Publications
Home
Centers & Programs
Mathematical Challenges
Publications
- Title
- Blow-up Dynamics for Radial Self-Dual Chern-Simons-Schrödinger Equation with Prescribed Asymptotic Profile
- KIAS Author
- Oh, Sung-Jin,Oh, Sung-Jin
- Journal
- COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2026
- Archive
-
- Abstract
- We construct finite energy blow-up solutions for the radial self-dual Chern-Simons-Schr & ouml;dinger equation with a continuum of blow-up rates. Our result stands in stark contrast to the rigidity of blow-up of H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{3}$$\end{document} solutions proved by the first author for equivariant index m >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 1$$\end{document}, 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 H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{1}$$\end{document} soliton resolution for the self-dual Chern-Simons-Schr & ouml;dinger 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 Schr & ouml;dinger case. However, the Schr & ouml;dinger nature of the equation (in particular, the lack of finite speed of propagation) and the optimal range (up to the H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{1}$$\end{document}-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.
