Abstract
In this paper, we consider the small initial data global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in \(\mathbb {R}^3\). In addition, we also establish the temporal decay estimates for the weak solutions. With these estimates in hand, we study the algebraic time decay for higher-order Sobolev norms of small initial data solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Acheritogaray, M., Degond, P., Frouvelle, A., Liu, J.-G.: Kinetic formulation and global existence for the Hall-Magnetohydrodynamics system. Kinet. Relat. Models 4, 901–918 (2011)
Bjorland, C., Schonbek, M.: Poincaré’s inequality and diffusive evolution equations. Adv. Differ. Equ. 14, 241–260 (2009)
Brandolese, L.: Characterization of solutions to dissipative systems with sharp algebraic decay. SIAM J. Math. Anal. 48, 1616–1633 (2016)
Brandolese, L., Schonbek, M.: Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Am. Math. Soc. 364, 5057–5090 (2012)
Chae, D., Degond, P., Liu, J.-G.: Well-posedness for Hall-magnetohydrodynamics. Inst. H. Poincaré Anal. Non Linéaire 31, 555–565 (2014)
Chae, D., Schonbek, M.: On the temporal decay for the Hall-magnetohydrodynamic equations. J. Differ. Equ. 255, 3971–3982 (2013)
Chae, D., Wan, R., Wu, J.: Local well-posedness for Hall-MHD equations with fractional magnetic diffusion. J. Math. Fluid Mech. 17, 627–638 (2015)
Dai, M., Qing, J., Schonbek, M.: Asymptotic behavior of solutions to liquid crystal systems in \(\mathbb{R}^3\). Commun. Partial Differ. Equ. 37, 2138–2164 (2012)
Fan, J., Ahmad, B., Hayat, T., Zhou, Y.: On blow-up criteria for a new Hall-MHD system. Appl. Math. Comput. 274, 20–24 (2016)
Fan, J., Alsaedi, A., Hayat, T., Nakamura, G., Zhou, Y.: On strong solutions to the compressible Hall-magnetohydrodynamic system. Nonlinear Anal. Real World Appl. 22, 423–434 (2015)
Fan, J., Ahmad, B., Hayat, T., Zhou, Y.: On well-posedness and blow-up for the full compressible Hall-MHD system. Nonlinear Anal. Real World Appl. 31, 569–579 (2016)
Fan, J., Fukumoto, Y., Nakamura, G., Zhou, Y.: Regularity criteria for the incompressible Hall-MHD system. Z. Angew. Math. Mech. 95, 1156–1160 (2015)
Fan, J., Jia, X., Nakamura, G., Zhou, Y.: On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects. Z. Angew. Math. Phys. 66, 1695–1706 (2015)
Gala, S., Ragusa, M.A.: On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in \(\mathbb{R}^3\). Z. Angew. Math. Phys. 67, 18 (2016)
Jia, X., Zhou, Y.: On regularity criteria for the 3D incompressible MHD equations involving one velocity component. J. Math. Fluid Mech. 18, 187–206 (2016)
Jiang, Z., Zhu, M.: Regularity criteria for the 3D generalized MHD and Hall-MHD systems. Bull. Malays. Math. Sci. 41, 105–122 (2018). https://doi.org/10.1007/s40840-015-0243-9
Jiu, Q., Yu, H.: Decay of solutions to the three-dimensional generalized Navier–Stokes equations. Asymptot. Anal. 94, 105–124 (2015)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)
Kenig, C., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)
Maiellaro, M.: Uniqueness of MHD thermodiffusive mixture flows with Hall and ion-slip effects. Meccanica 12, 9–14 (1977)
Mulone, G., Solonnikov, V.A.: On an initial boundary-value problem for the equation of magnetohydrodynamics with the Hall and ion-slip effects. J. Math. Sci. 87, 3381–3392 (1997)
Mulone, G., Salemi, F.: Some continuous dependence theorems in MHD with Hall and ion-slip currents in unbound domains. Rend. Ac. Sci. Fis. Mat. Napoli. 55, 139–152 (1988)
Niche, C.J., Schonbek, M.: Decay characterization of solutions to dissipative equations. J. Lond. Math. Soc. 91(2), 573–595 (2015)
Schonbek, M.: \(L^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88(2), 209–222 (1985)
Schonbek, M.: Large time behaviour of solutions to the Navier–Stokes equations. Commun. Partial Differ. Equ. 11(7), 733–763 (1986)
Wan, R., Zhou, Y.: On global existence, energy decay and blow-up criteria for the Hall-MHD system. J. Differ. Equ. 259, 5982–6008 (2015)
Wan, R., Zhou, Y.: Low regularity well-posedness for the 3D generalized hall-MHD system. Acta Appl. Math. 147, 95–111 (2017)
Weng, S.: On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system. J. Differ. Equ. 260, 6504–6524 (2016)
Zhao, X.: Decay of solutions to a new Hall-MHD system in \(\mathbb{R}^3\). C. R. Acad. Sci. Paris Ser. I. 355, 310–317 (2017)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhao, X., Zhu, M. Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects. Z. Angew. Math. Phys. 69, 22 (2018). https://doi.org/10.1007/s00033-018-0907-z
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-018-0907-z