Stability of 2D MHD with Only Horizontal Dissipation near a Background Magnetic Field on ℝ2
Abstract
Whether the two-dimensional Navier--Stokes equations with dissipation in only one direction are stable in H2 on the whole plane is a well-known open problem: without transverse dissipation, a cubic interaction fails to be time-integrable. We show that a magnetic field removes this obstruction. For the 2D incompressible MHD equations on ℝ2 in which both the viscosity ν and the magnetic diffusivity μ act only in the horizontal direction, we prove the global nonlinear H2 stability of every background field γe2 aligned with the non-dissipative direction, with a stability threshold that is explicit in ν, μ and γ. The stabilizing mechanism is genuinely nonlinear: at the linear level the background field produces no damping at all in the vertical direction. Instead, in Elsässer variables the obstruction becomes an interaction between Alfvén waves travelling in opposite directions, and we identify nonlinear dissipative quantities that measure the overlap of these waves and are time-integrable although neither wave decays. We then sharpen the threshold by a space--time div--curl estimate for balance laws in the (t,x2)-plane. For ν=μ≤1≤γ, the admissible perturbations are of size ν2/3γ1/3, improving the direct scale ν and growing without bound as the background field strengthens.
Related
- Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation
- Global Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation
- Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping
- Stabilization by a background magnetic field: global well-posedness of the full compressible viscous non-resistive MHD system without heat-conductivity
- Global well-posedness for the 2D primitive equations with subcritical horizontal dissipation
BibTeX
@misc{wu2026stability,
title = {{Stability of 2D MHD with Only Horizontal Dissipation near a Background Magnetic Field on $\mathbbR^2$}},
author = {Jiahong Wu and Mengxin Yan and Ning Zhu and Yi Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28308},
eprint = {2610.28308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28308},
}
RIS
TY - UNPB
TI - Stability of 2D MHD with Only Horizontal Dissipation near a Background Magnetic Field on ℝ^2
AU - Jiahong Wu
AU - Mengxin Yan
AU - Ning Zhu
AU - Yi Zhu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28308
UR - https://arxiv.org/abs/2610.28308
AB - Whether the two-dimensional Navier--Stokes equations with dissipation in only one direction are stable in H^2 on the whole plane is a well-known open problem: without transverse dissipation, a cubic interaction fails to be time-integrable. We show that a magnetic field removes this obstruction. For the 2D incompressible MHD equations on ℝ^2 in which both the viscosity ν and the magnetic diffusivity μ act only in the horizontal direction, we prove the global nonlinear H^2 stability of every background field γe_2 aligned with the non-dissipative direction, with a stability threshold that is explicit in ν, μ and γ. The stabilizing mechanism is genuinely nonlinear: at the linear level the background field produces no damping at all in the vertical direction. Instead, in Elsässer variables the obstruction becomes an interaction between Alfvén waves travelling in opposite directions, and we identify nonlinear dissipative quantities that measure the overlap of these waves and are time-integrable although neither wave decays. We then sharpen the threshold by a space--time div--curl estimate for balance laws in the (t,x_2)-plane. For ν=μ≤1≤γ, the admissible perturbations are of size ν^2/3γ^1/3, improving the direct scale ν and growing without bound as the background field strengthens.
ER -