Stability for a system of the 2D incompressible magneto-micropolar fluid equations with partial mixed dissipation. (English) Zbl 1542.35322
Summary: This paper focuses on the 2D incompressible anisotropic magneto-micropolar fluid equations with vertical dissipation, horizontal magnetic diffusion, and horizontal vortex viscosity. The goal is to investigate the stability of perturbations near a background magnetic field in the 2D magneto-micropolar fluid equations. Two main results are obtained. The first result is based on the linear system. Global existence for any large initial data and asymptotic linear stability are established. The second result explores stability for the nonlinear system. It is proven that if the initial data are sufficiently small, then the solution for some perturbations near a background magnetic field remains small. Additionally, the long-time behaviour of the solution is presented. The most challenging terms in the proof are the linear terms in the velocity equation and the micro-rotation equation that will grow with respect to time \(t\). We are able to find some background fields to control the growth of the linear terms. Our results reveal that some background fields can stabilise electrically conducting fluids.
{© 2024 IOP Publishing Ltd & London Mathematical Society}
{© 2024 IOP Publishing Ltd & London Mathematical Society}
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76E25 | Stability and instability of magnetohydrodynamic and electrohydrodynamic flows |
| 76U05 | General theory of rotating fluids |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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