September  2012, 11(5): 2037-2054. doi: 10.3934/cpaa.2012.11.2037

Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities

1. 

Department of Mathematics, Koç University, Istanbul, Turkey

2. 

Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  May 2011 Revised  December 2011 Published  March 2012

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with a nonlinearity of arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.
Citation: Varga K. Kalantarov, Sergey Zelik. Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2037-2054. doi: 10.3934/cpaa.2012.11.2037
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.

[2]

A. O. Çelebi, V. K. Kalantarov and D. Ugurlu, Continuous dependence for the convective Brinkman-Forchheimer equations, Appl. Anal., 84 (2005), 877-888.

[3]

A. O. Çelebi, V. K. Kalantarov and D. Ugurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Appl. Math. Lett., 19 (2006), 801-807.

[4]

P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.

[5]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen," Akademie-Verlag, Berlin, 1974.

[6]

R. G. Gordeev, The existence of a periodic solution in a certain problem of tidal dynamics, In "Probles of Mathematical Analysis, No. 4: Integral and Differential Operators. Differential Equations," pp. 3-9, 142-143, Leningrad. Univ., Leningrad, 1973.

[7]

J. Hale, "Asymptotic Behavior of Dissipative Systems," AMS Mathematical Surveys and Monographs no. 25, Providence, RI, 1988.

[8]

I. Kuzin and S. Pohozaev, "Entire Solutions of Semilinear Elliptic Equations," Progress in Nonlinear Differential Equations and their Applications, 33. Birkhöuser Verlag, Basel, 1997.

[9]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly damped wave equation, JDE, 247 (2009), 1120-155. doi: 10.1016/j.jde.2009.04.010.

[10]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Gordon and Breach Science Publishers, New York, 1969.

[11]

O. A. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Accademia Nazionae dei Lincei series, Cambridge University press, Cambridge, 1991.

[12]

Y. Liu and C. Lin, Structural stability for Brinkman-Forchheimer equations, Electron. J. Differential Equations, 2 (2007), 1-8.

[13]

A. L. Likhtarnikov, Existence and stability of bounded and periodic solutions in a nonlinear problem of tidal dynamics, In "The Direct Method in the Theory of Stability and its Applications" (Irkutsk, 1979), pp. 83-91, 276, "Nauka" Sibirsk. Otdel., Novosibirsk, 1981.

[14]

P. Lindqvist, On the equation $div(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.1090/S0002-9939-1990-1007505-7.

[15]

M. Marion, Attractors for reaction-diffusion equations: existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147. doi: 10.1080/00036818708839678.

[16]

D. Nield and A. Bejan, "Convection in Porous Media," Springer, 2006.

[17]

Y. Ouyang and L. Yan, A note on the existence of a global attractor for the BrinkmanForchheimer equations, Nonlinear Analysis, 70 (2009), 2054-2059.

[18]

L. E. Payne and B. Straughan, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Studies in Applied Mathematics, 10 (1999), 419-439.

[19]

M. Röckner and X. Zhang, Tamed 3D Navier-Stokes equation: existence, uniqueness and regularity, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 525-549.

[20]

A. Shenoy, Non-Newtonian fluid heat transfer in porous media, Adv. Heat Transfer, 24 (1994), 101-190. doi: 10.1016/S0065-2717(08)70233-8.

[21]

B. Straughan, "Stability and Wave Motion in Porous Media," Applied Mathematical Sciences, Springer, 2008. doi: 10.1007/978-1-4684-0313-8.

[22]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer Verlag, 1988.

[23]

D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 68 (2008), 1986-1992. doi: 10.1016/j.na.2007.01.025.

[24]

B. Wang and S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Meth. Appl. Sci., 31 (2008), 1479-1495.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.

[2]

A. O. Çelebi, V. K. Kalantarov and D. Ugurlu, Continuous dependence for the convective Brinkman-Forchheimer equations, Appl. Anal., 84 (2005), 877-888.

[3]

A. O. Çelebi, V. K. Kalantarov and D. Ugurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Appl. Math. Lett., 19 (2006), 801-807.

[4]

P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.

[5]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen," Akademie-Verlag, Berlin, 1974.

[6]

R. G. Gordeev, The existence of a periodic solution in a certain problem of tidal dynamics, In "Probles of Mathematical Analysis, No. 4: Integral and Differential Operators. Differential Equations," pp. 3-9, 142-143, Leningrad. Univ., Leningrad, 1973.

[7]

J. Hale, "Asymptotic Behavior of Dissipative Systems," AMS Mathematical Surveys and Monographs no. 25, Providence, RI, 1988.

[8]

I. Kuzin and S. Pohozaev, "Entire Solutions of Semilinear Elliptic Equations," Progress in Nonlinear Differential Equations and their Applications, 33. Birkhöuser Verlag, Basel, 1997.

[9]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly damped wave equation, JDE, 247 (2009), 1120-155. doi: 10.1016/j.jde.2009.04.010.

[10]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Gordon and Breach Science Publishers, New York, 1969.

[11]

O. A. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Accademia Nazionae dei Lincei series, Cambridge University press, Cambridge, 1991.

[12]

Y. Liu and C. Lin, Structural stability for Brinkman-Forchheimer equations, Electron. J. Differential Equations, 2 (2007), 1-8.

[13]

A. L. Likhtarnikov, Existence and stability of bounded and periodic solutions in a nonlinear problem of tidal dynamics, In "The Direct Method in the Theory of Stability and its Applications" (Irkutsk, 1979), pp. 83-91, 276, "Nauka" Sibirsk. Otdel., Novosibirsk, 1981.

[14]

P. Lindqvist, On the equation $div(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.1090/S0002-9939-1990-1007505-7.

[15]

M. Marion, Attractors for reaction-diffusion equations: existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147. doi: 10.1080/00036818708839678.

[16]

D. Nield and A. Bejan, "Convection in Porous Media," Springer, 2006.

[17]

Y. Ouyang and L. Yan, A note on the existence of a global attractor for the BrinkmanForchheimer equations, Nonlinear Analysis, 70 (2009), 2054-2059.

[18]

L. E. Payne and B. Straughan, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Studies in Applied Mathematics, 10 (1999), 419-439.

[19]

M. Röckner and X. Zhang, Tamed 3D Navier-Stokes equation: existence, uniqueness and regularity, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 525-549.

[20]

A. Shenoy, Non-Newtonian fluid heat transfer in porous media, Adv. Heat Transfer, 24 (1994), 101-190. doi: 10.1016/S0065-2717(08)70233-8.

[21]

B. Straughan, "Stability and Wave Motion in Porous Media," Applied Mathematical Sciences, Springer, 2008. doi: 10.1007/978-1-4684-0313-8.

[22]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer Verlag, 1988.

[23]

D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 68 (2008), 1986-1992. doi: 10.1016/j.na.2007.01.025.

[24]

B. Wang and S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Meth. Appl. Sci., 31 (2008), 1479-1495.

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