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- W4328117319 abstract "We consider the self-dual Chern-Simons-Schrodinger 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^{igamma_{ast}}}{T-t}QBig(frac{r}{T-t}Big)to u^{ast}quadtext{as }tto T^{-}. ] When the equivariance index $mgeq1$, we construct a codimension one blow-up manifold, i.e. a codimension one set of initial data yielding pseudoconformal blow-up solutions. Moreover, when $mgeq3$, 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, Raphael, Rodnianski, and others. One of our crucial inputs is a remarkable conjugation identity, which enables the method of supersymmetric conjugates as like Schrodinger maps and wave maps. It suggests how we define adapted derivatives. More interestingly, it shows a deep connection with the Schrodinger 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." @default.
- W4328117319 created "2023-03-22" @default.
- W4328117319 creator A5022259689 @default.
- W4328117319 creator A5067102283 @default.
- W4328117319 date "2020-09-07" @default.
- W4328117319 modified "2023-10-18" @default.
- W4328117319 title "Construction of blow-up manifolds to the equivariant self-dual Chern-Simons-Schrodinger equation" @default.
- W4328117319 doi "https://doi.org/10.48550/arxiv.2009.02943" @default.
- W4328117319 hasPublicationYear "2020" @default.
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