论文标题
构造爆炸歧管，为自动偶偶联的Chern-simons-schrödinger方程
Construction of blow-up manifolds to the equivariant self-dual Chern-Simons-Schrödinger equation
论文作者
论文摘要
我们考虑在均衡对称性下的自我偶义Chern-simons-Schrödinger方程（CSS）。除其他外，（CSS）具有静态解决方案$ q $和伪形式对称性。我们研究伪形式爆炸解决方案$ u $的条件稳定性，以便\ [u（t，r） - \ frac {e^{iγ_{\ ast}}} {t-t-t} q \ big（\ big big（\ frac {r} } t \ to t^{ - }。 \]当均衡索引$ m \ geq1 $时，我们构造了一个编纂一个爆炸歧管，即一组编成一组初始数据，产生了伪符号的爆炸解决方案。此外，当$ M \ geq3 $时，我们建立了构造的爆破歧管（条件稳定性）的Lipschitz规律性。 与作者先前的作品[25]（ARXIV：1909.01055）相比，这是爆炸解决方案的正向结构，这是具有规定的渐近型材料的向后构造。鉴于[25]的不稳定性结果，编辑态度有望是最佳的。 我们通过Merle，Raphaël，Rodnianski等开发的强大能量方法进行调制分析。我们的关键输入之一是一个显着的共轭身份，它可以像Schrödinger图和波浪图一样进行超对称共轭物的方法。它表明我们如何定义适应的衍生物。更有趣的是，它与线性化级别的Schrödinger图显示了深厚的联系，并使我们能够找到排斥性结构。 在许多地方，非本地非线性成为障碍。例如，我们需要从非本地非线性中捕获非扰动的贡献，并以[25]的精神将其吸收到相位校正中。更重要的是，我们需要采取非线性途径来构建修改的配置文件。这是从[25]提出的，并且由于自我为偶性而获得。从中，我们也认识到稳定的模式和不稳定模式。
We consider the self-dual Chern-Simons-Schrödinger 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 \[ u(t,r)-\frac{e^{iγ_{\ast}}}{T-t}Q\Big(\frac{r}{T-t}\Big)\to u^{\ast}\quad\text{as }t\to T^{-}. \] 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, Raphaël, Rodnianski, and others. One of our crucial inputs is a remarkable conjugation identity, which enables the method of supersymmetric conjugates as like Schrödinger maps and wave maps. It suggests how we define adapted derivatives. More interestingly, it shows a deep connection with the Schrödinger 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.
