A Liouville theorem for the Chern–Simons–Schrödinger equation

Benjamin Dodson

doi:10.3934/dcds.2023110

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2024, Volume 44, Issue 2: 447-461. Doi: 10.3934/dcds.2023110

This issue Previous Article Multilinear Wiener-Wintner type ergodic averages and its application Next Article Anisotropic singular Trudinger-Moser inequalities on the whole Euclidean space

A Liouville theorem for the Chern–Simons–Schrödinger equation

Benjamin Dodson

Johns Hopkins University, USA

Received: February 2023

Revised: September 2023

Early access: September 2023

Published: February 2024

The first author is supported by [NSF Grant DMS-2153750].

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Abstract

In this paper we prove a Liouville theorem for the Chern–Simons–Schrödinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [10] for the soliton resolution result in a weighted space.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q51.

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References

[1]	L. Bergé, A. de Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrodinger equation, Nonlinearity, 8 (1995), 235-253. doi: 10.1088/0951-7715/8/2/007.
[2]	G. Dunne, Self-Dual Chern-Simons Theories, Volume 36, Springer Science & Business Media, Berlin, 2009. doi: 10.1007/978-3-540-44777-1.
[3]	H. Huh, Energy solution to the Chern-Simons-Schrödinger equation, Abstract and Applied Analysis, 2013, Hidawi, Art. ID 590653, 7 pp. doi: 10.1155/2013/590653.
[4]	R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Physical Review D, 42 (1990), 3500-3513. doi: 10.1103/PhysRevD.42.3500.
[5]	R. Jackiw and S.-Y. Pi, Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Physical Review Letters, 64 (1990), 2969-2972. doi: 10.1103/PhysRevLett.64.2969.
[6]	R. Jackiw and S.-Y. Pi, Time-dependent Chern-Simons solitons and their quantization, Physical Review D, 44 (1991), 2524-2532. doi: 10.1103/PhysRevD.44.2524.
[7]	K. Kim and S. Kwon, On Pseudoconformal Blow-Up Solutions to the Self-Dual Chern-Simons-Schrödinger Equation: Existence, Uniqueness, and Instability, Mem. Amer. Math. Soc., 284 (2023), vi+128 pp, arXiv: 1909.01055. doi: 10.1090/memo/1409.
[8]	K. Kim and S. Kwon, Construction of blow-up manifolds to the equivariant self-dual Chern-Simons-Schrödinger equation, Ann. PDE, 9 (2023), Paper No. 6,129 pp, arXiv: 2009.02943. doi: 10.1007/s40818-023-00147-8.
[9]	K. Kim, S. Kwon and S.-J. Oh, Blow-up dynamics for smooth finite energy radial data solutions to the self-dual Chern-Simons-Schrödinger equation, preprint, 2020, arXiv: 2010.03252.
[10]	K. Kim, S. Kwon and S.-J. Oh, Soliton resolution for equivariant self-dual Chern-Simons-Schrödinger equation in weighted Sobolev class, preprint, 2022, arXiv: 2202.07314.
[11]	Z. M. Lim, Large data well-posedness in the energy space of the Chern–Simons–Schrödinger system, Journal of Differential Equations, 264 (2018), 2553-2597. doi: 10.1016/j.jde.2017.10.026.
[12]	Z. Li and B. Liu, On threshold solutions of the equivariant Chern–Simons–Schrödinger equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 39 (2022), 371–417. arXiv: 2010.09045. doi: 10.4171/aihpc/10.
[13]	B. Liu and P. Smith, Global wellposedness of the equivariant Chern–Simons–Schrödinger equation, Revista Matemática Iberoamericana, 32 (2016), 751-794. doi: 10.4171/RMI/898.
[14]	B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern–Simons–Schrödinger, International Mathematics Research Notices, 2014 (2014), 6341-6398. doi: 10.1093/imrn/rnt161.
[15]	Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg–de Vries equation, Journal de Mathématiques Pures et Appliquées, 79 (2000), 339-425. doi: 10.1016/S0021-7824(00)00159-8.
[16]	F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Mathematical Journal, 69 (1993), 427-454. doi: 10.1215/S0012-7094-93-06919-0.
[17]	S.-J. Oh and F. Pusateri, Decay and scattering for the Chern–Simons–Schrödinger equations, International Mathematics Research Notices, 2015 (2015), 13122-13147. doi: 10.1093/imrn/rnv093.