Partial regularity for the 3D magneto-hydrodynamics system with hyper-dissipation. (English) Zbl 1320.35129
Summary: We prove that for the 3D MHD equations with hyper-dissipations \((-\Delta)^{\alpha} (1 < \alpha < 5/4)\) the Hausdorff dimension of singular set at the first blowing up time is at most \(5 - 4\alpha\), by means of physical and frequency localization, Bony’s paraproduct and Littlewood-Paley theory.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |
| 35Q30 | Navier-Stokes equations |
| 35R11 | Fractional partial differential equations |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
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