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Control of blow-up profiles for the mass-critical focusing nonlinear Schrödinger equation on bounded domains

Ludovick Gagnon, Kévin Le Balc'h

TL;DR

The paper investigates control of blow-up profiles for the mass-critical focusing nonlinear Schrödinger equation on bounded two-dimensional domains with Dirichlet boundary conditions. It introduces a localized nonlinear feedback control supported near prescribed blow-up points that prevents blow-up and yields a global solution, and it proves a null-controllability result for blow-up profiles under a small-time controllability assumption for the linear Schrödinger equation. The approach combines blow-up profile decompositions, rapid stabilization in the control zone, and fixed-point arguments to manage exterior errors, with extensions to three dimensions discussed via Kato-type methods and open questions. The findings highlight the potential of localized control to steer and prevent singular behavior in nonlinear dispersive equations, offering a baseline for further research on control of blow-up phenomena in related PDE models.

Abstract

In this paper, we consider the mass-critical focusing nonlinear Schrödinger on bounded two-dimensional domains with Dirichlet boundary conditions. In the absence of control, it is well-known that free solutions starting from initial data sufficiently large can blow-up. More precisely, given a finite number of points, there exists particular profiles blowing up exactly at these points at the blow-up time. For pertubations of these profiles, we show that, with the help of an appropriate nonlinear feedback law located in an open set containing the blow-up points, the blow-up can be prevented from happening. More specifically, we construct a small-time control acting just before the blow-up time. The solution may then be extended globally in time. This is the first result of control for blow-up profiles for nonlinear Schrödinger type equations. Assuming further a geometrical control condition on the support of the control, we are able to prove a null-controllability result for such blow-up profiles. Finally, we discuss possible extensions to three-dimensional domains.

Control of blow-up profiles for the mass-critical focusing nonlinear Schrödinger equation on bounded domains

TL;DR

The paper investigates control of blow-up profiles for the mass-critical focusing nonlinear Schrödinger equation on bounded two-dimensional domains with Dirichlet boundary conditions. It introduces a localized nonlinear feedback control supported near prescribed blow-up points that prevents blow-up and yields a global solution, and it proves a null-controllability result for blow-up profiles under a small-time controllability assumption for the linear Schrödinger equation. The approach combines blow-up profile decompositions, rapid stabilization in the control zone, and fixed-point arguments to manage exterior errors, with extensions to three dimensions discussed via Kato-type methods and open questions. The findings highlight the potential of localized control to steer and prevent singular behavior in nonlinear dispersive equations, offering a baseline for further research on control of blow-up phenomena in related PDE models.

Abstract

In this paper, we consider the mass-critical focusing nonlinear Schrödinger on bounded two-dimensional domains with Dirichlet boundary conditions. In the absence of control, it is well-known that free solutions starting from initial data sufficiently large can blow-up. More precisely, given a finite number of points, there exists particular profiles blowing up exactly at these points at the blow-up time. For pertubations of these profiles, we show that, with the help of an appropriate nonlinear feedback law located in an open set containing the blow-up points, the blow-up can be prevented from happening. More specifically, we construct a small-time control acting just before the blow-up time. The solution may then be extended globally in time. This is the first result of control for blow-up profiles for nonlinear Schrödinger type equations. Assuming further a geometrical control condition on the support of the control, we are able to prove a null-controllability result for such blow-up profiles. Finally, we discuss possible extensions to three-dimensional domains.

Paper Structure

This paper contains 16 sections, 17 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. The mass-critical focusing nonlinear Schrödinger equation and blow-up
  3. Control of the mass-critical focusing nonlinear Schrödinger equation
  4. Comments and generalizations
  5. Extension to the 3-dimensional case
  6. Strategy of the proofs of the main results \ref{['tm:mainresult1']} and \ref{['tm:mainresult2']}
  7. Proof of the main results
  8. Useful lemmas
  9. Rapid stabilization of the blow-up profile in the control zone
  10. Extension to a global solution: end of the proof of \ref{['tm:mainresult1']}
  11. Local null-controllability result: end of the proof of \ref{['tm:mainresult2']}
  12. Proofs of the results in the $3$-d case
  13. An open-loop null-control
  14. Appendix
  15. The controlled linear system.
  16. ...and 1 more sections

Key Result

Theorem 1.1

For any $R>0$, there exists a time $T>0$ such that for every $\psi_0 \in B_R := \{ \psi_0 \in H^2(\Omega) \cap H_0^1(\Omega)\ ;\ \|\psi_0\|_{H^2(\Omega)} \leq R\},$ the problem eq:L2CriticalNLS has a unique solution in $C([0,T];H^2(\Omega)\cap H^1_0(\Omega))$. In addition, the flow map $\psi_0 \to \

Theorems & Definitions (31)

  • Theorem 1.1: Lemma 2.1 of BGT03
  • Theorem 1.2: Theorem 3.6.1 of Caz03
  • Theorem 1.3: BGT03God11
  • Theorem 1.4: PR07, BG80
  • Theorem 1.5
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • proof
  • ...and 21 more