Well-posedness for Hall-magnetohydrodynamics
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, p. 555-565

We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resistive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.

DOI : https://doi.org/10.1016/j.anihpc.2013.04.006
Classification:  35L60,  35K55,  35Q80
Keywords: Hall-MHD, Smooth solutions, Well-posedness, Liouville theorem
@article{AIHPC_2014__31_3_555_0,
     author = {Chae, Dongho and Degond, Pierre and Liu, Jian-Guo},
     title = {Well-posedness for Hall-magnetohydrodynamics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {3},
     year = {2014},
     pages = {555-565},
     doi = {10.1016/j.anihpc.2013.04.006},
     zbl = {1297.35064},
     mrnumber = {3208454},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_3_555_0}
}
Chae, Dongho; Degond, Pierre; Liu, Jian-Guo. Well-posedness for Hall-magnetohydrodynamics. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, pp. 555-565. doi : 10.1016/j.anihpc.2013.04.006. http://www.numdam.org/item/AIHPC_2014__31_3_555_0/

[1] M. Acheritogaray, P. Degond, A. Frouvelle, J.-G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system, Kinet. Relat. Models 4 (2011), 901 -918 | MR 2861579 | Zbl 1251.35076

[2] S.A. Balbus, C. Terquem, Linear analysis of the Hall effect in protostellar disks, Astrophys. J. 552 (2001), 235 -247

[3] L.M.B.C. Campos, On hydromagnetic waves in atmospheres with application to the sun, Theor. Comput. Fluid Dyn. 10 (1998), 37 -70 | Zbl 0911.76099

[4] F. Charles, B. Després, B. Perthame, R. Sentis, Nonlinear stability of a Vlasov equation for magnetic plasmas, Kinet. Relat. Models 6 (2013), 269 -290 | MR 3030713 | Zbl 1262.35197

[5] J.-Y. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford (1998) | MR 1688875

[6] G. Duvaut, J.L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal. 46 (1972), 241 -279 | MR 346289 | Zbl 0264.73027

[7] T.G. Forbes, Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn. 62 (1991), 15 -36

[8] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. II, Springer (1994) | MR 1284206 | Zbl 0949.35005

[9] H. Homann, R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Phys. D 208 (2005), 59 -72 | MR 2167907 | Zbl 1154.76392

[10] M.J. Lighthill, Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Philos. Trans. R. Soc. Lond. Ser. A 252 (1960), 397 -430 | MR 148337 | Zbl 0097.20806

[11] A.J. Majda, A.L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press (2001) | MR 1867882

[12] P.D. Mininni, D.O. Gòmez, S.M. Mahajan, Dynamo action in magnetohydrodynamics and Hall magnetohydrodynamics, Astrophys. J. 587 (2003), 472 -481

[13] J.M. Polygiannakis, X. Moussas, A review of magneto-vorticity induction in Hall-MHD plasmas, Plasma Phys. Control. Fusion 43 (2001), 195 -221

[14] D.A. Shalybkov, V.A. Urpin, The Hall effect and the decay of magnetic fields, Astron. Astrophys. (1997), 685 -690

[15] M.E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, American Mathematical Society (2000) | MR 1766415 | Zbl 0963.35211

[16] H. Triebel, Theory of Function Spaces I, Birkhäuser Basel (1983) | MR 781540

[17] M. Wardle, Star formation and the Hall effect, Astrophys. Space Sci. 292 (2004), 317 -323