Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE AND DECAY PROPERTIES OF WEAK SOLUTIONS TO THE INHOMOGENEOUS HALL-MAGNETOHYDRODYNAMIC EQUATIONS

  • HAN, PIGONG (ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE, CHINESE ACADEMY OF SCIENCES) ;
  • LEI, KEKE (ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE, CHINESE ACADEMY OF SCIENCES) ;
  • LIU, CHENGGANG (SCHOOL OF STATISTICS AND MATHEMATICS, ZHONGNAN UNIVERSITY OF ECONOMICS AND LAW) ;
  • WANG, XUEWEN (ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE, CHINESE ACADEMY OF SCIENCES)
  • Received : 2022.05.04
  • Accepted : 2022.06.10
  • Published : 2022.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study the temporal decay of global weak solutions to the inhomogeneous Hall-magnetohydrodynamic (Hall-MHD) equations. First, an approximation problem and its weak solutions are obtained via the Caffarelli-Kohn-Nirenberg retarded mollification technique. Then, we prove that the approximate solutions satisfy uniform decay estimates. Finally, using the weak convergence method, we construct weak solutions with optimal decay rates to the inhomogeneous Hall-MHD equations.

Keywords

Acknowledgement

This work is supported by the National Key R&D Program of China (2021YFA1000800), the National Natural Science Foundation of China under Grant No. 11871457, the K.C.Wong Education Foundation, Chinese Academy of Sciences.

References

  1. D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations, 256 (2014), 3835-3858. https://doi.org/10.1016/j.jde.2014.03.003
  2. J. Fan and T. Ozawa, Regularity criteria for the density-dependent Hall-magnetohydrodynamics, Appl. Math. Lett., 36 (2014), 14-18. https://doi.org/10.1016/j.aml.2014.04.010
  3. D. Chae, R. Wan and J. Wu, Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion, J. Math. Fluid Mech., 17 (2015), 627-638. https://doi.org/10.1007/s00021-015-0222-9
  4. M. E. Schonbek, L 2 decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 88 (1985), 209-222. https://doi.org/10.1007/BF00752111
  5. D. Chae and M. E. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), 3971-3982. https://doi.org/10.1016/j.jde.2013.07.059
  6. H. O. Bae and H. J. Choe, Decay rate for the incompressible flows in half spaces, Math. Z., 238 (2001), 799-816. https://doi.org/10.1007/s002090100276
  7. H. O. Bae and B. J. Jin, Temporal and spatial decays for the Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 461-477. https://doi.org/10.1017/S0308210505000247
  8. H. O. Bae and B. J. Jin, Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations, J. Differential Equations, 209 (2005), 365-391. https://doi.org/10.1016/j.jde.2004.09.011
  9. H. O. Bae and B. J. Jin, Asymptotic behavior for the Navier-Stokes equations in 2D exterior domains, J. Funct. Anal., 240 (2006), 508-529. https://doi.org/10.1016/j.jfa.2006.04.029
  10. H. O. Bae and B. J. Jin, Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains, Bull. Korean Math. Soc., 44 (2007), 547-567. https://doi.org/10.4134/BKMS.2007.44.3.547
  11. L. Brandolese, Space-time decay of Navier-Stokes flows invariant under rotations, Math. Ann., 329 (2004), 685-706. https://doi.org/10.1007/s00208-004-0533-2
  12. C. He and D. Zhou, Existence and asymptotic behavior for an incompressible Newtonian flow with intrinsic degree of freedom, Math. Methods Appl. Sci., 37 (2014), 1191-1205. https://doi.org/10.1002/mma.2880
  13. M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations in Hm spaces, Comm. Partial Differential Equations, 20 (1995), 103-117. https://doi.org/10.1080/03605309508821088
  14. P. L. Lions, Mathematical topics in fluid mechanics, Vol. 1, Incompressible models, Oxford University Press, New York, 1996.
  15. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Steady-state problems, Second edition, Springer Monographs in Mathematics, Springer, New York, 2011.
  16. L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. https://doi.org/10.1002/cpa.3160350604
  17. P. Han, C. Liu, K. Lei and X. Wang, Asymptotic behavior of weak solutions to the inhomogeneous Navier-Stokes equations, J. Math. Fluid Mech., https://doi.org/10.1007/s00021-021-00636-5.