三维广义 Navier-Stokes-Coriolis方程组在 Besov 空间中的整体适定性
Global Well-Posedness of theThree-Dimensional Generalized Navier-Stokes-Coriolis Equations in Besov Spaces
DOI: 10.12677/PM.2024.148301, PDF,    科研立项经费支持
作者: 买园伟*, 王伟宁:西北师范大学数学与统计学院,甘肃 兰州
关键词: 广义 Navier-Stokes 方程组Coriolis 力整体适定性Generalized Navier-Stokes Equations Coriolis Force Global Well-Posedness
摘要: 通过在分数阶拉普拉斯耗散的正则化效应和 Coriolis 力的色散效应之间建立新的平衡,我们证明了三维广义 Navier-Stokes-Coriolis 方程组柯西问题在 Besov 空间中的整体适定性。特别地,当旋转速度足够快时,允许初速度任意大。
Abstract: By striking new balances between the regularizing effects of the fractional Lapla- cian dissipation and the dispersive effects of Coriolis force, we prove the global well- posedness of Cauchy problem for the three-dimensional generalized Navier-Stokes- Coriolis equations in Besov spaces. Particularly, it is shown that initial velocity can be arbitrarily large provided that the speed of rotation is sufficiently high.
文章引用:买园伟, 王伟宁. 三维广义 Navier-Stokes-Coriolis方程组在 Besov 空间中的整体适定性[J]. 理论数学, 2024, 14(8): 31-45. https://doi.org/10.12677/PM.2024.148301

