论文标题
构建$ l^2 $至关重要的半波方程的多可能性爆破解决方案
Construction of multi-bubble blow-up solutions to the $L^2$-critical half-wave equation
论文作者
论文摘要
本文涉及$ l^2 $至关重要的半波动方程的冒泡现象。鉴于任意有限的许多不同的奇异性，我们构建了精确地集中在这些奇点上的爆炸解决方案。这为半波方程式提供了多泡解决方案的第一个示例。特别是，该解决方案表现出质量量化特性。我们的证明策略借鉴了单泡案件的\ cite {k-l-r}中的调制方法，并探索了\ cite {csz21，rsz21}中的本地化技术，用于非线性schrödinger方程（NLS）的气泡解决方案。但是，与单泡或NLS病例不同，不同的气泡在维度1中表现出最强的相互作用。 In order to get sharp estimates to control strong interactions, as well as nonlocal effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half-wave operator, and find that there exists a narrow room between the orders $|t|^{2+}$ and $|t|^{3-}$ for the remainder in the geometrical decomposition.基于此，引入了一种新型的自举方案来解决多泡的非本地结构。
This paper concerns the bubbling phenomena for the $L^2$-critical half-wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow-up solutions concentrating exactly at these singularities. This provides the first examples of multi-bubble solutions for the half-wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in \cite{K-L-R} for the single-bubble case, and explores the localization techniques in \cite{CSZ21,RSZ21} for bubbling solutions to nonlinear Schrödinger equations (NLS). However, unlike the single-bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control strong interactions, as well as nonlocal effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half-wave operator, and find that there exists a narrow room between the orders $|t|^{2+}$ and $|t|^{3-}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi-bubble non-local structure.
