三维稳态磁流体动力学方程的Liouville定理

田琴; 向长林; 别群益

doi:10.21656/1000-0887.430375

三维稳态磁流体动力学方程的Liouville定理

doi: 10.21656/1000-0887.430375

田琴^{1, 2,},
向长林^{1, 2, ,},
别群益^{1, 2,}

1.
三峡大学理学院，湖北宜昌 443002
2.
三峡大学三峡数学研究中心，湖北宜昌 443002

基金项目:

国家自然科学基金项目 11871305

国家自然科学基金项目 12271296

详细信息

作者简介:
田琴(1998—)，女，硕士生(E-mail: 1321540194@qq.com)

别群益(1970—)，男，教授，博士，博士生导师(E-mail: qybie@126.com)

通讯作者:
向长林(1984—)，男，副教授，博士，博士生导师(通讯作者. E-mail: changlin.xiang@ctgu.edu.cn)

中图分类号: O175.2
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- 被引次数: 0
出版历程
- 收稿日期: 2022-11-22
- 修回日期: 2023-03-04
- 刊出日期: 2023-10-31

On the Liouville Theorems for 3D Stationary Magnetohydrodynamic Equations

TIAN Qin^{1, 2
,},
XIANG Changlin^{1, 2
, ,},
BIE Qunyi^{1, 2
,}

1.
College of Science, China Three Gorges University, Yichang, Hubei 443002, P.R. China
2.
Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, Hubei 443002, P.R. China

摘要

摘要: 研究了三维稳态磁流体动力学方程的Liouville定理. 首先由能量估计建立了一个Caccioppoli型不等式，再结合Sobolev嵌入得到了Liouville定理成立的3个充分条件，其中一个充分条件表明：若三维稳态磁流体动力学方程的光滑解( u , b )∈L^p，3/2 < p < 3，则 u = b ≡ 0 . 该结果在不需要有限Dirichlet积分的条件下，将Lebesgue空间中可积指标的下界从2扩展至3/2，改进和推广了已有关于磁流体动力学方程Liouville定理的一些结论.
- 磁流体动力学方程 /
- Liouville定理 /
- Caccioppoli型不等式
Abstract: The Liouville theorems for 3D stationary magnetohydrodynamic equations were studied. First, a Caccioppoli type inequality was obtained with the energy method, then 3 sufficient conditions for the Liouville theorems were obtained based on the Sobolev embedding theorems, of which 1 sufficient condition indicates that, given a smooth solution to the 3D stationary magnetohydrodynamic equation satisfying( u , b )∈L^p, 3/2 < p < 3, equality u = b ≡ 0 will be tenable. This work extends the lower bound of the integrable index in the Lebesgue space from 2 to 3/2 without the finite Dirichlet integral condition, which improves and generalizes some conclusions about the Liouville theorems for stationary magnetohydrodynamic equations.
- magnetohydrodynamic equations /
- Liouville theorems /
- Caccioppoli type inequality

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参考文献(17)

[1]	施惟慧, 王曰朋, 沈春. Navier-Stokes方程与Euler方程的稳定性比较[J]. 应用数学和力学, 2006, 27(9): 1101-1107. doi: 10.3321/j.issn:1000-0887.2006.09.013 SHI Weihui, WANG Yuepeng, SHEN Chun. Comparison of stability between Navier-Stokes and Euler equations[J]. Applied Mathematics and Mechanics, 2006, 27(9): 1101-1107. (in Chinese) doi: 10.3321/j.issn:1000-0887.2006.09.013
[2]	谢洪燕, 李杰, 贺方毅. 关于轴对称Navier-Stokes方程正则性的一个注记[J]. 应用数学和力学, 2017, 38(3): 276-283. doi: 10.21656/1000-0887.370192 XIE Hongyan, LI Jie, HE Fangyi. A note on the regularity of axisymmetric Navier-Stokes equations[J]. Applied Mathematics and Mechanics, 2017, 38(3): 276-283. (in Chinese) doi: 10.21656/1000-0887.370192
[3]	王小霞. 含非线性阻尼的2D g-Navier-Stokes方程解的一致渐近性[J]. 应用数学和力学, 2022, 43(4): 416-423. doi: 10.21656/1000-0887.410398 WANG Xiaoxia. Uniform asymptotic behavior of solutions of 2D g-Navier-Stokes equations with nonlinear damping[J]. Applied Mathematics and Mechanics, 2022, 43(4): 416-423. (in Chinese) doi: 10.21656/1000-0887.410398
[4]	CHAE D, WENG S. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations[J]. Discrete & Continuous Dynamical Systems, 2016, 36(10): 5267-5285.
[5]	CHAE D. Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations[J]. Communications in Mathematical Physics, 2014, 326(1): 37-48. doi: 10.1007/s00220-013-1868-x
[6]	CHAE D, WOLF J. On Liouville type theorems for the steady Navier-Stokes equations in R³[J]. Journal of Differential Equations, 2016, 261(10): 5541-5560. doi: 10.1016/j.jde.2016.08.014
[7]	GIAQUINTA M. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems: Vol 105 [M]. Princeton University Press, 2016.
[8]	SEREGIN G. Remarks on Liouville-type theorems for steady-state Navier-Stokes equations[J]. Petersburg Mathematical Journal, 2019, 30(2): 321-328. doi: 10.1090/spmj/1544
[9]	CHAE D, YONEDA T. On the Liouville theorem for the stationary Navier-Stokes equations in a critical space[J]. Journal of Mathematical Analysis and Applications, 2013, 405(2): 706-710. doi: 10.1016/j.jmaa.2013.04.040
[10]	GALDI G. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems[M]. Springer Science & Business Media, 2011.
[11]	KOZONO H, TERASAWA Y, WAKASUGI Y. A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions[J]. Journal of Functional Analysis, 2017, 272(2): 804-818. doi: 10.1016/j.jfa.2016.06.019
[12]	SEREGIN G, WANG W. Sufficient conditions on Liouville-type theorems for the 3D steady Navier-Stokes equations[J]. Petersburg Mathematical Journal , 2020, 31(2): 387-393. doi: 10.1090/spmj/1603
[13]	SCHULZ S. Liouville-type theorem for the stationary equations of magnetohydrodynamics[J]. Acta Mathematica Scientia, 2019, 39(2): 491-497. doi: 10.1007/s10473-019-0213-7
[14]	YUAN B, XIAO Y. Liouville-type theorems for the 3D stationary Navier-Stokes, MHD and Hall-MHD equations[J]. Journal of Mathematical Analysis and Applications, 2020, 491(2): 124343. doi: 10.1016/j.jmaa.2020.124343
[15]	周艳平, 别群益, 王其如, 等. 三维稳态MHD方程和Hall-MHD方程的Liouville型定理[J]. 中国科学: 数学, 2023, 53(3): 431-440. https://www.cnki.com.cn/Article/CJFDTOTAL-JAXK202303002.htm ZHOU Yanping, BIE Qunyi, WANG Qiru, et al. On Liouville type theorems for three-dimensional stationary MHD and Hall-MHD equations[J]. Scientia Sinica: Mathematica, 2023, 53(3): 431-440. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JAXK202303002.htm
[16]	CHEN X, LI S, WANG W. Remarks on Liouville-type theorems for the steady MHD and Hall-MHD equations[J]. Journal of Nonlinear Science, 2022, 32(1): 1-20. doi: 10.1007/s00332-021-09760-y
[17]	STEIN E M. Singular Integrals and Differentiability Properties of Functions[M]. Princeton University Press, 1970.