Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 395-413, 2023


Upper and lower convergence rates for strong solutions of the 3D non-Newtonian flows associated with Maxwell equations under large initial perturbation

Jae-Myoung Kim

Received June 28, 2021.   Published online February 14, 2023.

Abstract:  We show the upper and lower bounds of convergence rates for strong solutions of the 3D non-Newtonian flows associated with Maxwell equations under a large initial perturbation.
Keywords:  non-Newtonian fluid; MHD equation; decay estimate; large initial perturbation
Classification MSC:  35Q30, 76A05, 35B35


References:
[1] G. Astarita, G. Marrucci: Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, London (1974).
[2] H.-O. Bae, B. J. Jin: Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations. J. Differ. Equations 209 (2005), 365-391. DOI 10.1016/j.jde.2004.09.011 | MR 2110209 | Zbl 1062.35058
[3] M. J. Benvenutti, L. C. F. Ferreira: Existence and stability of global large strong solutions for the Hall-MHD system. Differ. Integral Equ. 29 (2016), 977-1000. MR 3513590 | Zbl 1389.35255
[4] M. D. Gunzburger, O. A. Ladyzhenskaya, J. S. Peterson: On the global unique solvability of initial-boundary value problems for the coupled modified Navier-Stokes and Maxwell equations. J. Math. Fluid Mech. 6 (2004), 462-482. DOI 10.1007/s00021-004-0107-9 | MR 2101892 | Zbl 1064.76118
[5] B. Guo, P. Zhu: Algebraic $L^2$ decay for the solution to a class system of non-Newtonian fluid in $\mathbb R^n$. J. Math. Phys. 41 (2000), 349-356. DOI 10.1063/1.533135 | MR 1738602 | Zbl 0989.35108
[6] K. Kang, J.-M. Kim: Existence of solutions for the magnetohydrodynamics with power- law type nonlinear viscous fluid. NoDEA, Nonlinear Differ. Equ. Appl. 26 (2019), Article ID 11, 24 pages. DOI 10.1007/s00030-019-0557-7 | MR 3924622 | Zbl 1417.76045
[7] G. Karch, D. Pilarczyk: Asymptotic stability of Landau solutions to Navier-Stokes system. Arch. Ration. Mech. Anal. 202 (2011), 115-131. DOI 10.1007/s00205-011-0409-z | MR 2835864 | Zbl 1256.35061
[8] G. Karch, D. Pilarczyk, M. E. Schonbek: $L^2$-asymptotic stability of singular solutions to the Navier-Stokes system of equations in $\mathbb{R}^3$. J. Math. Pures Appl. (9) 108 (2017), 14-40. DOI 10.1016/j.matpur.2016.10.008 | MR 3660767 | Zbl 1368.35207
[9] J.-M. Kim: Temporal decay of strong solutions to the magnetohydrodynamics with power-law type nonlinear viscous fluid. J. Math. Phys. 61 (2020), Article ID 011504, 6 pages. DOI 10.1063/1.5128708 | MR 4047930 | Zbl 1432.76289
[10] J.-M. Kim: Time decay rates for the coupled modified Navier-Stokes and Maxwell equations on a half space. AIMS Math. 6 (2021), 13423-13431. DOI 10.3934/math.2021777 | MR 4332321 | Zbl 07533493
[11] H. Kozono: Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations. J. Funct. Anal. 176 (2000), 153-197. DOI 10.1006/jfan.2000.3625 | MR 1784412 | Zbl 0970.35106
[12] T. Miyakawa: On upper and lower bounds of rates of decay for nonstationary Navier- Stokes flows in the whole space. Hiroshima Math. J. 32 (2002), 431-462. DOI 10.32917/hmj/1151007491 | MR 1954053 | Zbl 1048.35063
[13] Š. Nečasová, P. Penel: $L^2$ decay for weak solution to equations of non-Newtonian incompressible fluids in the whole space. Nonlinear Anal., Theory Methods Appl., Ser. A 47 (2001), 4181-4191. DOI 10.1016/S0362-546X(01)00535-1 | MR 1972358 | Zbl 1042.76504
[14] M. Oliver, E. S. Titi: Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in $\mathbb R^n$. J. Funct. Anal. 172 (2000), 1-18. DOI 10.1006/jfan.1999.3550 | MR 1749867 | Zbl 0960.35081
[15] V. N. Samokhin: A magnetohydrodynamic-equation system for a nonlinearly viscous liquid. Differ. Equations 27 (1991), 628-636; translation from Differ. Uravn. 27 (1991), 886-896. MR 1117118 | Zbl 0795.76094
[16] M. E. Schonbek: Large time behaviour of solutions to the Navier-Stokes equations. Commun. Partial Differ. Equations 11 (1986), 733-763. DOI 10.1080/03605308608820443 | MR 0837929 | Zbl 0607.35071
[17] P. Secchi: $L^2$ stability for weak solutions of the Navier-Stokes equations in $\mathbb R^3$. Indiana Univ. Math. J. 36 (1987), 685-691. DOI 10.1512/iumj.1987.36.36039 | MR 0905619 | Zbl 0635.35076
[18] M. Wiegner: Decay results for weak solutions of the Navier-Stokes equations on $\mathbb R^n$. J. Lond. Math. Soc., II. Ser. 35 (1987), 303-313. DOI 10.1112/jlms/s2-35.2.303 | MR 0881519 | Zbl 0652.35095
[19] W. L. Wilkinson: Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer. International Series of Monographs on Chemical Engineering 1. Pergamon Press, New York (1960). MR 0110392 | Zbl 0124.41802
[20] Q. Xie, Y. Guo, B.-Q. Dong: Upper and lower convergence rates for weak solutions of the 3D non-Newtonian flows. J. Math. Anal. Appl. 494 (2021), Article ID 124641, 21 pages. DOI 10.1016/j.jmaa.2020.124641 | MR 4161399 | Zbl 1457.76027
[21] Y. Zhou: Asymptotic stability for the 3D Navier-Stokes equations. Commun. Partial Differ. Equations 30 (2005), 323-333. DOI 10.1081/PDE-200037770 | MR 2131057 | Zbl 1142.35548
[22] Y. Zhou: Asymptotic stability for the Navier-Stokes equations in $L^n$. Z. Angew. Math. Phys. 60 (2009), 191-204. DOI 10.1007/s00033-008-7045-y | MR 2486152 | Zbl 1293.76049

Affiliations:   Jae-Myoung Kim, Department of Mathematics Education, Andong National University, 1375 Gyeongdong-ro, Songcheon-dong, Andong 36729, Republic of Korea, e-mail: jmkim02@anu.ac.kr


 
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