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Stability of Couette flow for the 2D magnetohydrodynamic equations with vertical dissipation
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 15 July 2026, pp. 1-29
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This paper examines the stability of Couette flow in the two-dimensional incompressible magnetohydrodynamics system with only vertical dissipation. Our analysis departs from previous works by avoiding coordinate transformations and instead exploiting the special structure of the magnetic stream function. A key point is that the equation for the magnetic stream function
$\phi$ avoids unfavourable linear terms present in the magnetic field formulation, which allows us to uncover new stability mechanisms specific to the anisotropically dissipative regime. Making use of this special structure, we establish global uniform regularity of solutions near background magnetic fields and provide a rigorous resolution of the associated stability problem.
