On the global solution of a 3-D MHD system with initial data near equilibrium. (English) Zbl 1372.35229
Summary: In this paper, we prove the global existence of smooth solutions to the three-dimensional incompressible magnetohydrodynamical system with initial data close enough to the equilibrium state, \((e_{3},0)\). Compared with previous works by F. Lin et al. [J. Differ. Equations 259, No. 10, 5440–5485 (2015; Zbl 1321.35138)] and by L. Xu and P. Zhang [SIAM J. Math. Anal. 47, No. 1, 26–65 (2015; Zbl 1352.35099)], here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is nondegenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in the earlier works. By using the Frobenius theorem and anisotropic Littlewood-Paley theory for the Lagrangian formulation of the system, we achieve the global \(L^1\)-in-time Lipschitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large-time decay rate of the solution is also obtained.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
global existence; smooth solutions; incompressible magnetohydrodynamical system; Littlewood-Paley theory; large-time decayReferences:
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