数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 1081-1104.doi: 10.1007/s10473-023-0306-1
Xueke Pu, Wenli Zhou†
收稿日期:2021-09-18
修回日期:2022-10-17
出版日期:2023-06-25
发布日期:2023-06-06
通讯作者:
† Wenli Zhou, E-mail: wywlzhou@163.com
作者简介:Xueke Pu, E-mail: puxueke@gmail.com
基金资助:Xueke Pu, Wenli Zhou†
Received:2021-09-18
Revised:2022-10-17
Online:2023-06-25
Published:2023-06-06
Contact:
† Wenli Zhou, E-mail: wywlzhou@163.com
About author:Xueke Pu, E-mail: puxueke@gmail.com
Supported by:摘要： In this paper, we rigorously derive the governing equations describing the motion of a stable stratified fluid, from the mathematical point of view. In particular, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ratio goes to zero, and the rate of convergence is of the same order as the aspect ratio. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.
Xueke Pu, Wenli Zhou. ON THE RIGOROUS MATHEMATICAL DERIVATION FOR THE VISCOUS PRIMITIVE EQUATIONS WITH DENSITY STRATIFICATION*[J]. 数学物理学报(英文版), 2023, 43(3): 1081-1104.
Xueke Pu, Wenli Zhou. ON THE RIGOROUS MATHEMATICAL DERIVATION FOR THE VISCOUS PRIMITIVE EQUATIONS WITH DENSITY STRATIFICATION*[J]. Acta mathematica scientia,Series B, 2023, 43(3): 1081-1104.
| [1] Azérad P, Guillén F.Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics. SIAM J Math Anal, 2001, 33(4): 847-859 [2] Bardos C, Lopes Filho M C, Niu D, et al. Stability of two-dimensional viscous incompressible flows under three-dimensional perturbations and inviscid symmetry breaking. SIAM J Math Anal, 2013, 45(3): 1871-1885 [3] Bresch D, Guillén-González F, Masmoudi N, Rodríguez-Bellido M A. On the uniqueness of weak solutions of the two-dimensional primitive equations. Differ Integral Equ, 2003, 16(1): 77-94 [4] Cao C, Ibrahim S, Nakanishi K, Titi E S.Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics. Commun Math Phys, 2015, 337(2): 473-482 [5] Cao C, Li J, Titi E S.Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity. Arch Ration Mech Anal, 2014, 214(1): 35-76 [6] Cao C, Li J, Titi E S.Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity. J Differ Equ, 2014, 257(11): 4108-4132 [7] Cao C, Li J, Titi E S.Global well-posedness of the three-dimensional primitive equations with only hori- zontal viscosity and diffusion. Commun Pure Appl Math, 2016, 69(8): 1492-1531 [8] Cao C, Li J, Titi E S.Strong solutions to the 3D primitive equations with horizontal dissipation: near H1 initial data. J Funct Anal, 2017, 272(11): 4606-4641 [9] Cao C, Li J, Titi E S.Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity. Phys D, 2020, 412: 132606 [10] Cao C, Titi E S.Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann Math, 2007, 166(1): 245-267 [11] Cao C, Titi E S.Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion. Commun Math Phys, 2012, 310(2): 537-568 [12] Cao C, Titi E S.Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model. Commun Pure Appl Math, 2003, 56(2): 198-233 [13] Fang D, Han B.Global well-posedness for the 3D primitive equations in anisotropic framework. J Math Anal Appl, 2020, 484(2): 123714 [14] Furukawa K, Giga Y, Hieber M, et al.Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equations. Nonlinearity, 2020, 33(12): 6502-6516 [15] Giga Y, Gries M, Hieber M, et al.The hydrostatic Stokes semigroup and well-posedness of the primitive equations on spaces of bounded functions. J Funct Anal, 2020, 279(3): 108561 [16] Han-Kwan D, Nguyen T.Ill-posedness of the hydrostatic Euler and singular Vlasov equations. Arch Ration Mech Anal, 2016, 221(3): 1317-1344 [17] Hieber M, Hussein A, Kashiwabara T.Global strong Lp well-posedness of the 3D primitive equations with heat and salinity diffusion. J Differ Equ, 2016, 261(12): 6950-6981 [18] Hieber M, Kashiwabara T.Global strong well-posedness of the three dimensional primitive equations in Lp-spaces. Arch Ration Mech Anal, 2016, 221(3): 1077-1115 [19] Ibrahim S, Lin Q, Titi E S.Finite-time blowup and ill-posedness in Sobolev spaces of the inviscid primitive equations with rotation. J Differ Equ, 2021, 286: 557-577 [20] Ju N.On H2 solutions and z-weak solutions of the 3D primitive equations. Indiana Univ Math J, 2017, 66(3): 973-996 [21] Kobelkov G M.Existence of a solution in the large for the 3D large-scale ocean dynamics equations. C R Math Acad Sci Paris, 2006, 343(4): 283-286 [22] Kukavica I, Pei Y, Rusin W, Ziane M.Primitive equations with continuous initial data. Nonlinearity, 2014, 27(6): 1135-1155 [23] Kukavica I, Ziane M.The regularity of solutions of the primitive equations of the ocean in space dimension three. C R Math Acad Sci Paris, 2007, 345(5): 257-260 [24] Kukavica I, Ziane M.On the regularity of the primitive equations of the ocean. Nonlinearity, 2007, 20(12): 2739-2753 [25] Li J, Titi E S.The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: rigorous justification of the hydrostatic approximation. J Math Pures Appl, 2019, 124: 30-58 [26] Li J, Titi E S.Existence and uniqueness of weak solutions to viscous primitive equations for a certain class of discontinuous initial data. SIAM J Math Anal, 2017, 49(1): 1-28 [27] Li J, Titi E S, Yuan G.The primitive equations approximation of the anisotropic horizontally viscous Navier-Stokes equations. J Differ Equ, 2022, 306: 492-524 [28] Li J, Yuan G.Global well-posedness of z-weak solutions to the primitive equations without vertical diffu- sivity. J Math Phys, 2022, 63(2): 021501 [29] Lions J L, Temam R, Wang S.New formulations of the primitive equations of atmosphere and applications. Nonlinearity, 1992, 5(2): 237-288 [30] Lions J L, Temam R,Wang S.On the equations of the large scale ocean. Nonlinearity, 1992, 5(5): 1007-1053 [31] Lions J L, Temam R, Wang S.Mathematical theory for the coupled atmosphere-ocean models. J Math Pures Appl, 1995, 74(2): 105-163 [32] Majda A.Introduction to PDEs and Waves for the Atmosphere and Ocean. Providence, RI: American Mathematical Society, 2003 [33] Pedlosky J. Geophysical Fluid Dynamics. New York: Springer, 1987 [34] Pu X, Zhou W. Rigorous derivation of the full primitive equations by the scaled Boussinesq equations with rotation. Bull Malays Math Sci Soc, 2023, 46(3): Art 88 [35] Renardy M.Ill-posedness of the hydrostatic Euler and Navier-Stokes equations. Arch Ration Mech Anal, 2009, 194(3): 877-886 [36] Robinson J C, Rodrigo J L, Sadowski W.The Three-Dimensional Navier-Stokes Equations: Classical Theory. Cambridge: Cambridge University Press, 2016 [37] Seidov D.An intermediate model for large-scale ocean circulation studies. Dynam Atmos Oceans, 1996, 25(1): 25-55 [38] Tachim Medjo T.On the uniqueness of z-weak solutions of the three-dimensional primitive equations of the ocean. Nonlinear Anal Real World Appl, 2010, 11(3): 1413-1421 [39] Temam R.Navier Stokes Equations: Theory and Numerical Analysis. Amsterdam: North-Holland Pub- lishing Company, 1977 [40] Vallis G K.Atmospheric and Oceanic Fluid Dynamics. Cambridge: Cambridge University Press, 2006 [41] Washington W M, Parkinson C L.An Introduction to Three Dimensional Climate Modeling. Oxford: Oxford University Press, 1986 [42] Wong T K.Blowup of solutions of the hydrostatic Euler equations. Proc Amer Math Soc, 2015, 143(3): 1119-1125 |