Stability for a system of 2D incompressible anisotropic magnetohydrodynamic equations. (English) Zbl 1514.35358
Summary: This paper investigates the global well-posedness and the stability of perturbations near a background magnetic field on the 2D incompressible anisotropic magnetohydrodynamic equations. More precisely, we consider the system with partial mixed velocity dissipations and horizontal magnetic diffusion. Two goals are achieved. First, we obtain the global well-posedness and the \(H^2\)-stability for the nonlinear MHD equations. The approach is to apply the bootstrapping argument. Efforts are devoted to the a priori estimate of a energy functional. Second, we establish the long-time behavior of explicit decay rates of the solution in homogeneous Sobolev spaces \(\dot{H}^s\) for the linear system under the suitable assumptions for the initial data. Our work proves that any perturbations near a background magnetic field remain asymptotically stable.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76E25 | Stability and instability of magnetohydrodynamic and electrohydrodynamic flows |
| 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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