Blow-up dynamics for smooth finite energy radial data solutions to the self-dual Chern-Simons-Schr\"odinger equation

Abstract

We consider the finite-time blow-up dynamics of solutions to the self-dual Chern-Simons-Schr\"odinger (CSS) equation (also referred to as the Jackiw-Pi model) near the radial soliton $Q$ with the least $L^{2}$-norm (ground state). While a formal application of pseudoconformal symmetry to $Q$ gives rise to an $L^{2}$-continuous curve of initial data sets whose solutions blow up in finite time, they all have infinite energy due to the slow spatial decay of $Q$. In this paper, we exhibit initial data sets that are smooth finite energy radial perturbations of $Q$, whose solutions blow up in finite time. Interestingly, their blow-up rate differs from the pseudoconformal rate by a power of logarithm. Applying pseudoconformal symmetry in reverse, this also yields a first example of an infinite-time blow-up solution, whose blow-up profile contracts at a logarithmic rate. Our analysis builds upon the ideas of previous works of the first two authors on (CSS) [21,22], as well as the celebrated works on energy-critical geometric equations by Merle, Rapha\"el, and Rodnianski [33,38]. A notable feature of this paper is a systematic use of nonlinear covariant conjugations by the covariant Cauchy-Riemann operators in all parts of the argument. This not only overcomes the nonlocality of the problem, which is the principal challenge for (CSS), but also simplifies the structure of nonlinearity arising in the proof.