分数阶各向异性 Navier-Stokes 方程初值问题解的唯一性
Uniqueness of Solutions to Initial Value Problems of Fractional Anisotropic Navier-Stokes Equations
DOI: 10.12677/PM.2021.1112218, PDF, HTML,    国家自然科学基金支持
作者: 刘敉秀, 孙小春*：西北师范大学数学与统计学院，甘肃 兰州
关键词: 分数阶各向异性 Navier-Stokes 方程；Sobolev 空间；乘积公式；Fractional Anisotropic Navier-Stokes Equations； Sobolev Space； Product Formula
摘要: 该文证明了仅有水平分数阶耗散的不可压缩 Navier-Stokes 方程初值问题在各向异性 Sobolev 函数空间 H2α−2,s( ℝ3) 中解的唯一性，其中1/2 < α ≤ 1, α/2 < s < 2α −α/2。 证明的关键是给出 (α,s,t)满足适当范围的函数乘积公式，进而利用 Fourier 分析技巧得出结论。
Abstract: In this paper, we proved the uniqueness of the solution of the initial value problem of incompressible Navier-Stokes equation with only horizontal fractional dissipation in the anisotropic Sobolev function space H2α−2,s( ℝ3), where 1/2 < α ≤ 1, α/2 < s < 2α −α/2. The key of the proof is to give the product formula of the function when (α, s, t) satisfies the appropriate range, and then the conclusion is obtained by using Fourier analysis technique.
文章引用：刘敉秀, 孙小春. 分数阶各向异性 Navier-Stokes 方程初值问题解的唯一性[J]. 理论数学, 2021, 11(12): 1957-1966. https://doi.org/10.12677/PM.2021.1112218

参考文献

[1] Chemin, J.-Y., Desjardins, B., Gallagher, I. and Grenier, E. (2000) Fluids with Anisotropic Viscosity. ESAIM: Mathematical Modelling and Numerical Analysis, 34, 315-335.
https://doi.org/10.1051/m2an:2000143
[2] Iftimie, D. (2002) A Uniqueness Result for the Navier-Stokes Equations with Vanishing Vertical Viscosity. SIAM Journal on Mathematical Analysis, 33, 1483-1493.
https://doi.org/10.1137/S0036141000382126
[3] Iftimie, D. (1999) The 3D Navier-Stkoes Equation Seen as a Perturbation of the 2D Navier- Stkoes Equations. Bulletin de la Soci´et´e Math´ematique de France, 127, 473-517.
https://doi.org/10.24033/bsmf.2358
[4] Paicu, M. (2005) E´ quation anisotrope de Navier-Stokes dans des espaces critiques. Revista Matema´tica Iberoamericana, 21, 179-235.
https://doi.org/10.4171/RMI/420
[5] Chemin, J.-Y. and Zhang, P. (2007) On the Global Well Posedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations. Communications in Mathematical Physics, 272, 529-566.
https://doi.org/10.1007/s00220-007-0236-0
[6] Paicu, M. and Majdoub, M. (2009) Uniform Local Existence for Inhomogeneous Rotating Fluid Equations. Journal of Dynamics and Differential Equations, 21, 21-44.
https://doi.org/10.1007/s10884-008-9120-7
[7] Paicu, M. and Zhang, P. (2011) Global Solutions to the 3-D Incompressible Anisotropic Navier- Stokes System in the Critical Spaces. Communications in Mathematical Physics, 307, 713-759.
https://doi.org/10.1007/s00220-011-1350-6
[8] Ding, Y. and Sun, X. (2015) Uniqueness of Weak Solutions for Fractional Navier-Stokes Equa- tions. Frontiers of Mathematics in China, 10, 33-51.
https://doi.org/10.1007/s11464-014-0370-x
[9] de Oliveira, H.B. (2019) Generalized Nacier-Stokes Equations with Nonlinear Anisotropic Vis- cosity. Analysis and Applications (Singap.), 17, 977-1003.
https://doi.org/10.1142/S021953051950009X
[10] Liu, Y., Paicu, M. and Zhang, P. (2020) Global Well-Posedness of 3-D Anisotropic Navier-Stokes System with Small Unidirectional Derivative. Archive for Rational Mechanics and Anal- ysis, 238, 805-843.
https://doi.org/10.1007/s00205-020-01555-x
[11] Sun, X. and Liu, H. (2021) Uniqueness of the Weak Solution to the Fractional Anisotropic Navier-Stokes Equations. Mathematical Methods in the Applied Sciences, 44, 253-264.
https://doi.org/10.1002/mma.6727
[12] Li, F. and Yuan, B. (2021) Global Well-Posedness of the 3D Generalized Navier-Stokes E-quations with Fractional Partial Dissipation. Acta Applicandae Mathematicae, 171, 16 p.
https://doi.org/10.1007/s10440-021-00388-4
[13] Abidin, M. and Chen, J. (2021) Global Well-Posedness for Fractional Navier-Stokes Equations in Variable Exponent Fourier-Besov-Morrey Spaces. Acta Mathematica Scientia, 41, 164-176.
https://doi.org/10.1007/s10473-021-0109-1
[14] Lou, Z., Yang, Q., He, J. and He, K. (2021) Uniform Analytic Solutions for Fractional Navier- Stokes Equations. Applied Mathematics Letters, 112, 106784, 7 p.
https://doi.org/10.1016/j.aml.2020.106784

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