BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity

By Zhen Lei, Yi Zhou. 2009

193 citations recorded by the source corpus.

In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/950edbe57d507479b86425fa22b0feb901fb794e

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In Applied Mathematics

1. Zhen Lei
2. Jiahong Wu
3. Nader Masmoudi
4. Song Jiang
5. Chongsheng Cao
6. D. Regmi
7. Baoquan Yuan
8. Linna Bai

In Computational Mechanics

1. Song Jiang
2. Jiahong Wu
3. Fucai Li
4. Qiangchang Ju
5. Nader Masmoudi
6. Zhen Lei
7. Baoquan Yuan
8. Chongsheng Cao

In Mathematical Physics

1. Zhen Lei
2. Jiahong Wu
3. Chongsheng Cao
4. Nader Masmoudi
5. D. Regmi
6. Baoquan Yuan
7. Linna Bai
8. Song Jiang

Citations in the corpus

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• Remarks on the blowup criteria for Oldroyd models (2009, unknown): “+kfkL2 +kfk BMO ln(e+kfkC˙β ) , RT 0 k∇g(s,·)kL∞ds≤ C 1+ RT 0 kg(s,·)kL2ds + sup q RT 0 k∆ ∇g(s,·)kL∞dsln e+ RT 0 k∇g(s,·)kC˙βds . (2.3) Proof. The first inequality is well-known. For example, see [2, 13, 15]. To prove the 5 second inequality, we use the Littlewood-Paley theory to compute that Z T ”
• Global regularity results for the 2D Boussinesq equations with vertical dissipation (2011, unknown): “To obtain higher global regularity, we adopt an argument of [27].”
• The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion (2013, background): “A partial answer for the case when ν1 = ν2 = 0, η1 > 0 and η2 > 0 was obtained in [1] and [5].”
• Global well posedness for the Maxwell-Navier-Stokes system in 2D (2010, background): “Moreover, most of the mathematical works dealing with existence, uniqueness [13,30,1,2], blow-up criterion [6,20], regularity criteria [15,33,32], long time behavior or inviscid and non-resistive limits [12] or related problems [22] also make this assumption and hence remove the displacement current”
• On axially symmetric incompressible magnetohydrodynamics in three dimensions (2012, background): “lso proved the global well-posedness in 2D case. A nontrivial blowup criterion for the perfect resistive MHD (1.7) was established in terms of only L1 t(BMO) norm of vorticity of the velocity field in [10]. Recently, Lin, Xu and Zhang [11] made an important progress on the global well-posedness of cl”
• Incompressible Limit of the Compressible Magnetohydrodynamic Equations with Periodic Boundary Conditions (2009, background): “incompressible MHD equations correspond to the case of turbulent flow with very high Reynolds number (where the viscosity of flow can be ignored, see [16]).”