四维不可压缩Navier-Stokes方程的能量守恒

王斌; 周艳平; 别群益

doi:10.21656/1000-0887.430370

四维不可压缩Navier-Stokes方程的能量守恒

doi: 10.21656/1000-0887.430370

三峡大学理学院，湖北宜昌 443002

基金项目:

国家自然科学基金项目 11901346

国家自然科学基金项目 11871305

详细信息

作者简介:
王斌(1998—)，女，硕士生(E-mail: 2895969956@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:
周艳平(1980—)，女，副教授，博士(通讯作者. E-mail: zhyp5208@163.com)

中图分类号: O175.2
计量
- 文章访问数: 349
- HTML全文浏览量: 123
- PDF下载量: 65
- 被引次数: 0
出版历程
- 收稿日期: 2022-11-16
- 修回日期: 2022-12-24
- 刊出日期: 2023-08-01

Energy Conservation of the 4 D Incompressible Navier-Stokes Equations

College of Science, China Three Gorges University, Yichang, Hubei 443002, P.R.China

摘要

摘要: 研究了四维不可压缩Navier-Stokes方程的能量守恒，当该方程的Leray-Hopf弱解(适当弱解)存在维数小于4的奇异集时，基于Wu在文章中关于四维不可压缩Navier-Stokes方程的部分正则性结果，得到了四维空间中$L^q\left([0, T] ; L^p\left(\mathbb{R}^4\right)\right)$条件，保证该方程能量守恒.
- Navier-Stokes方程 /
- 部分正则性 /
- 能量守恒
Abstract: The energy conservation of 4D incompressible Navier-Stokes equations was studied. In the case of a singular set with a dimension number less than 4 for the Leray-Hopf weak solution (suitable weak solution), the $L^q\left([0, T] ; L^p\left(\mathbb{R}^4\right)\right)$ condition in the 4D space was obtained based on Wu's partial regularity results about the 4D incompressible Navier-Stokes equations, to ensure the energy conservation.
- Navier-Stokes equation /
- partial regularity /
- energy conservation

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图 1 d=0

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图 2 0 < d < 2

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图 3 d=2

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图 4 2 < d < 4

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参考文献(26)

