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On the weak solutions and determining modes of the g-Benard problem

Year 2018, Volume: 47 Issue: 6, 1453 - 1466, 12.12.2018

Muharrem Özlük Meryem Kaya

Abstract

In this paper we study the existence and uniqueness of weak solutions of the g-Benard problem. Then, we investigate the long-term dynamics; specifically, we derive upper bounds for the number of determining modes for this system.

Keywords

B\'{e}nard convection existence and uniqueness of weak solutions determining modes

References

  • Bae, H., Roh, J., Existence of Solutions of the g-Navier-Stokes Equations, Taiwanese J. Math., 8, No. 1, 85-102, 2004.
  • Boland, J. and Layton, W., Error analysis for finite element methods for steady natural convection problems, Numer. Funct. Anal. Optim., 11:5-6, 449-483, 1990, DOI: 10.1080/01630569008816383.
  • Galdi,G.P., Lectures in Mathematical Fluid Dynamics, Birkhauser-Verlag, 2000.
  • Farhat, A., Jolly, M.S., and Titi, E.S., Continuous Data Assimilation for the 2D Benard Convection Through Velocity Measurements Alone, Physica D, 303, 59-66, 2015.
  • Foias, C., Jolly, M.S., Kravchenko, R., and Titi, E.S., A unified approach to determining forms for the 2D Navier-Stokes equations -- the general interpolants case, Russ. Math. Surv., 69, No. 2, 359-381, 2014.
  • Foias, C., Manley, O., Rosa, R. and Temam, R., Navier - Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications, vol. 83, Cambridge University Press, 2004.
  • Foias, C., Manley, O., Temam, R., Attractors for the Benard problem: Existence and physical bounds on their fractal dimension, Nonlinear Anal. Theory, Methods & Applications, 11, 939-967, 1987.
  • Foias, C., Prodi, G., Sur le comportement global des solutions non stationnaires des equations de Navier-Stokes en dimension two, Rend. Sem. Mat. Univ., Padova, 39, 1-34, 1967.
  • Hale, J.K., Raugel, G., A damped hyperbolic equation on thin domains, Trans. Amer. Math. Soc., 329, 185-219, 1992.
  • Hale, J.K., Raugel, G., Partial differential equations on thin domains, Differ. Eq. Math. Phys., Birmingham, AL, 1990, Academic Press, Boston, 63-97, 1992.
  • Hale, J. K., Raugel, G., Reaction - Diffusion equation on thin domains, J. Math. Pures Appl., 71, 33-95, 1992.
  • Hoang, L.T., Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (I), J. Math. Fluid Mech., 12, No. 3, 435-472, 2010.
  • Hoang, L.T., Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (II), J. Math. Fluid Mech., 15, 361-395, 2013.
  • Hu, C., Navier-Stokes equations in 3D thin domains with Navier friction boundary condition, J. Differ. Equ., 236, No. 1, 133-163, 2007.
  • Hu, C., Global strong solutions of Navier-Stokes equations with interface boundary in three-dimensional thin domains, Nonlinear Anal. 74, No. 12, 3964-3997, 2011.
  • Iftimie, D., The 3D Navier - Stokes equations seen as a perturbation of the 2D Navier - Stokes equations, Bull. Soc. Math., France, 127, 473-517, 1999.
  • Iftimie, D. and Raugel, G., Some results on the Navier-Stokes equations in thin 3D domains, J. Differ. Equ., 169, 281-331, 2001.
  • Iftimie, D., Raugel, G., Sell, G.R., Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J., 56, No. 3, 1083-1156, 2007.
  • Jones, D.A., and Titi, E.S., Determination of the solutions of the Navier - Stokes equations by finite volume elements, Phys. D, 60, 165-174, 1992.
  • Jones, D., Titi, E.S., Upper bounds on the number of determining modes, nodes, and volume elements for the Navier - Stokes equations, Indiana Univ. Math. J., 42, No. 3, 875-887, 1993.
  • Kagei, Y., On weak solutions of nonstationary Boussinesq equations, Diff. Integral Equ., 6, 587-611, 1993.
  • Kapustyan, O.V., Melnik, V.S., Valero, J., A weak attractor and properties of solutions for the three-dimensional Benard problem, Discr. Contin. Dyn. Syst. Ser. A, 18, 449-481, 2007.
  • Kapustyan, O.V., Pankov, A.V., Global $\varphi$-attractor for a modified 3D Benard system on channel-like domains, Nonauton. Dyn. Syst., 1, Issue 1, 1-9, 2014.
  • Kapustyan, O.V., Pankov, A.V., Valero, J., On global attractors of multivalued semiflows generated by the 3D Benard system, Set-Valued and Variat. Anal., 20, 445-465, 2012.
  • Kaya, M. and Çelebi, A.O., Existence of weak solutions of the g-Kelvin-Voight equation, Math. Comput. Modelling, 49, 497-504, 2009.
  • Kaya, M. and Çelebi, A.O., Global attractor for the regularized Benard problem, Appl. Anal., 93, Issue 9, 1989-2001, 2014.
  • Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, 2nd Edition, Gordon and Breach, New York, 1969.
  • Moise, I., Temam, R., Ziane, M., Asymptotic analysis of the Navier - Stokes equations in thin domains, Topol. Methods Nonlinear Anal., 10, 249-282, 1997.
  • Montgomery, S., Global regularity of the Navier-Stokes equations on thin three-dimensional domains with periodic boundary conditions, Electron. J. Differ. Equ., 11, 1-19, 1999.
  • Morimoto, H., Non-stationary Boussinesq equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 9, 61-75, 1992.
  • Olson, E., Titi, E.S., Determining modes for continuous data assimilation in 2D turbulence, J. Stat. Physics, 113, No.516, 799-840, 2003.
  • Olson, E., Titi, E.S., Determining modes and Grashof number in 2D turbulence: a numerical case study, Theor. Comput. Fluid Dyn., 22, Issue 5, 327-339, 2008.
  • Raugel, G., Sell, G.R., Navier - Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6, 503-568, 1993.
  • Raugel, G., Sell, G.R., Navier - Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions, Nonlinear Partial Differential Equations and Their Applications, College de France Seminar, vol. XI, Longman, Harlow, , 205-247, 1994.
  • Roh, J., g-Navier-Stokes equations, Thesis, University of Minnesota, 2001.
  • Roh, J., Dynamics of the g-Navier-Stokes equations, J. Differ. Equ., 211, No. 2, 452-484, 2005.
  • Roh, J., Geometry of $L^2(\Omega,g)$, J. Chungcheong Math. Soc., 19, No.3, 283-289, 2006.
  • Temam, R., Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 41, SIAM, Philadelphia, 1983.
  • Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis, vol. 2 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 3rd edition, 1984.
  • Temam, R. and Ziane, M., Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differ. Equ., 1, 499-546, 1996.
  • Temam, R. and Ziane, M., Navier-Stokes equations in thin spherical domains, Contemp. Math., 209, 281-314, 1997.
  • Wu, D., On the Dimension of the Pullback Attractors for g-Navier-Stokes Equations, Discrete Dyn. Nature Soc., 2010, Article ID 893240, 16 pages.

