Construction of multi-bubble blow-up solutions to the L 2 $L^2$ -critical half-wave equation

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 Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single-bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non-linear 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 these interactions, as well as non-local 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 ${\left|t\right|}^{2+}$ and ${\left|t\right|}^{3-}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi-bubble non-local structure.