# zbMATH — the first resource for mathematics

On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics. (English) Zbl 1295.35122
Summary: In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corresponding result in [D. Chae et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 3, 555–565 (2014; Zbl 1297.35064)], the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estimate in terms of $$\dot B_{2,1}^s$$ norm is essential for this result. Contrary to the usual MHD the global well-posedness in the $$2\frac{1}{2}$$ dimensional Hall-MHD is wide open.

##### MSC:
 35B44 Blow-up in context of PDEs 35L60 First-order nonlinear hyperbolic equations 35K55 Nonlinear parabolic equations 76W05 Magnetohydrodynamics and electrohydrodynamics 35Q35 PDEs in connection with fluid mechanics 35Q60 PDEs in connection with optics and electromagnetic theory
well-posedness
Full Text:
##### References:
 [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] Balbus, S. A.; Terquem, C., Linear analysis of the Hall effect in protostellar disks, Astrophys. J., 552, 235-247, (2001) [3] Berselli, L. C., On a regularity criterion for the 3D Navier-Stokes equations, Differential Integral Equations, 15, 1129-1137, (2002) · Zbl 1034.35087 [4] Chae, D.; Degond, P.; Liu, J.-G., Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincare Anal. Non Lineaire, (2013), in press [5] Chae, D.; Lee, J., Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Comm. Math. Phys., 233, 297-311, (2003) · Zbl 1019.86002 [6] Chae, D.; Schonbek, M., On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255, 3971-3982, (2013) · Zbl 1291.35212 [7] Forbes, T. G., Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 62, 15-36, (1991) [8] Homann, H.; Grauer, R., Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Phys. D, 208, 59-72, (2005) · Zbl 1154.76392 [9] Kenig, C.; Ponce, G.; Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4, 323-347, (1991) · Zbl 0737.35102 [10] Kozono, H.; Taniuchi, Y., Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235, 173-194, (2000) · Zbl 0970.35099 [11] Lighthill, M. J., Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Philos. Trans. R. Soc. Lond. Ser. A, 252, 397-430, (1960) · Zbl 0097.20806 [12] Majda, A. J.; Bertozzi, A. L., Vorticity and incompressible flow, Cambridge Texts Appl. Math., (2002), Cambridge · Zbl 0983.76001 [13] Mininni, P. D.; Gómez, D. O.; Mahajan, S. M., Dynamo action in magnetohydrodynamics and Hall magnetohydrodynamics, Astrophys. J., 587, 472-481, (2003) [14] Ogawa, T., Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34, 1318-1330, (2003) · Zbl 1036.35082 [15] Serrin, J., The initial value problem for the Navier-Stokes equations, (Langer, R. E, Nonlinear Probl., Proc. Sympos., Madison 1962, (1963), Univ. Wisconsin Press Madison), 69-98 [16] Shalybkov, D. A.; Urpin, V. A., The Hall effect and the decay of magnetic fields, Astron. Astrophys., 321, 685-690, (1997) [17] Triebel, H., Theory of function spaces I, (1983), Birkhäuser Basel [18] Wardle, M., Star formation and the Hall effect, Astrophys. Space Sci., 292, 317-323, (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.