points, and as an application consider the two-dimensional Navier-Stokes equation under a fully discrete approximation. Finally, we apply our theory to the persistence of inertial manifolds for those PDEs which possess them. te

Persistence of invariant manifolds for nonlinear PDEs

TL;DR

This work extends Fenichel-type persistence theory to infinite-dimensional Hilbert manifolds, showing that overflowing invariant submanifolds persist under perturbations of PDE semigroups when normal hyperbolicity and normal-attraction conditions hold. The authors develop a tubular-neighborhood framework, graph transforms, and contraction arguments to obtain a perturbed manifold that converges to the original as , and they extend the theory to global unstable manifolds, maximum-perturbation analyses, and inertial manifolds. They apply the abstract results to the two-dimensional Navier–Stokes equations with fully discrete approximations, proving persistence of unstable manifolds under numerical schemes and establishing lower semi-continuity of the global unstable manifold and attractors in gradient-like settings. The paper also treats inertial-manifold persistence under a gap condition by introducing a dissipative truncation and proving convergence to the unperturbed inertial manifold. Collectively, these results connect long-time PDE dynamics with their numerical approximations, providing rigorous assurance that key invariant structures survive discretization and perturbation in this infinite-dimensional context.

Abstract

We prove that under certain stability and smoothing properties of the semi-groups generated by the partial differential equations that we consider, manifolds left invariant by these flows persist under perturbation. In particular, we extend well known finite-dimensional results to the setting of an infinite-dimensional Hilbert manifold with a semi-group that leaves a submanifold invariant. We then study the persistence of global unstable manifolds of hyperbolic fixed-points, and as an application consider the two-dimensional Navier-Stokes equation under a fully discrete approximation. Finally, we apply our theory to the persistence of inertial manifolds for those PDEs which possess them. te

Paper Structure

This paper contains 33 sections, 23 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Construction of the perturbed manifold
  3. Infinite-dimensional geometry and sufficient conditions for persistence
  4. Proof.
  5. Proof.
  6. Proof.
  7. Proof.
  8. Lipschitz negatively invariant manifolds for the perturbed mapping
  9. Proof.
  10. Proof.
  11. $C^1$ negatively invariant manifold for the perturbed mapping
  12. Proof.
  13. Proof.
  14. Proof.
  15. Proof.
  16. ...and 18 more sections

Key Result

Lemma 2.1

Let $X$ be a $C^r$ manifold, $r\ge 3$, that admits a partition of unity and let ${\mathcal{M}}$ be a closed submanifold. Then there exists a tubular neighborhood of ${\mathcal{M}}$ in $X$ of class $C^{r-2}$.

Figures (1)

  • Figure 7.1: Lower Semi-continuity of approximate unstable manifold to the true unstable manifold .

Theorems & Definitions (26)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.4
  • Definition 2.5
  • Corollary 2.6
  • Lemma 2.7
  • Lemma 2.9: Invertability
  • Lemma 2.10
  • Theorem 2.11
  • Definition 2.12
  • ...and 16 more