关于Boussinesq方程组无粘极限的研究

Abstract

Abstract:

In this paper, we investigate the inviscid limit of the 3D viscous Boussinesq equations with slip boundary condition. We establish the local well-posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, we establish the vanishing viscosity limit process and obtain a strong rate of convergence as the boundary of the domain is flat. In addition, the key observation is that the boundary term as $θ$ can be estimated by the part of high order of energy through the trace formula.

Key words: Boussinesq equations, Vanishing viscosity limit, Slip boundary conditions

CLC Number:

O175.2

Lianhong Guo. Research on the Inviscid Limit for Boussinesq Equations[J].Acta mathematica scientia,Series A, 2021, 41(1): 91-99.

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References 24

1	Constantin P , Foias C . Navier-Stokes Equations. Chicago: University of Chicago Press, 1988
2	Pedlosky J . Geophysical Fluid Dynamics. New York: Springer-Verlag, 1987
3	Majda A . Introduction to PDEs and Waves for the Atmosphere and Ocean. New York: Courant Institute of Mathematical Sciences, 2003
4	Cushman-Roisin B , Jean-Marie B . Introduction to Geophysical Fluid Dynamics, Physical and Numerical Aspects. New York: Academic Press, 2011
5	Rajagopal K R , Ruzicka M , Srinivasa A R . On the Oberbeck-Boussinesq approximation. Mathematical Models and Methods in Applied Sciences, 1996, 6 (8): 1157- 1167 doi: 10.1142/S0218202596000481
6	Chae D . Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv Math, 2006, 203: 497- 513 doi: 10.1016/j.aim.2005.05.001
7	Hou T Y , Li C . Global well-posedness of the viscous Boussinesq equations. Discrete Contin Dyn Syst, 2005, 12 (1): 1- 12 doi: 10.3934/dcds.2005.12.1
8	Berselli L C , Spirito S . On the vanishing viscosity limit for the Navier-Stokes equations under slip boundary conditions in general domains. Commun Math Phys, 2012, 316: 171- 198 doi: 10.1007/s00220-012-1581-1
9	Iftimie D , Planas G . Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity, 2006, 19 (4): 899- 918 doi: 10.1088/0951-7715/19/4/007
10	Xiao Y L , Xin Z P . On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Comm Pure Appl Math, 2007, 60: 1027- 1055 doi: 10.1002/cpa.20187
11	Xiao Y L , Xin Z P , Wu J H . Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition. J Funct Anal, 2009, 257: 3375- 3394 doi: 10.1016/j.jfa.2009.09.010
12	Xiao Y L , Xin Z P . Remarks on vanishing viscosity limits for 3D Navier-Stokes equations with a slip boundary condition. Chin Ann Math, 2011, 32: 321- 332 doi: 10.1007/s11401-011-0649-0
13	Berselli L C , Spirito S . On the Boussinesq system:regularity criteria and singular limits. Methods Appl Anal, 2011, 18: 391- 496
14	Jiang S , Zhang J W , Zhao J N . Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane. Journal of Differential Equations, 2011, 250 (10): 3907- 3936 doi: 10.1016/j.jde.2011.01.002
15	Berselli L C , Spirito S . An elementary approach to the inviscid limits for the 3D Navier-Stokes equations with slip boundary conditions and applications to the 3D Boussinesq equations. Nonlinear Differ Equ Appl, 2014, 21: 149- 166 doi: 10.1007/s00030-013-0242-1
16	Wang J , Xie F . Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain. Proc Roy Soc Edinburgh Sect A, 2015, 145 (3): 611- 637 doi: 10.1017/S0308210513000875
17	Li L R , Wang K , Hong M L . The Asymptotic limit for the 3D Boussinesq system. Chinese Quarrterly Journal of Mathematics, 2016, 31 (1): 51- 59
18	Li H P , Xu Z H , Zhu X L . The vanishing diffusivity limit for the 2-D Boussinesq equations with boundary effct. Nonlinear Analysis, 2016, 133: 144- 160 doi: 10.1016/j.na.2015.12.004
19	Zhang Z P . Vanishing diffusivity limit for the 3D heat-conductive Boussinesq equations with a slip boundary condition. J Math Anal Appl, 2018, 460 (1): 47- 57 doi: 10.1016/j.jmaa.2017.11.045
20	Navier C L M . Memoire sur les loir du mouvement des fluids. Memoires de l'Academie Royale des Sciences de l'Institut de France, 1823, 6: 389- 440
21	Beirão da Veiga H , Crispo F . Sharp inviscid limit results under Navier type boundary conditions:An $L^{p}$ theory. J Math Fluid Mech, 2010, 12: 397- 411 doi: 10.1007/s00021-009-0295-4
22	Beirão da Veiga H , Crispo F . Concerning the $ W^{k, p}$-inviscid limit for 3D flows under a slip boundary condition. J Math Fluid Mech, 2011, 13: 117- 135 doi: 10.1007/s00021-009-0012-3
23	Chen P F , Xiao Y H , Zhang H . Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equations with a slip boundary condition. Math Meth Appl Sci, 2017, 40: 5925- 5932 doi: 10.1002/mma.4443
24	Constantin P , Doering C R . Infinite Prandtl number convection. J Stat Phys, 1999, 94: 159- 172 doi: 10.1023/A:1004511312885

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[3]	SHEN Ji-Hong, ZHANG Ming-You, YANG Yan-Bing, LIU Bo-Wei, XU Run-Zhang. Global Well-posedness of Cauchy Problem for Damped Multidimensional Generalized Boussinesq Equations [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1173-1187.

Research on the Inviscid Limit for Boussinesq Equations

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