Effect of parallel magnetic field on the compressible Kelvin-Helmholtz problem

Binqiang Xie; Jiakai Ma; Binqiang Xie; Jiakai Ma

doi:10.3934/cam.2026014

Communications in Analysis and Mechanics

2026, Volume 18, Issue 2: 366-399. doi: 10.3934/cam.2026014

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Research article

Effect of parallel magnetic field on the compressible Kelvin-Helmholtz problem

Binqiang Xie ^,,
Jiakai Ma

School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510006, China

Received: 30 July 2025 Revised: 05 January 2026 Accepted: 26 January 2026 Published: 21 April 2026
76W05, 35Q35, 35D05, 76X05

In this paper, we present an analysis of the KH instability in 2D magnetohydrodynamic (MHD) flows, providing rigorous confirmation that a parallel magnetic field can have a destabilizing effect on this instability. When the Mach number $ M : = \frac{\dot{v}_{1}^{+}}{c} $ lies strictly between $ M_{\mathrm{low}} : = \sqrt{1-\sqrt{\frac{1-\beta}{1+\beta}}} + \epsilon_0 $ and $ M_{\mathrm{upp}} : = \sqrt{1+\sqrt{\frac{1-\beta}{1+\beta}}} - \epsilon_0 $, where $ \beta : = \frac{c_A^2}{c^2} $ and $ \epsilon_0 > 0 $ is a small but fixed constant, we prove the linear and nonlinear ill-posedness of the KH problem for compressible MHD flows.
- free surface,
- Kelvin-Helmholtz instability,
- two-dimensional MHD flows,
- parallel magnetic field
Citation: Binqiang Xie, Jiakai Ma. Effect of parallel magnetic field on the compressible Kelvin-Helmholtz problem[J]. Communications in Analysis and Mechanics, 2026, 18(2): 366-399. doi: 10.3934/cam.2026014

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Abstract

In this paper, we present an analysis of the KH instability in 2D magnetohydrodynamic (MHD) flows, providing rigorous confirmation that a parallel magnetic field can have a destabilizing effect on this instability. When the Mach number $ M : = \frac{\dot{v}_{1}^{+}}{c} $ lies strictly between $ M_{\mathrm{low}} : = \sqrt{1-\sqrt{\frac{1-\beta}{1+\beta}}} + \epsilon_0 $ and $ M_{\mathrm{upp}} : = \sqrt{1+\sqrt{\frac{1-\beta}{1+\beta}}} - \epsilon_0 $, where $ \beta : = \frac{c_A^2}{c^2} $ and $ \epsilon_0 > 0 $ is a small but fixed constant, we prove the linear and nonlinear ill-posedness of the KH problem for compressible MHD flows.

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