Construction of blow-up manifolds to the equivariant self-dual Chern-Simons-Schr\"odinger equation

Abstract

We consider the self-dual Chern-Simons-Schr\"odinger equation (CSS) under equivariance symmetry. Among others, (CSS) has a static solution $Q$ and pseudoconformal symmetry. We study the conditional stability of pseudoconformal blow-up solutions $u$ such that \[ u(t,r)-\frac{e^{i\gamma_{\ast}}}{T-t}Q\Big(\frac{r}{T-t}\Big)\to u^{\ast}\quad\text{as }t\to T^{-}. \] When the equivariance index $m\geq1$, we construct a codimension one blow-up manifold, i.e. a codimension one set of initial data yielding pseudoconformal blow-up solutions. Moreover, when $m\geq3$, we establish the Lipschitz regularity of the constructed blow-up manifold (the conditional stability). This is a forward construction of blow-up solutions, as opposed to authors' previous work [25] (arXiv:1909.01055), which is a backward construction of blow-up solutions with prescribed asymptotic profiles. In view of the instability result of [25], the codimension one condition is expected to be optimal. We perform the modulation analysis with a robust energy method developed by Merle, Rapha\"el, Rodnianski, and others. One of our crucial inputs is a remarkable conjugation identity, which enables the method of supersymmetric conjugates as like Schr\"odinger maps and wave maps. It suggests how we define adapted derivatives. More interestingly, it shows a deep connection with the Schr\"odinger maps at the linearized level and allows us to find a repulsivity structure. The nonlocal nonlinearities become obstacles in many places. For instance, we need to capture non-perturbative contributions from the nonlocal nonlinearities and absorb them into phase corrections in a spirit of [25]. More importantly, we need to take a nonlinear pathway to construct modified profiles. This is suggested from [25] and becomes available thanks to the self-duality. From this, we also recognize the stable modes and unstable modes.