Construction of Blow-Up Manifolds to the Equivariant Self-dual Chern–Simons–Schrödinger Equation

Abstract

We consider the self-dual Chern–Simons–Schrödinger equation (CSS) under equivariance symmetry. Among others, (CSS) has a static solution Q and the pseudoconformal symmetry. We study the quantitative description of pseudoconformal blow-up solutions u such that

$$\begin{aligned} u(t,r)-\frac{e^{i\gamma _{*}}}{T-t}Q\Big (\frac{r}{T-t}\Big )\rightarrow u^{*}\quad \text {as }t\rightarrow T^{-}. \end{aligned}$$

When the equivariance index \(m\ge 1\), we construct a set of initial data (under a codimension one condition) yielding pseudoconformal blow-up solutions. Moreover, when \(m\ge 3\), we establish the codimension one property and Lipschitz regularity of the initial data set, which we call the blow-up manifold. This is a forward construction of blow-up solutions, as opposed to authors’ previous work [25], 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 established in this paper is expected to be optimal. We perform the modulation analysis with a robust energy method developed by Merle, Raphaël, Rodnianski, and others. One of our crucial inputs is a remarkable conjugation identity, which (with self-duality) enables the method of supersymmetric conjugates as like Schrödinger maps and wave maps. It suggests how we proceed to higher order derivatives while keeping the Hamiltonian form and construct adapted function spaces with their coercivity relations. More interestingly, it shows a deep connection with the Schrödinger maps at the linearized level and allows us to find a repulsivity structure for higher order derivatives.