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On global existence, energy decay and blow-up criteria for the Hall-MHD system. (English) Zbl 1328.35185
In this article the authors study the 3D incompressible Hall-Magnetohydrodynamics system. They show global existence of a unique solution in homogeneous Sobolev or Besov framework for small data together with various time decay properties of the solution. They also give two “Osgood type” blow-up criteria for local solutions on a finite interval \((0,T_*)\) when \(t\to T_*\) and they finally prove two Beale-Kato-Majda criteria for local solutions of a related model.

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: DOI
[1] Acheritogaray, M.; Degond, P.; Frouvelle, A.; Liu, J.-G., Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system, Kinet. Relat. Models, 4, 901-918, (2011) · Zbl 1251.35076
[2] Chae, D.; Degond, P.; Liu, J.-G., Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31, 555-565, (2014) · Zbl 1297.35064
[3] Chae, D.; Lee, J., On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations, 256, 3835-3858, (2014) · Zbl 1295.35122
[4] Chae, D.; Schonbek, M., On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255, 3971-3982, (2013) · Zbl 1291.35212
[5] Chae, D.; Wan, R.; Wu, J., Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion · Zbl 1327.35314
[6] Chae, D.; Weng, S., Singularity formation for the incompressible Hall-MHD equations without resistivity, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2015), in press · Zbl 1347.35199
[7] R. Wan, Global regularity for generalized Hall-MHD system, preprint.
[8] R. Wan, Y. Zhou, Low regularity well-posedness for the 3D generalized Hall-MHD system, preprint. · Zbl 1365.35219
[9] Benvenutti, M.; Ferreira, L., Existence and stability of global large strong solutions for the Hall-MHD system · Zbl 1389.35255
[10] Dumas, E.; Sueur, F., On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-magnetohydrodynamic equations, Comm. Math. Phys., 330, 1179-1225, (2014) · Zbl 1294.35094
[11] Fan, J.; Huang, S.; Nakamura, G., Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations, Appl. Math. Lett., 26, 963-967, (2013) · Zbl 1315.35164
[12] Fan, J.; Ozawa, T., Regularity criteria for Hall-magnetohydrodynamics and the space-time monopole equation in Lorenz gauge, Contemp. Math., 612, 81-89, (2014) · Zbl 1297.35068
[13] Grafakos, L., Modern Fourier analysis, Grad. Texts in Math., vol. 250, (2008), Springer
[14] Bahouri, H.; Chemin, J.; Danchin, R., Fourier analysis and nonlinear partial differential equations, (2011), Springer · Zbl 1227.35004
[15] Majda, A.; Bertozzi, A., Vorticity and incompressible flow, (2001), Cambridge University Press Cambridge, UK
[16] Stein, E., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton · Zbl 0207.13501
[17] Schonbek, M., \(L^2\) decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88, 209-222, (1985) · Zbl 0602.76031
[18] Schonbek, M., Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11, 733-763, (1986) · Zbl 0607.35071
[19] Danchin, R., Local theory in critical spaces for the compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26, 7-8, 1183-1233, (2001) · Zbl 1007.35071
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