On the regularity of weak solutions to the magnetohydrodynamic equations

By Cheng He, Zhouping Xin. 2005

497 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a199d3784b1b8f367264caadb58794e2d7339597

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In Applied Mathematics

1. Jiahong Wu
2. Changxing Miao
3. Sadek Gala
4. Zhifei Zhang
5. Zhen Lei
6. Qionglei Chen
7. Dongho Chae
8. Baoquan Yuan

In Mathematical Physics

1. Jiahong Wu
2. Changxing Miao
3. Sadek Gala
4. Zhifei Zhang
5. Zhen Lei
6. Qionglei Chen
7. Dongho Chae
8. Baoquan Yuan

Citations in the corpus

• Regularity criteria for the solutions to the 3D MHD equations in the multiplier space (2010, methodology): “as 0 ≤ r 4 ,11].”
• Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations (2015, background): “his solution is zero or not. We first present the following simple Liouville theorem for MHD equations, by prescribing the L3 integrability only on u. Similar regularity criteria has been developed in [11] and [20] for unsteady MHD equations. Theorem 3.1. Let (u,h) be a smooth solution to (3.1) in R3”
• Regularity criteria for the generalized viscous MHD equations (2007, background): “So a lot of literatures are devoted to find regularity criteria or prove partial regularity for these equations, such as [1–4,7,9,12,14,17–20,22] for Navier–Stokes equations, and [5,10,16,21] for MHD equations.”
• On axially symmetric incompressible magnetohydrodynamics in three dimensions (2012, background): “We also emphasis the partial regularity theory and Serrin type criterions in [6, 7], and various blowup criterions in [1, 4] (see also the reference therein).”
• Singularity formation for the incompressible Hall-MHD equations without resistivity (2013, background): “There are many mathematical results on MHD system, for the existence of global weak solutions [9, 18], regularity criterion [11, 12] and global smooth small solutions [17, 21].”
• On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space (2010, background): “For the magneto-hydrodynamic equations, He and Xin [12] first studied and established regularity criteria imposed only on the velocity field u or its gradient ∇ u .”
• Regularity Criteria for the Generalized MHD Equations (2008, background): “Noticing the experimental and numerical results that appear to indicate the dominant role of the velocity field (Hasegawa, 1985; Politano et al., 1995), He and Xin were able to derive several regularity criteria for the 3D MHD equations that put no constraint on b (He and Xin, 2005).”
• On the Regularity Criterion of Weak Solution for the 3D Viscous Magneto-Hydrodynamics Equations (2007, methodology): “Recently, He, Xin[12], and Zhou[24] have proved some regularity criteria to the MHD equations which do not impose any condition on the magnetic field b. Precisely, they showed that the weak solution remains smooth on (0, T ] × R3 if the velocity u satisfies one of the following conditions u ∈ Lq(0, ”
• Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows (2012, unknown): “Following He-Xin [10], we prove (3.1).”
• Global Regularity for a Class of Generalized Magnetohydrodynamic Equations (2011, unknown): “, [3,4,8–11,13–15,17,20–23,25]).”
• The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations (2007, background): “The authors wish to thank Prof.Zhouping Xin for stimulating discussion about this problem.”