参考文献

[1] Chemin, J.Y., Desjardins, B., Gallagher, I. and Grenier, E. (2006) Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier-Stokes Equations. Oxford University Press, 32.
[2] Metzler, R. and Klafter, J. (2000) The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach. Physics Reports, 339, 1-77.
https://doi.org/10.1016/s0370-1573(00)00070-3
[3] Leray, J. (1934) Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Mathemat- ica, 63, 193-248.
https://doi.org/10.1007/bf02547354
[4] Fujita, H. and Kato, T. (1964) On the Navier-Stokes Initial Value Problem. I. Archive for Rational Mechanics and Analysis, 16, 269-315.
https://doi.org/10.1007/bf00276188
[5] Kato, T. (1984) Strong Lp-Solutions of the Navier-Stokes Equation in Rm, with Applications to Weak Solutions. Mathematische Zeitschrift, 187, 471-480.
https://doi.org/10.1007/bf01174182
[6] Cannone, M. (1997) A Generalization of a Theorem by Kato on Navier-Stokes Equations. Revista Matematica Iberoamericana, 13, 515-541.
https://doi.org/10.4171/rmi/229
[7] Koch, H. and Tataru, D. (2001) Well-Posedness for the Navier-Stokes Equations. Advances in Mathematics, 157, 22-35.
https://doi.org/10.1006/aima.2000.1937
[8] Mercado, J.R., Guido, E.P., SSnchez-Sesma, A.J.,I~niguez, M. and Gonzalez, A. (2012) Analysis of the Blasius' Formula and the Navier-Stokes Fractional Equation. In: Klapp, J., Medina, A., Cros, A. and Vargas, C., Eds., Fluid Dynamics in Physics, Engineering and Environmental Applications, Springer Berlin Heidelberg, 475-480.
https://doi.org/10.1007/978-3-642-27723-8 44
[9] Lions, J.L. (1969) Quelques methodes de resolution des problemes aux limites non lineaires. Dunod.
[10] Wu, J. (2003) Generalized MHD Equations. Journal of Differential Equations, 195, 284-312.
https://doi.org/10.1016/j.jde.2003.07.007
[11] Wu, J. (2005) Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces. Communications in Mathematical Physics, 263, 803-831.
https://doi.org/10.1007/s00220-005-1483-6
[12] Yu, X. and Zhai, Z. (2012) Well-Posedness for Fractional Navier-Stokes Equations in the Largest Critical Spaces B∞,∞-(2β-1)(Rn). Mathematical Methods in the Applied Sciences, 35, 676- 683.
https://doi.org/10.1002/mma.1582
[13] Babin, A., Mahalov, A. and Nicolaenko, B. (1997) Regularity and Integrability of 3D Euler and Navier-Stokes Equations for Rotating Fluids. Asymptotic Analysis, 15, 103-150.
https://doi.org/10.3233/asy-1997-15201
[14] Nicolaenko, B., Babin, A. and Mahalov, A. (1999) Global Regularity of 3D Rotating Navier- Stokes Equations for Resonant Domains. Indiana University Mathematics Journal, 48, 1133- 1176.
https://doi.org/10.1512/iumj.1999.48.1856
[15] Chemin, J.-Y., Desjardins, B., Gallagher, I. and Grenier, E. (2002) Anisotropy and Dispersion in Rotating Fluids. In: Studies in Mathematics and Its Applications, Vol. 31, Elsevier, 171-192.
https://doi.org/10.1016/s0168-2024(02)80010-8
[16] Iwabuchi, T. and Takada, R. (2013) Global Solutions for the Navier-Stokes Equations in the Rotational Framework. Mathematische Annalen, 357, 727-741.
https://doi.org/10.1007/s00208-013-0923-4
[17] Koh, Y., Lee, S. and Takada, R. (2014) Dispersive Estimates for the Navier-Stokes Equations in the Rotational Framework. Advances in Differential Equations, 19, 857-878.
https://doi.org/10.57262/ade/1404230126
[18] Sun, J., Yang, M. and Cui, S. (2016) Existence and Analyticity of Mild Solutions for the 3D Rotating Navier-Stokes Equations. Annali di Matematica Pura ed Applicata, 196, 1203-1229.
https://doi.org/10.1007/s10231-016-0613-4
[19] Sun, X. and Ding, Y. (2019) Dispersive Effect of the Coriolis Force and the Local Well- Posedness for the Fractional Navier-Stokes-Coriolis System. Journal of Evolution Equations, 20, 335-354.
https://doi.org/10.1007/s00028-019-00531-7
[20] Ahn, J., Kim, J. and Lee, J. (2021) Coriolis Effect on Temporal Decay Rates of Global Solutions to the Fractional Navier-Stokes Equations. Mathematische Annalen, 383, 259-289.
https://doi.org/10.1007/s00208-020-02122-1
[21] Kishimoto, N. and Yoneda, T. (2017) Global Solvability of the Rotating Navier-Stokes Equations with Fractional Laplacian in a Periodic Domain. Mathematische Annalen, 372, 743-779.
https://doi.org/10.1007/s00208-017-1605-4
[22] Bahouri, H., Chemin, J.Y. and Danchin, R. (2011) Fourier Analysis and Nonlinear Partial Differential Equations. In: Grundlehren der mathematischen Wissenschaften, Vol. 343, Springer- Verlag.
[23] Abidi, H., Gui, G. and Zhang, P. (2013) Well-Posedness of 3-D Inhomogeneous Navier-Stokes Equations with Highly Oscillatory Initial Velocity Field. Journal de Mathematiques Pures et Appliquees, 100, 166-203.
https://doi.org/10.1016/j.matpur.2012.10.015
[24] Hieber, M. and Shibata, Y. (2009) The Fujita-Kato Approach to the Navier-Stokes Equations in the Rotational Framework. Mathematische Zeitschrift, 265, 481-491.
https://doi.org/10.1007/s00209-009-0525-8
[25] Zhai, Z. (2010) Global Well-Posedness for Nonlocal Fractional Keller-Segel Systems in Critical Besov Spaces. Nonlinear Analysis: Theory, Methods Applications, 72, 3173-3189.
https://doi.org/10.1016/j.na.2009.12.011
[26] Tomas, P.A. (1975) A Restriction Theorem for the Fourier Transform. Bulletin of the American Mathematical Society, 81, 477-478.
https://doi.org/10.1090/s0002-9904-1975-13790-6
[27] Strichartz, R.S. (1977) Restrictions of Fourier Transforms to Quadratic Surfaces and Decay of Solutions of Wave Equations. Duke Mathematical Journal, 44, 705-714.
https://doi.org/10.1215/s0012-7094-77-04430-1

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