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Blow-up dynamics for radial self-dual Chern-Simons-Schrödinger equation with prescribed asymptotic profile

Kihyun Kim, Soonsik Kwon, Sung-Jin Oh

TL;DR

The paper proves that in the radial, $m=0$ setting of the self-dual Chern-Simons-Schrödinger equation, finite-energy blow-up can occur with a continuum of rates driven by a prescribed asymptotic radiation profile $z^{*}(r)=q r^{\nu}\chi(r)$ with $\Re\nu>0$. The authors develop a backward blow-up construction combined with precise modulation analysis, constructing a radiation $z(t,r)$ from the asymptotic profile and coupling it to a modulated soliton $Q_{\lambda,\gamma}$ via a refined set of modulation parameters $\bm{\zeta}$ and $\bm{b}$, controlled by a nonlinear energy functional $\mathcal{E}$. This framework yields blow-up rates of the form $\lambda(t) \sim |\log(T-t)|^{-1}(T-t)^{p}$ for any $p>1$ (and the associated phase and modulation data), illustrating a nonrigid, highly interactive dynamics between the soliton and radiation in the Schrödinger setting. The results rely on delicate construction of approximate radiation, invariant-subspace decomposition of the linearized flow, and bootstrap arguments to propagate estimates backward to the blow-up time, with implications for the understanding of nonlocal, mass-critical quantum gauge theories.

Abstract

We construct finite energy blow-up solutions for the radial self-dual Chern-Simons-Schrödinger equation with a continuum of blow-up rates. Our result stands in stark contrast to the rigidity of blow-up of $H^{3}$ solutions proved by the first author for equivariant index $m \geq 1$, where the soliton-radiation interaction is too weak to admit the present blow-up scenarios. It is optimal (up to an endpoint) in terms of the range of blow-up rates and the regularity of the asymptotic profiles in view of the authors' previous proof of $H^{1}$ soliton resolution for the self-dual Chern-Simons-Schrödinger equation in any equivariance class. Our approach is a backward construction combined with modulation analysis, starting from prescribed asymptotic profiles and deriving the corresponding blow-up rates from their strong interaction with the soliton. In particular, our work may be seen as an adaptation of the method of Jendrej-Lawrie-Rodriguez (developed for energy critical equivariant wave maps) to the Schrödinger case. However, the Schrödinger nature of the equation (in particular, the lack of finite speed of propagation) and the optimal range (up to the $H^{1}$-endpoint) of our blow-up construction give rise to new challenges. Notably, the construction of (approximate) radiation from the prescribed asymptotic profile is one of our key novelties and might be of independent interest.

Blow-up dynamics for radial self-dual Chern-Simons-Schrödinger equation with prescribed asymptotic profile

TL;DR

The paper proves that in the radial, setting of the self-dual Chern-Simons-Schrödinger equation, finite-energy blow-up can occur with a continuum of rates driven by a prescribed asymptotic radiation profile with . The authors develop a backward blow-up construction combined with precise modulation analysis, constructing a radiation from the asymptotic profile and coupling it to a modulated soliton via a refined set of modulation parameters and , controlled by a nonlinear energy functional . This framework yields blow-up rates of the form for any (and the associated phase and modulation data), illustrating a nonrigid, highly interactive dynamics between the soliton and radiation in the Schrödinger setting. The results rely on delicate construction of approximate radiation, invariant-subspace decomposition of the linearized flow, and bootstrap arguments to propagate estimates backward to the blow-up time, with implications for the understanding of nonlocal, mass-critical quantum gauge theories.

Abstract

We construct finite energy blow-up solutions for the radial self-dual Chern-Simons-Schrödinger equation with a continuum of blow-up rates. Our result stands in stark contrast to the rigidity of blow-up of solutions proved by the first author for equivariant index , where the soliton-radiation interaction is too weak to admit the present blow-up scenarios. It is optimal (up to an endpoint) in terms of the range of blow-up rates and the regularity of the asymptotic profiles in view of the authors' previous proof of soliton resolution for the self-dual Chern-Simons-Schrödinger equation in any equivariance class. Our approach is a backward construction combined with modulation analysis, starting from prescribed asymptotic profiles and deriving the corresponding blow-up rates from their strong interaction with the soliton. In particular, our work may be seen as an adaptation of the method of Jendrej-Lawrie-Rodriguez (developed for energy critical equivariant wave maps) to the Schrödinger case. However, the Schrödinger nature of the equation (in particular, the lack of finite speed of propagation) and the optimal range (up to the -endpoint) of our blow-up construction give rise to new challenges. Notably, the construction of (approximate) radiation from the prescribed asymptotic profile is one of our key novelties and might be of independent interest.
Paper Structure (33 sections, 31 theorems, 510 equations)

This paper contains 33 sections, 31 theorems, 510 equations.

Table of Contents

  1. Introduction
  2. Self-dual Chern--Simons--Schrödinger equation
  3. Equivariant self-dual Chern--Simons--Schrödinger equation
  4. Known results
  5. Main result
  6. Strategy of the proof
  7. Construction of the associated radiation
  8. Blow-up construction
  9. Basic preliminaries
  10. Notation
  11. Equivariant Sobolev spaces
  12. Decomposition of the nonlinearity
  13. Asymptotic profile and approximate radiation
  14. Equations for the radiation
  15. Statement of the main result
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $m\geq0$; let $u$ be a $H_{m}^{1}$-solution to eq:CSS-m-equiv which blows up forwards in time at $T\in(0,\infty)$. Then, $u(t)$ admits the decomposition for some $\lambda(t)\in(0,\infty)$, $\gamma(t)\in\mathbb{R}/2\pi\mathbb{Z}$, and $z^{\ast}\in L^{2}$ with the following properties:

Theorems & Definitions (70)

  • Theorem 1.1: Soliton resolution for $H_{m}^{1}$ finite-time blow-up solutions KimKwonOh2022arXiv1
  • Theorem 1.2: Exotic finite-time blow-up solutions
  • Corollary 1.3: Infinite-time blow-up
  • Remark 1.4: Optimality
  • Remark 1.5: Strongly interacting regime
  • Remark 1.6: On the continuum of blow-up rates and regularity class
  • Remark 1.7: Comparison with the Krieger--Schlag--Tataru approach
  • Remark 1.8: On uniqueness
  • Proposition 3.1: Properties of the radiation
  • Lemma 3.2: Fundamental systems for $\mathcal{A}_{\mathfrak{m},\nu}$
  • ...and 60 more