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November  2013, 12(6): 2615-2625. doi: 10.3934/cpaa.2013.12.2615

A stability result for the Stokes-Boussinesq equations in infinite 3d channels

Marta Lewicka 1, and  Mohammadreza Raoofi 2,

1. 

University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260, United States
2. 

Isfahan University of Technology, Isfahan, Iran

Received  August 2012 Revised  November 2013 Published  May 2013

We consider the Stokes-Boussinesq (and the stationary Na\-vier-Stokes-Boussinesq) equations in a slanted, i.e. not aligned with the gravity's direction, 3d channel and with an arbitrary Rayleigh number. For the front-like initial data and under the no-slip boundary condition for the flow and no-flux boundary condition for the reactant temperature, we derive uniform estimates on the burning rate and the flow velocity, which can be interpreted as stability results for the laminar front.
Keywords: reaction-advection-diffusion equations, Boussinesq equations, combustion in a channel, Navier-Stokes equations, parabolic equations..
Mathematics Subject Classification: Primary: 76B03; Secondary: 76V0.
Citation: Marta Lewicka, Mohammadreza Raoofi. A stability result for the Stokes-Boussinesq equations in infinite 3d channels. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2615-2625. doi: 10.3934/cpaa.2013.12.2615
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), 35-92.  Google Scholar

[2]

Henri Berestycki, "Some Nonlinear PDE's in the Theory of Flame Propagation," ICIAM 99 (Edinburgh), 1322, Oxford Univ. Press, Oxford, 2000.  Google Scholar

[3]

Henri Berestycki, Peter Constantin and Lenya Ryzhik, Non-planar fronts in Boussinesq reactive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 407-437.  Google Scholar

[4]

Peter Constantin, Alexander Kiselev and Lenya Ryzhik, Fronts in reactive convection: bounds, stability, and instability, Comm. Pure Appl. Math., 56 (2003), 1781-1804.  Google Scholar

[5]

Peter Constantin, Alexander Kiselev, Lenya Ryzhik and Andrej Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math., 168 (2008), 643-674.  Google Scholar

[6]

Peter Constantin, Marta Lewicka and Lenya Ryzhik, Travelling waves in two-dimensional reactive Boussinesq systems with no-slip boundary conditions, Nonlinearity, 19 (2006), 2605-2615.  Google Scholar

[7]

Peter Constantin, Alexei Novikov and Lenya Ryzhik, Relaxation in reactive flows, Geom. Funct. Anal., 18 (2008), 1145-1167.  Google Scholar

[8]

Marta Lewicka, Existence of traveling waves in the Stokes-Boussinesq system for reactive flows, J. Differential Equations, 237 (2007), 343-371.  Google Scholar

[9]

Jian-Guo Liu, Jie Liu and Robert L. Pego, Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure Appl. Math., 60 (2007), 1443-1487.  Google Scholar

[10]

Marta Lewicka and Piotr B. Mucha, On the existence of traveling waves in the 3D Boussinesq system, Comm. Math. Phys., 292 (2009), 417-429.  Google Scholar

[11]

Rozenn Texier-Picard and Vitaly Volpert, Problèmes de réaction-diffusion-convection dans des cylindres non bornés, C. R. Acad. Sci. Paris S\'er. I Math., 333 (2001), 1077-1082.  Google Scholar

[12]

Wenzheng Xie, A sharp pointwise bound for functions with $L^2$-Laplacians on arbitrary domains and its applications, Bull. Amer. Math. Soc. (N.S.), 26 (1992), 294-298.  Google Scholar

[13]

Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, "The Mathematical Theory of Combustion and Explosions," Consultants Bureau [Plenum], New York, 1985. Translated from the Russian by Donald H. McNeill.  Google Scholar

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), 35-92.  Google Scholar

[2]

Henri Berestycki, "Some Nonlinear PDE's in the Theory of Flame Propagation," ICIAM 99 (Edinburgh), 1322, Oxford Univ. Press, Oxford, 2000.  Google Scholar

[3]

Henri Berestycki, Peter Constantin and Lenya Ryzhik, Non-planar fronts in Boussinesq reactive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 407-437.  Google Scholar

[4]

Peter Constantin, Alexander Kiselev and Lenya Ryzhik, Fronts in reactive convection: bounds, stability, and instability, Comm. Pure Appl. Math., 56 (2003), 1781-1804.  Google Scholar

[5]

Peter Constantin, Alexander Kiselev, Lenya Ryzhik and Andrej Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math., 168 (2008), 643-674.  Google Scholar

[6]

Peter Constantin, Marta Lewicka and Lenya Ryzhik, Travelling waves in two-dimensional reactive Boussinesq systems with no-slip boundary conditions, Nonlinearity, 19 (2006), 2605-2615.  Google Scholar

[7]

Peter Constantin, Alexei Novikov and Lenya Ryzhik, Relaxation in reactive flows, Geom. Funct. Anal., 18 (2008), 1145-1167.  Google Scholar

[8]

Marta Lewicka, Existence of traveling waves in the Stokes-Boussinesq system for reactive flows, J. Differential Equations, 237 (2007), 343-371.  Google Scholar

[9]

Jian-Guo Liu, Jie Liu and Robert L. Pego, Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure Appl. Math., 60 (2007), 1443-1487.  Google Scholar

[10]

Marta Lewicka and Piotr B. Mucha, On the existence of traveling waves in the 3D Boussinesq system, Comm. Math. Phys., 292 (2009), 417-429.  Google Scholar

[11]

Rozenn Texier-Picard and Vitaly Volpert, Problèmes de réaction-diffusion-convection dans des cylindres non bornés, C. R. Acad. Sci. Paris S\'er. I Math., 333 (2001), 1077-1082.  Google Scholar

[12]

Wenzheng Xie, A sharp pointwise bound for functions with $L^2$-Laplacians on arbitrary domains and its applications, Bull. Amer. Math. Soc. (N.S.), 26 (1992), 294-298.  Google Scholar

[13]

Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, "The Mathematical Theory of Combustion and Explosions," Consultants Bureau [Plenum], New York, 1985. Translated from the Russian by Donald H. McNeill.  Google Scholar

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