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

doi:10.1007/s10473-024-0614-0

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (6): 2296-2306.doi: 10.1007/s10473-024-0614-0

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

Jae-Myoung KIM, Seungchan KO^†

1. Department of Mathematics Education, Andong National University, Gyeongdong-ro,Andong-si, 36729, Gyeongsangbuk-do, Republic of Korea;
2. Department of Mathematics, Inha University, Inha-ro, Michuhol-gu, 22212, Incheon-si, Republic of Korea

收稿日期:2023-07-24 修回日期:2024-05-02 发布日期:2024-12-06
通讯作者: † Seungchan KO, E-mail: scko@inha.ac.kr
作者简介:Jae-Myoung KIM, E-mail: jmkim02@andong.ac.kr
基金资助:
Seungchan Ko's research was supported by Inha University Research Grant and National Research Foundation of Korea Grant funded by the Korean Government (RS-2023-00212227). Jae-Myoung Kim's research was supported by National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1C1C1A01006521).

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

Jae-Myoung KIM, Seungchan KO^†

1. Department of Mathematics Education, Andong National University, Gyeongdong-ro,Andong-si, 36729, Gyeongsangbuk-do, Republic of Korea;
2. Department of Mathematics, Inha University, Inha-ro, Michuhol-gu, 22212, Incheon-si, Republic of Korea

Received:2023-07-24 Revised:2024-05-02 Published:2024-12-06
Contact: † Seungchan KO, E-mail: scko@inha.ac.kr
About author:Jae-Myoung KIM, E-mail: jmkim02@andong.ac.kr
Supported by:
Seungchan Ko's research was supported by Inha University Research Grant and National Research Foundation of Korea Grant funded by the Korean Government (RS-2023-00212227). Jae-Myoung Kim's research was supported by National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1C1C1A01006521).

摘要/Abstract

摘要： 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.

关键词: stationary magneto-micropolar equations, Liouville-type theorem

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

中图分类号:

35Q30

Jae-Myoung KIM, Seungchan KO. SOME LIOUVILLE-TYPE THEOREMS FOR THE STATIONARY 3D MAGNETO-MICROPOLAR FLUIDS[J]. 数学物理学报(英文版), 2024, 44(6): 2296-2306.

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.

参考文献

[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

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

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

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	Lanxi XU, Fangfang GUAN. THE NONLINEAR STABILITY OF PLANE PARALLEL SHEAR FLOWS WITH RESPECT TO TILTED PERTURBATIONS[J]. 数学物理学报(英文版), 2024, 44(3): 1036-1045.
[2]	Zhenhua Guo, Xueyao Zhang. INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^*[J]. 数学物理学报(英文版), 2024, 44(1): 247-274.
[3]	Thieu Huy NGUYEN, Truong Xuan PHAM, Thi Ngoc Ha VU, The Sac LE. EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS[J]. 数学物理学报(英文版), 2022, 42(5): 1875-1901.
[4]	黄丙远, 丁时进, 伍日清. CLASSICAL SOLUTIONS OF THE 3D COMPRESSIBLE FLUID-PARTICLE SYSTEM WITH A MAGNETIC FIELD[J]. 数学物理学报(英文版), 2022, 42(4): 1585-1606.
[5]	王梅, 李自来, 郭真华. GLOBAL SOLUTIONS TO A 3D AXISYMMETRIC COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2022, 42(2): 521-539.
[6]	于海波. THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THE 3D COMPRESSIBLE ISOTHERMAL NAVIER-STOKES EQUATIONS[J]. 数学物理学报(英文版), 2022, 42(1): 233-246.
[7]	苏奕帆, 郭真华. ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2021, 41(5): 1635-1658.
[8]	栗付才, 张志朋. ZERO KINEMATIC VISCOSITY-MAGNETIC DIFFUSION LIMIT OF THE INCOMPRESSIBLE VISCOUS MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2021, 41(5): 1503-1536.
[9]	陈拓炜, 张永前. A STRONG SOLUTION OF NAVIER-STOKES EQUATIONS WITH A ROTATION EFFECT FOR ISENTROPIC COMPRESSIBLE FLUIDS[J]. 数学物理学报(英文版), 2021, 41(5): 1579-1605.
[10]	文娟, 黄鹏展, 何雅玲. THE TWO-LEVEL STABILIZED FINITE ELEMENT METHOD BASED ON MULTISCALE ENRICHMENT FOR THE STOKES EIGENVALUE PROBLEM[J]. 数学物理学报(英文版), 2021, 41(2): 381-396.
[11]	Viet Duoc TRINH. TIME GLOBAL MILD SOLUTIONS OF NAVIER-STOKES-OSEEN EQUATIONS[J]. 数学物理学报(英文版), 2021, 41(2): 450-460.
[12]	Srđan TRIFUNOVIĆ, 王亚光. WEAK SOLUTION TO THE INCOMPRESSIBLE VISCOUS FLUID AND A THERMOELASTIC PLATE INTERACTION PROBLEM IN 3D[J]. 数学物理学报(英文版), 2021, 41(1): 19-38.
[13]	Muhammad Zainul ABIDIN, 陈杰诚. GLOBAL WELL-POSEDNESS FOR FRACTIONAL NAVIER-STOKES EQUATIONS IN VARIABLE EXPONENT FOURIER-BESOV-MORREY SPACES[J]. 数学物理学报(英文版), 2021, 41(1): 164-176.
[14]	Hakho HONG. THE EXISTENCE AND STABILITY OF STATIONARY SOLUTIONS OF THE INFLOW PROBLEM FOR FULL COMPRESSIBLE NAVIER-STOKES-POISSON SYSTEM[J]. 数学物理学报(英文版), 2021, 41(1): 319-336.
[15]	李思然. ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE L^q-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS[J]. 数学物理学报(英文版), 2020, 40(6): 1700-1708.