Blowup dynamics for mass critical Half-wave equation in 2D

TL;DR

This paper studies the blow-up behavior of solutions to a 2D mass-critical half-wave equation, establishing non-radial ground state solutions that blow up at a specific rate as time approaches zero.

Contribution

It constructs non-radial ground state blow-up solutions for the 2D mass-critical half-wave equation with a precise blow-up speed.

Findings

Existence of non-radial ground state blow-up solutions.

Blow-up speed characterized as  D^{1/2} u(t)_{L^2}  1/|t| as t  0^-.

Solutions are constructed for initial data in H^s with s in (/4, 1).

Abstract

We consider the half-wave equation $iu_t=Du-|u|u$ in two dimensions. For the initial data $u_0(x)\in H^{s}(\mathbb{R}^2)$, $s\in\left(\frac{3}{4},1\right)$, we obtain the non-radial ground state mass blow-up solutions with the blow-up speed $\|D^{\frac{1}{2}}u(t)\|_{L^2}\sim\frac{1}{|t|}$ as $t\to 0^-$.