Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

By Chongsheng Cao, Jiahong Wu. 2009

450 citations recorded by the source corpus.

Abstract Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5ec8f3ae3cbc50e2ab187ec47c62fa13e92c7b49

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In Applied Mathematics

1. Zhifei Zhang
2. Zhen Lei
3. Ping Zhang
4. Jiahong Wu
5. Chongsheng Cao
6. Hammadi Abidi
7. D. Regmi
8. Baoquan Yuan

In Control and Systems Engineering

1. Zhen Lei
2. Zhifei Zhang
3. Jiahong Wu
4. D. Regmi
5. Ping Zhang
6. Hammadi Abidi
7. Bo‐Qing Dong
8. Chongsheng Cao

In Mathematical Physics

1. Zhifei Zhang
2. Zhen Lei
3. Jiahong Wu
4. Chongsheng Cao
5. Hammadi Abidi
6. Ping Zhang
7. D. Regmi
8. Baoquan Yuan

Citations in the corpus

• Global small solutions to 2-D incompressible MHD system (2013, unknown): “With mixed partial dissipation and additional (artificial) magnetic diffusion in the 2-D MHD system, Cao and Wu [5] (see also [6]) proved its global wellposedness for any data in H2(R(2)).”
• Global regularity of the 2D micropolar fluid flows with zero angular viscosity (2010, result): “6) with zero magnetic diffusivity κ = 0 or zero kinetic viscosity ν = 0 is still open although partial results are recently obtained by Cao and Wu [4].”
• Remarks on global regularity of 2D generalized MHD equations (2013, background): “2) has a global H 1 weak solution [2, 5].”
• The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion (2013, background): “In order to obtain suitable bounds for the terms on the right, we need the anisotropic Sobolev inequalities stated in the following lemma (see [1]).”
• On the Global Solution of a 3‐D MHD System with Initial Data near Equilibrium (2015, background): “With mixed partial dissipation and additional magnetic diffusion in the two-dimensional MHD system, Cao and Wu [5] (see also [6]) proved that such a system is globally well-posed for any data in H2(R(2)).”
• Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation (2011, unknown): “This lemma generalizes an inequality in [6].”
• Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium (2019, unknown): “, [7, 8]).”
• On axially symmetric incompressible magnetohydrodynamics in three dimensions (2012, background): “Cao and Wu [2] proved the global regularity of 2D resistive MHD with partial viscosity and resistivity (see also [3] and the references therein).”
• Global Well-Posedness of the Incompressible Magnetohydrodynamics (2016, background): “[14] obtained the global well-posedness in the two-dimensional case for small initial u and H − e in appropriate Sobolev spaces (see also some further results in [5,8,19,21]).”