Acta Mathematica Scientia

, Volume 39, Issue 1, pp 119–126 | Cite as

Global Exponential Nonlinear Stability for Double Diffusive Convection in Porous Medium

  • Lanxi Xu (许兰喜)Email author
  • Ziyi Li (李紫奕)


Nonlinear stability of the motionless double-diffusive solution of the problem of an infinite horizontal fluid layer saturated porous medium is studied. The layer is heated and salted from below. By introducing two balance fields and through defining new energy functionals it is proved that for CLeR, Le ≤ 1 the motionless double-diffusive solution is always stable and for CLe < R, Le < 1 the solution is globally exponentially and nonlinearly stable whenever R < 4π2+LeC, where Le, C and R are the Lewis number, Rayleigh number for solute and heat, respectively. Moreover, the nonlinear stability proved here is global and exponential, and the stabilizing effect of the concentration is also proved.

Key words

energy method energy functional nonlinear stability diffusive convection porous medium 

2010 MR Subject Classification

76E06 76E15 


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  1. [1]
    Gilman A, Bear J. The influence of free convection on soil salinization in arid regions. Transport Porous MED, 1996, 23: 275–301CrossRefGoogle Scholar
  2. [2]
    Pieters G J M, Van Duijn C J. Transient growth in linearly stable gravity-driven flow in porous media. Eur J Mech B/Fluids, 2006, 25: 83–94MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Payne L E, Straughan B. A naturally efficient numerical technique for porous convection stability with non-trivial boundary conditions. Int J Num Anal Meth Geomech, 2000, 24(10): 815–836CrossRefzbMATHGoogle Scholar
  4. [4]
    Nield D, Bejan A D. Convection in Porous Media. 5th ed. New York: Springer, 2017CrossRefzbMATHGoogle Scholar
  5. [5]
    Capone F, Rionero S. Nonlinear stability of a convective motion in a porous layer driven by a horizontally periodic temperature gradient. Continuum Mech Thermodyn, 2003, 15: 529–538MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Lombardo S, Mulone G. Necessary and sufficient conditions for global nonlinear stability for rotating double-diffusive convection in a porous medium. Continuum Mech Thermodyn, 2002, 14: 527–540MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Lombardo S, Mulone G. Necessary and sufficient stability conditions via the eigenvalues-eigenvectors method: an application to the magnetic Bénard problem. Nonlinear Anal, 2005, 63(5/7): e2091–e2101CrossRefzbMATHGoogle Scholar
  8. [8]
    Payne L E, Straughan B. Unconditional nonlinear stability in temperature-dependent viscosity flow in a porous medium. Stud Appl Math, 2000, 105: 59–81MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Straughan B. The Energy Method, Stability, and Nonlinear Convection. 2nd ed. New York: Springer, 2004CrossRefzbMATHGoogle Scholar
  10. [10]
    Mulone G, Rionero S. The rotating Bénard problem: new stability results for any Prandtl and Taylor numbers. Continuum Mech Thermodyn, 1997, 9: 347–363MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Joseph D D. Global stability of the conduction-diffusion solution. Arch Rational Mech Anal, 1970, 36: 285–292MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Lombardo S, Mulone G, Straughan B. Stability in the Bénard problem for a double-diffusive mixture in a porous medium. Math Meth Appl Sci, 2001, 24: 1229–1246CrossRefzbMATHGoogle Scholar
  13. [13]
    Galdi G P, Padula M. A new approach to energy theory in the stability of fluid motion. Arch Rational Mech Anal, 1990, 110: 187–286MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Xu L X, Lan W L. On the nonlinear stability of parallel shear flow in the presence of a coplanar magnetic field. Nonlinear Anal, 2014, 95: 93–98MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Mulone G, Straughan B. An operative method to obtain necessary and sufficient stability conditions for double diffusive convection in porous media. Z Angew Math Mech, 2006, 86(7): 507–520MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Schmitt B J, Von Wahl W. Decomposition of solenoidal fields into poloidal fields, toroidal fields and mean flow. Applications to the Boussinesq equations// Heywood J G, Masuda K, Rautmann R, Solonnikov S A, eds. The Navier-Stokes Equations II-Theory and Numerical Methods. Lecture Notes in Mathematics 1530, Berlin, Heidelberg, New York: Springer, 1992: 291–305CrossRefGoogle Scholar
  17. [17]
    Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. Oxford: Oxford University Press, 1961zbMATHGoogle Scholar
  18. [18]
    Yang Z X, Zhang G B. Global stability of traveling wavefronts for nonlocal rection-diffusion equations with time delay. Acta Math Sci, 2018, 38B(1): 291–304Google Scholar
  19. [19]
    He L, Tang S J, Wang T. Stability of viscous shock waves for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Acta Math Sci, 2016, 36B(1): 36–50MathSciNetzbMATHGoogle Scholar

Copyright information

© Wuhan Institutes of Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of MathematicsBeijing University of Chemical TechnologyBeijingChina

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