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Existence, uniqueness and decay rates of a certain type of 3D Hall-MHD equations with power-law type

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Abstract

We investigate the local-in-time existence results of classical solutions to the 3D (Hall-)MHD equations with power-law type nonlinear viscous fluid when magnetic resistance is vanished and also show the global-in-time existence of classical solutions under small initial data. Moreover, we prove the space-time decay property for these solutions.

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References

  1. Bohme, G.: Non-Newtonian Fluid Mechanics. North-Holland Series in Applied Mathematics and Mechanics (1987)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Cramer, K., Pai, S.: Magnetofluid Dynamics for Engineers and Applied Physicists. McGraw-Hill, New York (1973)

    Google Scholar 

  5. Dai, M., Liu, H.: Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion. J. Differ. Equ. 266(11), 7658–7677 (2019)

    Article  MathSciNet  Google Scholar 

  6. Fefferman, C.L., McCormick, D.S., Robinson, J.C., Rodrigo, J.L.: Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. J. Funct. Anal. 267, 1035–1056 (2014)

    Article  MathSciNet  Google Scholar 

  7. Gunzburger, M.D., Ladyzhenskaya, O.A., Peterson, J.S.: On the global unique solvability of initial-boundary value problems for the coupled modified Navier–Stokes and Maxwell equations. J. Math. Fluid Mech. 6, 462–482 (2004)

    Article  MathSciNet  Google Scholar 

  8. Guo, B., Zhu, P.: Algebraic \(L^2\) decay for the solution to a class system of non-Newtonian fluid in \(\mathbb{R} ^n\). J. Math. Phys. 41(1), 349–356 (2000)

    Article  MathSciNet  Google Scholar 

  9. Kang, K., Kim, H.-K., Kim, J.-M.: Existence of regular solutions for a certain type of non-Newtonian Navier–Stokes equations. Z. Angew. Math. Phys. 70(4), Paper No. 124 (2019)

  10. Kang, K., Kim, J.-M.: Existence of solutions for the magnetohydrodynamics with power-law type nonlinear viscous fluid. NoDEA Nonlinear Differ. Equ. Appl. 26(2), Art. 11 (2019)

  11. Kang, K., Kim, H.-K., Kim, J.-M.: Existence and temporal decay of regular solutions to non-Newtonian fluids coupled with Maxwell equations. Nonlinear Anal. 180, 284–307 (2019)

    Article  MathSciNet  Google Scholar 

  12. Kim, J.-M.: Local existence of solutions to the non-resistive 3D MHD equations with power-law type. J. Math. Fluid Mech. 25(2), 31 (2023)

    Article  MathSciNet  Google Scholar 

  13. Kukavica, I.: Space-time decay for solutions of the Navier–Stokes equations. Indiana Univ. Math. J. 50(1), 205–222 (2001)

    Article  MathSciNet  Google Scholar 

  14. Kukavica, I.: On the weighted decay for solutions of the Navier–Stokes system. Nonlinear Anal. Theory Methods Appl. 70, 2466–2470 (2009)

    Article  MathSciNet  Google Scholar 

  15. Kukavica, I., Torres, J.J.: Weighted Lp decay for solutions of the Navier–Stokes equations. Commun. Partial Differ. Equ. 32, 819–831 (2007)

    Article  Google Scholar 

  16. Ladyzhenskaya, O.A.: New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems. Trudy Mat. Inst. Steklov. 102, 85–104 (1967)

    MathSciNet  Google Scholar 

  17. Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach, New York (1969)

    Google Scholar 

  18. Málek, J., Nečas, J., Rokyta, M., Ružička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996)

    Book  Google Scholar 

  19. Miyakawa, T.: On space-time decay properties of nonstationary incompressible Navier–Stokes flows in \(\mathbb{R} ^n\). Funkc. Ekvac. 43, 541–558 (2000)

    Google Scholar 

  20. Nečasová, Š, Penel, P.: \(L^2\) decay for weak solution to equations of non-Newtionian incompressible fluids in the whole space. Nonlinear Anal. 47, 4181–4191 (2001)

    Article  MathSciNet  Google Scholar 

  21. Pokorný, M.: Cauchy problem for the non-Newtonian viscous incompressible fluid. Appl. Math. 41, 169–201 (1996)

    Article  MathSciNet  Google Scholar 

  22. Samokhin, V.N.: On a system of equations in the magnetohydrodynamics of nonlinearly viscous media. Differ. Equ. 27, 628–636 (1991)

    MathSciNet  Google Scholar 

  23. Sapunkov, Y.G.: Self-similar solutions of non-Newtonian fluid boundary layer in MHD. Fluid Dyn. 2, 42–47 (1967)

    Google Scholar 

  24. Sarpakaya, T.: Flow of non-Newtonian fluids in a magnetic field. AIChE J. 7(7), 324–328 (1961)

    Article  Google Scholar 

  25. Sato, H.: The Hall effects in the viscous flow of ionized gas between parallel plates under transverse magnetic field. J. Phys. Soc. Jpn. 16, 1427–1433 (1961)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. 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)

    Article  MathSciNet  Google Scholar 

  28. Weng, S.: Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations. J. Funct. Anal. 270(6), 2168–2187 (2016)

    Article  MathSciNet  Google Scholar 

  29. Wilkinson, W.L.: Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer. Pergamon Press, London (1960)

    Google Scholar 

  30. Yuan, B., Liu, Y.: Global existence and decay rate of strong solution to incompressible Oldroyd type model equations. Rocky Mt. J. Math. 48, 1703–1720 (2018)

    Article  MathSciNet  Google Scholar 

  31. Zhao, X.: Decay of solutions to a new Hall-MHD system in \(\mathbb{R} ^3\). C. R. Math. Acad. Sci. Paris 355, 310–317 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to appreciate the anonymous referee for valuable comments. Jae-Myoung Kim was supported by National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1C1C1A01006521).

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  1. Department of Mathematics Education, Andong National University, Andong, Republic of Korea

    Jae-Myoung Kim

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Kim, JM. Existence, uniqueness and decay rates of a certain type of 3D Hall-MHD equations with power-law type. Anal.Math.Phys. 14, 22 (2024). https://doi.org/10.1007/s13324-024-00882-6

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