SOME LIOUVILLE-TYPE THEOREMS FOR THE STATIONARY 3D MAGNETO-MICROPOLAR FLUIDS

doi:10.1007/s10473-024-0614-0

Abstract

Abstract: In this paper, we prove some Liouville-type theorems for the stationary magneto-micropolar fluids under suitable conditions in three space dimensions. We first prove that the solutions are trivial under the assumption of certain growth conditions for the mean oscillations of the potentials. And then we show similar results assuming that the solutions are contained in $L^p(\R^3)$ with $p\in[2,9/2)$. Finally, we show the same result for lower values of $p\in[1,9/4)$ with the further assumption that the solutions vanish at infinity.

Key words: stationary magneto-micropolar equations, Liouville-type theorem

CLC Number:

35Q30

Jae-Myoung KIM, Seungchan KO. SOME LIOUVILLE-TYPE THEOREMS FOR THE STATIONARY 3D MAGNETO-MICROPOLAR FLUIDS[J].Acta mathematica scientia,Series B, 2024, 44(6): 2296-2306.

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References

[1] Ahmadi G, Shahinpoor M.Universal stability of magneto-micropolar fluid motions. Internat J Engrg Sci, 1974, 12: 657-663
[2] Rojas-Medar M A. Magneto-micropolar fluid motion: existence and uniqueness of strong solution. Math Nachr, 1997, 188: 301-319
[3] Rojas-Medar M A, Boldrini J L. Magneto-micropolar fluid motion: existence of weak solutions. Rev Mat Complut, 1998, 11(2): 443-460
[4] Ortega-Torres E E, Rojas-Medar M A. Magneto-micropolar fluid motion: global existence of strong solutions. Abstr Appl Anal, 1999, 4(2): 109-125
[5] Galdi G P.An Introduction to the Mathematical Theory of the Navier-Stokes Equations. 2nd ed. New York: Springer, 2011
[6] Chae D, Yoneda T.On the Liouville theorem for the stationary Navier-Stokes equations in a critical space. J Math Anal Appl, 2013, 405(2): 706-710
[7] Chae D.Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations. Comm Math Phys, 2014, 326(1): 37-48
[8] Chae D, Wolf J.On Liouville type theorems for the steady Navier-Stokes equations in $\Bbb{R}^3$. J Differential Equations, 2016, 261(10): 5541-5560
[9] Korobkov M, Pileckas K, Russo R.The Liouville theorem for the steady-state Navier-Stokes problem for axially symmetric 3D solutions in absence of swirl. J Math Fluid Mech, 2015, 17(2): 287-293
[10] Seregin G, Wang W.Sufficient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations. St Petersburg Math J, 2020, 31: 387-393
[11] Zhang Q S.A review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang. arXiv:2101.04905
[12] Koch G, Nadirashvili N, Seregin G A, Šák V.Liouville theorems for the Navier-Stokes equations and applications. Acta Math, 2009, 203(1): 83-105
[13] Seregin G.Liouville type theorem for stationary Navier-Stokes equations. Nonlinearity, 2016, 29(8): 2191-2195
[14] Seregin G.Remarks on Liouville type theorems for steady-state Navier-Stokes equations. St Petersburg Math J, 2019, 30: 321-328
[15] Chae D, Wolf J. On Liouville type theorem for the stationary Navier-Stokes equations. Calc Var Partial Differential Equations, 2019, 58(3): Art 111
[16] Chae D, Kim J, Wolf J. On Liouville-type theorems for the stationary MHD and the Hall-MHD systems in $\Bbb R^3$. Z Angew Math Phys, 2022, 73(2): Art 66
[17] Zhang Z, Yang X, Qiu S.Remarks on Liouville type result for the 3D Hall-MHD system. J Partial Differ Equ, 2015, 28(3): 286-290
[18] Schulz S.Liouville type theorem for the stationary equations of magneto-hydrodynamics. Acta Math Sci, 2019, 39B(2): 491-497
[19] Yuan B, Xiao Y.Liouville-type theorems for the 3D stationary Navier-Stokes, MHD and Hall-MHD equations. J Math Anal Appl, 2020, 491(2): 124343
[20] Liu P, Liu G.Some Liouville-type theorems for the stationary density-dependent Navier-Stokes equations. J Math Phys, 2022, 63(1): 013101
[21] Kozono H, Terasawa Y, Wakasugi Y.A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions. J Funct Anal, 2017, 272(2): 804-818
[22] Chamorro D, Jarrín O, Lemarié-Rieusset P-G. Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces. Ann Inst H Poincaré C Anal Non Linéaire, 2021, 38(3): 689-710
[23] Giaquinta M.Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. New York: Princeton University Press, 1983
[24] Chae D, Kim J, Wolf J.On Liouville type theorems for the stationary non-newtonian fluid equations. arXiv:2107.09867