Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TIME PERIODIC SOLUTION FOR THE COMPRESSIBLE MAGNETO-MICROPOLAR FLUIDS WITH EXTERNAL FORCES IN ℝ3

  • Qingfang Shi (School of Mathematics and Physics Qingdao University of Science and Technology) ;
  • Xinli Zhang (School of Mathematics and Physics Qingdao University of Science and Technology)
  • Received : 2022.06.09
  • Accepted : 2023.03.08
  • Published : 2023.05.01
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Abstract

In this paper, we consider the existence of time periodic solutions for the compressible magneto-micropolar fluids in the whole space ℝ3. In particular, we first solve the problem in a sequence of bounded domains by the topological degree theory. Then we obtain the existence of time periodic solutions in ℝ3 by a limiting process.

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References

  1. Y. Amirat and K. Hamdache, Weak solutions to the equations of motion for compressible magnetic fluids, J. Math. Pures Appl. (9) 91 (2009), no. 5, 433-467. https://doi.org/ 10.1016/j.matpur.2009.01.015 
  2. B. Berkovski and V. Bashtovoy, Magnetic Fluids and Applications Handbook, Begell House, New York, 1996. 
  3. J. Brezina and Y. Kagei, Spectral properties of the linearized compressible Navier-Stokes equation around time-periodic parallel flow, J. Differential Equations 255 (2013), no. 6, 1132-1195. https://doi.org/10.1016/j.jde.2013.04.036 
  4. H. Cai and Z. Tan, Periodic solutions to the compressible magnetohydrodynamic equations in a periodic domain, J. Math. Anal. Appl. 426 (2015), no. 1, 172-193. https://doi.org/10.1016/j.jmaa.2015.01.038 
  5. H. Cai and Z. Tan, Time periodic solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Discrete Contin. Dyn. Syst. 36 (2016), no. 4, 1847-1868. https://doi.org/10.3934/dcds.2016.36.1847 
  6. P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge Univ. Press, Cambridge, 2001. https://doi.org/10.1017/CBO9780511626333 
  7. A. C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966), 1-18. https://doi.org/10.1512/iumj.1967.16.16001 
  8. E. Feireisl, S. Matusu-Necasova, H. Petzeltova, and I. Straskraba, On the motion of a viscous compressible fluid driven by a time-periodic external force, Arch. Ration. Mech. Anal. 149 (1999), no. 1, 69-96. https://doi.org/10.1007/s002050050168 
  9. E. Feireisl, P. B. Mucha, A. Novotny, and M. Pokorny, Time-periodic solutions to the full Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal. 204 (2012), no. 3, 745-786. https://doi.org/10.1007/s00205-012-0492-9 
  10. P. M. Hatzikonstantinou and P. Vafeas, A general theoretical model for the magnetohydrodynamic flow of micropolar magnetic fluids. Application to Stokes flow, Math. Methods Appl. Sci. 33 (2010), no. 2, 233-248. https://doi.org/10.1002/mma.1170 
  11. C. Jin and T. Yang, Time periodic solution for a 3-D compressible Navier-Stokes system with an external force in ℝ3, J. Differential Equations 259 (2015), no. 7, 2576-2601. https://doi.org/10.1016/j.jde.2015.03.035 
  12. C. Jin and T. Yang, Time periodic solution to the compressible Navier-Stokes equations in a periodic domain, Acta Math. Sci. Ser. B (Engl. Ed.) 36 (2016), no. 4, 1015-1029. https://doi.org/10.1016/S0252-9602(16)30055-8 
  13. Y. Kagei and K. Tsuda, Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry, J. Differential Equations 258 (2015), no. 2, 399-444. https://doi.org/10.1016/j.jde.2014.09.016 
  14. H. Ma, S. Ukai, and T. Yang, Time periodic solutions of compressible Navier-Stokes equations, J. Differential Equations 248 (2010), no. 9, 2275-2293. https://doi.org/10.1016/j.jde.2009.11.031 
  15. R. Moreau, Magnetohydrodynamics, translated from the French by A. F. Wright, Fluid Mechanics and its Applications, 3, Kluwer Acad. Publ., Dordrecht, 1990. https://doi.org/10.1007/978-94-015-7883-7 
  16. P. K. Papadopoulos, P. Vafeas, and P. M. Hatzikonstantinou, Ferrofluid pipe flow under the influence of the magnetic field of a cylindrical coil, Phys. Fluids 24 (2012), 122002. 
  17. R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, New York, 1997. 
  18. Z. Tan and Q. Xu, On the motion of the 3D compressible micropolar fluids with time periodic external forces, J. Math. Phys. 59 (2018), no. 8, 081511, 28 pp. https://doi.org/10.1063/1.5051990 
  19. Z. Tan and H. Wang, Time periodic solutions of the compressible magnetohydrodynamic equations, Nonlinear Anal. 76 (2013), 153-164. https://doi.org/10.1016/j.na.2012.08.012 
  20. K. Tsuda, On the existence and stability of time periodic solution to the compressible Navier-Stokes equation on the whole space, Arch. Ration. Mech. Anal. 219 (2016), no. 2, 637-678. https://doi.org/10.1007/s00205-015-0902-x 
  21. A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 4, 607-647. 
  22. R. Y. Wei, B. Guo, and Y. Li, Global existence and optimal convergence rates of solutions for 3D compressible magneto-micropolar fluid equations, J. Differential Equations 263 (2017), no. 5, 2457-2480. https://doi.org/10.1016/j.jde.2017.04.002 
  23. J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magnetomicropolar fluid equations, Math. Methods Appl. Sci. 31 (2008), no. 9, 1113-1130. https://doi.org/10.1002/mma.967 
  24. P. Zhang, Blow-up criterion for 3D compressible viscous magneto-micropolar fluids with initial vacuum, Bound. Value Probl. 2013 (2013), 160, 16 pp. https://doi.org/10.1186/1687-2770-2013-160 
  25. P. Zhang, Decay of the compressible magneto-micropolar fluids, J. Math. Phys. 59 (2018), no. 2, 023102, 11 pp. https://doi.org/10.1063/1.5024795 
  26. X. Zhang and H. Cai, Existence and uniqueness of time periodic solutions to the compressible magneto-micropolar fluids in a periodic domain, Z. Angew. Math. Phys. 71 (2020), no. 6, Paper No. 184, 24 pp. https://doi.org/10.1007/s00033-020-01409-2