[1]	FEFFERMAN C L. Existence and smoothness of the Navier-Stokes equation[J]. The millennium prize problems, 2000, 57 : 67.
[2]	施惟慧. Navier-Stokes方程稳定性研究(Ⅰ)[J]. 应用数学和力学, 1994, 15(9): 821-822. http://www.applmathmech.cn/article/id/2971 SHI Weihui. Stability study of Navier-Stokes equation (Ⅰ)[J]. Applied Mathematics and Mechanics, 1994, 15(9): 821-822. (in Chinese) http://www.applmathmech.cn/article/id/2971
[3]	施惟慧, 方晓佐. Navier-Stokes方程稳定性研究(Ⅱ)[J]. 应用数学和力学, 1994, 15(10): 879-883. http://www.applmathmech.cn/article/id/2957 SHI Weihui, FANG Xiaozuo. Stability study of Navier-Stokes equation (Ⅱ)[J]. Applied Mathematics and Mechanics, 1994, 15(10): 879-883. (in Chinese) http://www.applmathmech.cn/article/id/2957
[4]	FEIREISL E, NOVOTNY A, PETZELTOVÁ H. On the existence of globally defined weak solutions to the Navier-Stokes equations[J]. Journal of Mathematical Fluid Mechanics, 2001, 3(4): 358-392. doi: 10.1007/PL00000976
[5]	王金城, 齐进, 吴锤结. 不可压缩Navier-Stokes方程最优动力系统建模和分析[J]. 应用数学和力学, 2020, 41(1): 1-15. doi: 10.21656/1000-0887.400279 WANG Jincheng, QI Jin, WU Chuijie. Modeling and analysis of the incompressible Navier-Stokes equation optimal dynamical system[J]. Applied Mathematics and Mechanics, 2020, 41(1): 1-15. (in Chinese) doi: 10.21656/1000-0887.400279
[6]	ZHANG Z, CHEN Q, MIAO C. On the uniqueness of weak solutions for the 3D Navier-Stokes equations[J]. Annales de l'Institut Henri Poincaré C, 2009, 26(6): 2165-2180. doi: 10.1016/j.anihpc.2009.01.008
[7]	LERAY J. Sur le mouvement d'un liquide visqueux emplissant l'espace[J]. Acta Mathematica, 1934, 63(1): 193-248.
[8]	HOPF E. Vber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Erhard Schmidt zu seinem 75. Geburtstag gewidmet[J]. Mathematische Nachrichten, 1950, 4 (1/6): 213-231.
[9]	MASUDA K. Weak solutions of Navier-Stokes equations[J]. Tohoku Mathematical Journal: Second Series, 1984, 36(4): 623-646.
[10]	FOIAS C. Une remarque sur l'unicité des solutions deséquations de Navier-Stokes en dimension n[J]. Bulletin de la Société Mathématique de France, 1961, 89 : 1-8.
[11]	SERRIN J. The Initial-Value Problem for the Navier-Stokes Equations[M]. Madison: The University of Wisconsion Press, 1963.
[12]	SCHEFFER V. Partial regularity of solutions to the Navier-Stokes equations[J]. Pacific Journal of Mathematics, 1976, 66(2): 535-552. doi: 10.2140/pjm.1976.66.535
[13]	SCHEFFER V. The Navier-Stokes equations in space dimension four[J]. Communications in Mathematical Physics, 1978, 61(1): 41-68. doi: 10.1007/BF01609467
[14]	SCHEFFER V. The Navier-Stokes equations on a bounded domain[J]. Communications in Mathematical Physics, 1980, 73(1): 1-42.
[15]	WU B. Partially regular weak solutions of the Navier-Stokes equations in R ⁴×[0, ∞][J]. Archive for Rational Mechanics and Analysis, 2021, 239(3): 1771-1808. doi: 10.1007/s00205-020-01603-6
[16]	DONG H, DU D. Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time[J]. Communications in Mathematical Physics, 2007, 273(3): 785-801. doi: 10.1007/s00220-007-0259-6
[17]	WANG Y, WU G. A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier-Stokes equations[J]. Journal of Differential Equations, 2014, 256(3): 1224-1249.
[18]	ONSAGER L. Statistical hydrodynamics[J]. Il Nuovo Cimento, 1949, 6(2): 279-287.
[19]	CAFFARELLI L, KOHN R, NIRENBERG L. Partial regularity of suitable weak solutions of the Navier-Stokes equations[J]. Communications on Pure and Applied Mathematics, 1982, 35(6): 771-831.
[20]	LIONS J L. Sur la régularité et l'unicité des solutions turbulentes des équations de Navier-Stokes[J]. Rendiconti del Seminario Matematico della Universita di Padova, 1960, 30 : 16-23.
[21]	LADYŽENSKAJA O A, SOLONNIKOV V A, URAL'CEVA N N. Linear and Quasilinear Equations of Parabolic Type[M]. American Mathematical Soc, 1988.
[22]	KUKAVICA I. Role of the pressure for validity of the energy equality for solutions of the Navier-Stokes equation[J]. Journal of Dynamics and Differential Equations, 2006, 18(2): 461-482.
[23]	SHINBROT M. The energy equation for the Navier-Stokes system[J]. SIAM Journal on Mathematical Analysis, 1974, 5(6): 948-954.
[24]	LESLIE T M, SHVYDKOY R. Conditions implying energy equality for weak solutions of the Navier-Stokes equations[J]. SIAM Journal on Mathematical Analysis, 2018, 50(1): 870-890.
[25]	SHVYDKOY R. On the energy of inviscid singular flows[J]. Journal of Mathematical Analysis and Applications, 2009, 349(2): 583-595.
[26]	EVANS L C, GARIEPY R F. Measure Theory and Fine Properties of Functions[M]. Routledge, 2018.