Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics
| dc.creator | Chen, Gui-Qiang | |
| dc.creator | Wang, Ya-Guang | |
| dc.date | 2006-10-07 | |
| dc.date.accessioned | 2026-07-07T07:28:50Z | |
| dc.date.available | 2026-07-07T07:28:50Z | |
| dc.description | Compressible vortex sheets are fundamental waves in entropy solutions to the multidimensional hyperbolic systems of conservation laws. For the Euler equations in 2-D gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number $M>\sqrt{2}$ and instability when $M<\sqrt{2}$; and Artola-Majda's analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when $M>\sqrt{2}$. For the Euler equations in 3-D, every compressible vortex sheet is violently unstable and this violent instability is the analogue of the Kelvin-Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in 3-D, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in 3-D magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss-Lopatinskii condition. In this paper, we develop a nonlinear approach to deal with these difficulties in 3-D MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients without loss of the order. Then we exploit these results to develop a suitable iteration scheme of Nash-Moser-Hörmander type and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in the 3-D MHD. | |
| dc.description | 27 pages | |
| dc.identifier | https://arxiv.org/abs/math/0610251 | |
| dc.identifier | http://arxiv.org/abs/math/0610251 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/117878 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Mathematical Physics | |
| dc.subject | 35L65;35L60;76W05;76E25;35R35;76N10;35L67 | |
| dc.title | Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics | |
| dc.type | text |