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On Pseudoconformal Blow-Up Solutions to the Self-Dual Chern-Simons-Schrödinger Equation: Existence, Uniqueness, and Instability

About this Title

Kihyun Kim and Soonsik Kwon

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 284, Number 1409
ISBNs: 978-1-4704-6120-1 (print); 978-1-4704-7447-8 (online)
DOI: https://doi.org/10.1090/memo/1409
Published electronically: March 21, 2023
Keywords: Chern-Simons-Schrödinger equation, blow-up, pseudoconformal, self-duality, rotational instability

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Table of Contents

Chapters

• 1. Introduction
• 2. Notations and preliminaries
• 3. Linearization of under equivariance
• 4. Profile $Q^{(\eta )}$
• 5. Setup for modulation analysis
• 6. Proof of bootstrap Lemma
• 7. Conditional uniqueness
• A. Equivariant Sobolev spaces
• B. Equivariant local theory

Abstract

We consider the self-dual Chern-Simons-Schrödinger equation (CSS), also known as a gauged nonlinear Schrödinger equation (NLS). CSS is $L^{2}$-critical, admits solitons, and has the pseudoconformal symmetry. These features are similar to the $L^{2}$-critical NLS. In this work, we consider pseudoconformal blow-up solutions under $m$-equivariance, $m\geq 1$. Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution $u$ with given asymptotic profile $z^{\ast }$: \[ \Big [u(t,r)-\frac {1}{|t|}Q\Big (\frac {r}{|t|}\Big )e^{-i\frac {r^{2}}{4|t|}}\Big ]e^{im\theta }\to z^{\ast }\qquad \text {in }H^{1} \] as $t\to 0^{-}$, where $Q(r)e^{im\theta }$ is a static solution. Secondly, we show that such blow-up solutions are unique in a suitable class. Lastly, yet most importantly, we exhibit an instability mechanism of $u$. We construct a continuous family of solutions $u^{(\eta )}$, $0\leq \eta \ll 1$, such that $u^{(0)}=u$ and for $\eta >0$, $u^{(\eta )}$ is a global scattering solution. Moreover, we exhibit a rotational instability as $\eta \to 0^{+}$: $u^{(\eta )}$ takes an abrupt spatial rotation by the angle \[ \Big (\frac {m+1}{m}\Big )\pi \] on the time interval $|t|\lesssim \eta$.

We are inspired by works in the $L^{2}$-critical NLS. In the seminal work of Bourgain and Wang (1997), they constructed such pseudoconformal blow-up solutions. Merle, Raphaël, and Szeftel (2013) showed an instability of Bourgain-Wang solutions. Although CSS shares many features with NLS, there are essential differences and obstacles over NLS. Firstly, the soliton profile to CSS shows a slow polynomial decay $r^{-(m+2)}$. This causes many technical issues for small $m$. Secondly, due to the nonlocal nonlinearities, there are strong long-range interactions even between functions in far different scales. This leads to a nontrivial correction of our blow-up ansatz. Lastly, the instability mechanism of CSS is completely different from that of NLS. Here, the phase rotation is the main source of the instability. On the other hand, the self-dual structure of CSS is our sponsor to overcome these obstacles. We exploited the self-duality in many places such as the linearization, spectral properties, and construction of modified profiles.