On the weakly nonlinear Kelvin–Helmholtz instability of current-vortex sheets

The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo (J Hyperbolic Differ Equ 8(4):691–726, 2011) have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. The local-in-time existence of smooth solutions to the Cauchy problem for such amplitude equation was already proven in Morando et al. (J Math Pures Appl 105(4):490–536, 2016; J Hyperbolic Differ Equ 14(2):193–248, 2017), under a suitable stability condition. The aim of the present note is to provide a new proof of the existence result of the solution, with optimal regularity with respect to the initial data. The existence of the solution follows from a fixed point argument, where the main ingredient is the a priori estimate of Qφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal Q\left[ \varphi \right] $$\end{document} proven in Morando et al. (J Hyperbolic Differ Equ 14(2):193–248, 2017).