Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unique continuation and stabilization for nonlinear Schrödinger equations under the Geometric Control Condition

Cristóbal Loyola

TL;DR

The paper addresses unique continuation and stabilization for nonlinear Schrödinger equations under the Geometric Control Condition on compact manifolds. It develops an abstract frequency-based reconstruction framework that splits solutions into low and high frequency components, uses Galerkin-type methods and holomorphic extensions to propagate analyticity in time from the observation region to the full domain, and establishes a robust high-frequency observability theory. Applying this to NLS with analytic nonlinearities, the authors prove global propagation of analyticity, obtain UCP in subcritical dimensions 2 and 3, and derive semiglobal stabilization and controllability results under GCC, resolving a long-standing nonlinear open question. The results combine observability, complex analysis in Banach spaces, and microlocal techniques, offering a flexible approach that extends to other conservative PDEs and potentially to broader geometric settings.

Abstract

In this article we prove global propagation of analyticity in finite time for solutions of semilinear Schrödinger equations with analytic nonlinearity from a region $ω$ where the Geometric Control Condition holds. Our approach refines a recent technique introduced by Laurent and the author, which combines control theory techniques and Galerkin approximation, to propagate analyticity in time from a zone where observability holds. As a main consequence, we obtain unique continuation for subcritical semilinear Schrödinger equations on compact manifolds of dimension $2$ and $3$ when the solution is assumed to vanish on $ω$. Furthermore, semiglobal control and stabilization follow only under the Geometric Control Condition on the observation zone. In particular, this answers in the affirmative an open question of Dehman, Gérard, and Lebeau from $2006$ for the nonlinear case.

Unique continuation and stabilization for nonlinear Schrödinger equations under the Geometric Control Condition

TL;DR

The paper addresses unique continuation and stabilization for nonlinear Schrödinger equations under the Geometric Control Condition on compact manifolds. It develops an abstract frequency-based reconstruction framework that splits solutions into low and high frequency components, uses Galerkin-type methods and holomorphic extensions to propagate analyticity in time from the observation region to the full domain, and establishes a robust high-frequency observability theory. Applying this to NLS with analytic nonlinearities, the authors prove global propagation of analyticity, obtain UCP in subcritical dimensions 2 and 3, and derive semiglobal stabilization and controllability results under GCC, resolving a long-standing nonlinear open question. The results combine observability, complex analysis in Banach spaces, and microlocal techniques, offering a flexible approach that extends to other conservative PDEs and potentially to broader geometric settings.

Abstract

In this article we prove global propagation of analyticity in finite time for solutions of semilinear Schrödinger equations with analytic nonlinearity from a region where the Geometric Control Condition holds. Our approach refines a recent technique introduced by Laurent and the author, which combines control theory techniques and Galerkin approximation, to propagate analyticity in time from a zone where observability holds. As a main consequence, we obtain unique continuation for subcritical semilinear Schrödinger equations on compact manifolds of dimension and when the solution is assumed to vanish on . Furthermore, semiglobal control and stabilization follow only under the Geometric Control Condition on the observation zone. In particular, this answers in the affirmative an open question of Dehman, Gérard, and Lebeau from for the nonlinear case.
Paper Structure (58 sections, 53 theorems, 324 equations)

This paper contains 58 sections, 53 theorems, 324 equations.

Table of Contents

  1. Introduction
  2. Main results on nonlinear Schrödinger equation
  3. Control and stabilization contributions
  4. Literature on unique continuation
  5. Abstract frequency-based reconstruction
  6. Outline of the article
  7. Acknowledgment
  8. Abstract analytic reconstruction
  9. Sketch of the strategy
  10. Preliminaries
  11. Step 1. Complex extension.
  12. Step 2. Holomorphicity of the extension.
  13. High-frequency linear observability problem
  14. Step 1. Boundedness.
  15. Step 2. Lipschitz-continuity.
  16. ...and 43 more sections

Key Result

Theorem 1.1

Let $d\in \mathbb{N}$ and $s>d/2$. Let $u\in C^0([0, T], H^{s}(\mathcal{M}))$ be a solution of eq:NLS. Assume that the above setting holds and assume moreover that: Then $t\mapsto u(t, \cdot)$ is analytic from $(0, T)$ into $H^2(\mathcal{M})$.

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 93 more