On the well-posedness and temporal decay for the 3D generalized incompressible Hall-MHD system

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In this paper, we prove a new result on the properties of decay character \(r^*\) (see Lemma 2.6) and then show a small data global well-posedness result for three-dimensional generalized incompressible Hall-MHD system. In the end, through Fourier splitting method, the properties of decay character \(r^*\) and mathematical induction, we study the decay rate of higher-order spatial and time derivatives of strong solutions to such system.

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  1. 1.

    Acheritogaray, M., Degond, P., Frouvelle, A., Liu, J.: Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system. Kinet. Relat. Models 4, 901–918 (2011)

  2. 2.

    Bjorland, C., Schonbek, M.E.: Poincaré’s inequality and diffusive evolution equations. Adv. Differ. Equ. 14, 241–260 (2009)

  3. 3.

    Bjorland, C., Schonbek, M.E.: On questions of decay and existence for the viscous Camassa–Holm equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 25, 907–936 (2008)

  4. 4.

    Brandolese, L.: Characterization of solutions to dissipative systems with sharp algebraic decay. SIAM J. Math. Anal. 48, 1616–1633 (2016)

  5. 5.

    Chae, D., Schonbek, M.E.: On the temporal decay for the Hall-magnetohydrodynamic equations. J. Differ. Equ. 255, 3971–3982 (2013)

  6. 6.

    Chae, D., Degond, P., Liu, J.: Well-posedness for Hall-magnetohydrodynamics. Ann. Inst. Henri Poincaré Anal. Non Linéaire 31, 555–565 (2014)

  7. 7.

    Chae, D., Lee, J.: On the blow-up criterion and small data global existence for the Hall-magneto-hydrodynamics. J. Differ. Equ. 256, 3835–3858 (2014)

  8. 8.

    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)

  9. 9.

    Chae, D., Wolf, J.: Regularity of the 3D stationary Hall magnetohydrodynamic equations on the plane. Commun. Math. Phys. 354, 213–230 (2017)

  10. 10.

    Dai, M.: Regularity criterion for the 3D Hall-magneto-hydrodynamics. J. Differ. Equ. 261, 573–591 (2016)

  11. 11.

    Fan, J., Alsaedi, A., Fukumoto, Y., Hayat, T., Zhou, Y.: A regularity criterion for the density-dependent Hall-magnetohydrodynamics. Z. Anal. Anwend. 34, 277–284 (2015)

  12. 12.

    Fan, J., Fukumoto, Y., Nakamura, G., Zhou, Y.: Regularity criteria for the incompressible Hall-MHD system. ZAMM Z. Angew. Math. Mech. 95, 1156–1160 (2015)

  13. 13.

    Jiang, Z., Zhu, M.: Regularity criteria for the 3D generalized MHD and Hall-MHD systems. Bull. Malays. Math. Sci. Soc. 41, 105–122 (2018)

  14. 14.

    Niche, C.J., Schonbek, M.E.: Decay characterization of solutions to dissipative equations. J. Lond. Math. Soc. 91(2), 573–595 (2015)

  15. 15.

    Pan, N., Ma, C., Zhu, M.: Global regularity for the 3D generalized Hall-MHD system. Appl. Math. Lett. 61, 62–66 (2016)

  16. 16.

    Schonbek, M.E.: \(L^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88, 209–222 (1985)

  17. 17.

    Schonbek, M.E.: Large time behaviour of solutions to the Navier–Stokes equations. Commun. Partial Differ. Equ. 11, 733–763 (1986)

  18. 18.

    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

  19. 19.

    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)

  20. 20.

    Wan, R., Zhou, Y.: Low regularity well-posedness for the 3D generalized Hall-MHD system. Acta Appl. Math. 147, 95–111 (2017)

  21. 21.

    Weng, S.: On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system. J. Differ. Equ. 260, 6504–6524 (2016)

  22. 22.

    Weng, S.: Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations. J. Funct. Anal. 270, 2168–2187 (2016)

  23. 23.

    Wu, X., Yu, Y., Tang, Y.: Global existence and asymptotic behavior for the 3D generalized Hall-MHD system. Nonlinear Anal. 151, 41–50 (2017)

  24. 24.

    Ye, Z.: Regularity criteria and small data global existence to the generalized viscous Hall-magnetohydrodynamics. Comput. Math. Appl. 70, 2137–2154 (2015)

  25. 25.

    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)

  26. 26.

    Zhao, X., Zhu, M.: Large time behavior of solutions to generalized Hall-MHD system in \({\mathbb{R}}^3\). J. Math. Phys. 59(7), 073502 (2018). 13 pp

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The authors are indebted to anonymous referees for their helpful comments. This work was partially supported by Natural Science Foundation of Anhui Province Higher School (Grant No: KJ2017A622) and National Natural Science Foundation of China (Grant No. 11771183).

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Correspondence to Mingxuan Zhu.

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Zhao, X., Zhu, M. On the well-posedness and temporal decay for the 3D generalized incompressible Hall-MHD system. Z. Angew. Math. Phys. 71, 27 (2020) doi:10.1007/s00033-020-1249-1

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  • Generalized Hall-MHD equations
  • Decay rate
  • Decay character
  • Mathematical induction

Mathematics Subject Classification

  • 35B40
  • 35Q35
  • 74W05