Global well-posedness of 3D axisymmetric MHD system with pure swirl magnetic field. (English) Zbl 1394.35327
Summary: In this paper, we consider the axisymmetric MHD system with nearly critical initial data having the special structure: \(u_0=u_0^r e_r+u^\theta_0 e_\theta +u_0^z e_z\), \(b_0=b_0^\theta e_\theta\). We prove that, this system is globally well-posed provided the scaling-invariant norms \(\| ru^\theta_0\|_{L^\infty}\), \(\| r^{-1}b^\theta_0\|_{L^{\frac{3}{2}}}\) are sufficiently small.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 35B07 | Axially symmetric solutions to PDEs |
| 42B25 | Maximal functions, Littlewood-Paley theory |
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