Continuous Dependence of Solutions to a Class of Double Diffusion Perturbation Models for Porous Media

SHI Jincheng; XIAO Shengzhong

doi:10.21656/1000-0887.410128

Volume 41 Issue 10

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SHI Jincheng, XIAO Shengzhong. Continuous Dependence of Solutions to a Class of Double Diffusion Perturbation Models for Porous Media[J]. Applied Mathematics and Mechanics, 2020, 41(10): 1092-1102. doi: 10.21656/1000-0887.410128

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Continuous Dependence of Solutions to a Class of Double Diffusion Perturbation Models for Porous Media

doi: 10.21656/1000-0887.410128

SHI Jincheng¹,
XIAO Shengzhong²

¹Huashang College, Guangdong University of Finance & Economics, Guangzhou 511300, P.R.China;²Scientific Research Office, Guangdong AIB Polytechnic,Guangzhou 510507, P.R.China

Funds: The National Natural Science Foundation of China(11371175)

Received Date: 2020-05-08
Rev Recd Date: 2020-06-09
Publish Date: 2020-10-01

Abstract

Abstract

The structural stability of solutions to a class of double diffusion perturbation models for porous media defined in bounded domains was studied. The nonhomogeneous Robin boundary conditions were imposed on the model. With the energy analysis method and the differential inequality technique, some useful a priori estimate of the solution was obtained, and in turn the differential inequality related to the solution was formulated. Then, integration of the differential inequality gave the result of continuous dependence on Lewis number L_e.The result shows that, the double diffusion perturbation model proposed to describe the fluid flow is accurate for porous media.
- double diffusion perturbation equations,
- continuous dependence,
- Rayleigh number,
- Lewis number

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References

[1]	AMES K A, STRAUGHAN B. Non-Standard and Improperly Pose Problems [M]. San Diego: Academic Press, 1997.
[2]	NIELD D A, BEJAN A. Convection in Porous Media [M]. New York: Springer, 1992.
[3]	STRAUGHAN B. Stability and Wave Motion in Porous Media [M]. New York: Springer, 2008.
[4]	PAYNE L E, SONG J C. Spatial decay in a double diffusive convection problem in Darcy flow[J]. Journal of Mathematical Analysis and Applications,2007,330: 864-875.
[5]	FRANCHI F, STRAUGHAN B. Continuous dependence and decay for the Forchheimer equations[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,2003,459: 3195-3202.
[6]	PAYNE L E, STRAUGHAN B. Structural stability for the Darcy equations of flow in porous media[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,1998,454: 1691-1698.
[7]	LIN C, PAYNE L E. Structural stability for a Brinkman fluid[J]. Mathematical Methods in the Applied Sciences,2007,30: 567-578.
[8]	CHEN W H, LIU Y. Structural stability for a Brinkman-Forchheimer type model with temperature dependent solubility[J]. Boundary Value Problems,2016,2016. DOI: 10.1186/s13661-016-0558-y.
[9]	CICHON M, STRAUGHAN B, YANTIR A. On continuous dependence of solutions of dynamic equations[J]. Applied Mathematics and Computation,2015,252: 473-483.
[10]	MA H P, LIU B. Exact controllability and continuous dependence of fractional neutral integro-differential equations with state-dependent delay[J]. Acta Mathematica Scientia(English Series),2017,37(1): 235-258.
[11]	WU H L, REN Y, HU F. Continuous dependence property of BSDE with constraints[J]. Applied Mathematics Letters,2015,45: 41-46.
[12]	HARFASH A J. Structural stability for two convection models in a reacting fluid with magnetic field effect[J]. Annales Henri Poincaré,2014,15:2441-2465.
[13]	LI Y, LIN C. Continuous dependence for the nonhomogeneous Brinkman-Forchheimer equations in a semi-infinite pipe[J]. Applied Mathematics and Computation,2014,244: 201-208.
[14]	LIU Y, XIAO S Z. Structural stability for the Brinkman fluid interfacing with a Darcy fluid in an unbounded domain[J]. Nonlinear Analysis: Real World Applications,2018,42: 308-333.
[15]	LIU Y, XIAO S Z, LIN Y W. Continuous dependence for the Brinkman-Forchheimer fluid interfacing with a Darcy fluid in a bounded domain[J]. Mathematics and Computers in Simulation,2018,150: 66-82.
[16]	LIU Y. Continuous dependence for a thermal convection model with temperature dependent solubility[J]. Applied Mathematics and Computation,2017,308: 18-30.
[17]	李远飞. 大尺度海洋大气动力学三维黏性原始方程对边界参数的连续依赖性[J]. 吉林大学学报(理学版), 2019,57(5): 1053-1059.(LI Yuanfei. Continuous dependence on boundary parameters for three-dimensional viscous primitive equation of large scale ocean atmospheric dynamics[J]. Journal of Jilin University(Science Edition),2019,57(5): 1053-1059.(in Chinese))
[18]	李远飞. 原始方程组对粘性系数的连续依赖性[J]. 山东大学学报(理学版), 2019,54(12): 12-23.(LI Yuanfei. Continuous dependence on the viscosity coefficient for the primitive equations[J]. Journal of Shandong University( Science Edition), 2019,54(12): 12-23.(in Chinese))
[19]	李远飞, 郭连红. 具有边界反应Brinkman-Forchheimer型多孔介质的结构稳定性[J]. 高校应用数学学报, 2019,34(3): 315-324.(LI Yuanfei, GUO Lianhong. Structural stability on boundary reaction terms in a porous medium of Brinkman-Forchheimer type[J]. Applied Mathematics: a Journal of Chinese Universities, 2019,34(3): 315-324.(in Chinese))
[20]	李远飞. 海洋动力学中二维黏性原始方程组解对热源的收敛性[J]. 应用数学和力学, 2020,41(3): 339-352.(LI Yuanfei. Convergence results on heat source for 2D viscous primitive equations of ocean dynamics[J]. Applied Mathematics and Mechanics,2020,41(3): 339-352.(in Chinese))
[21]	CIARLETTA M, STRAUGHAN B, TIBULLO V. Structural stability for a thermal convection model with temperature dependent solubility[J]. Nonlinear Analysis: Real World Applications,2015,22: 34-43.
[22]	STRAUGHAN B. Anisotropic inertia effect in microfluidic porous thermosolutal convection[J]. Microfluidics and Nanofluidics,2014,16: 361-368.
[23]	王双明. 一类具有时滞的周期流行病模型的动力学分析[J]. 山东大学学报(理学版), 2017,52(1): 81-87.(WANG Shuangming. Dynamical analysis of a class of periodic epidemic model with delay[J]. Journal of Shandong University (Natural Science),2017,52(1): 81-87.(in Chinese))
[24]	WEATHERBURN C E. Differential Geometry of Three Dimensions [M]. London: Cambridge University Press, 1980.
[25]	LIN C, PAYNE L E. Continuous dependence on the Soret coefficient for double diffusive convection in Darcy flow[J]. Journal of Mathematical Analysis and Applications,2008,342(1): 311-325.