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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 (33 sections, 23 theorems, 119 equations, 1 figure)

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