Year 2018, Volume: 47 Issue: 6, 1453 - 1466, 12.12.2018

Muharrem Özlük Meryem Kaya

Abstract

References

  • Bae, H., Roh, J., Existence of Solutions of the g-Navier-Stokes Equations, Taiwanese J. Math., 8, No. 1, 85-102, 2004.
  • Boland, J. and Layton, W., Error analysis for finite element methods for steady natural convection problems, Numer. Funct. Anal. Optim., 11:5-6, 449-483, 1990, DOI: 10.1080/01630569008816383.
  • Galdi,G.P., Lectures in Mathematical Fluid Dynamics, Birkhauser-Verlag, 2000.
  • Farhat, A., Jolly, M.S., and Titi, E.S., Continuous Data Assimilation for the 2D Benard Convection Through Velocity Measurements Alone, Physica D, 303, 59-66, 2015.
  • Foias, C., Jolly, M.S., Kravchenko, R., and Titi, E.S., A unified approach to determining forms for the 2D Navier-Stokes equations -- the general interpolants case, Russ. Math. Surv., 69, No. 2, 359-381, 2014.
  • Foias, C., Manley, O., Rosa, R. and Temam, R., Navier - Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications, vol. 83, Cambridge University Press, 2004.
  • Foias, C., Manley, O., Temam, R., Attractors for the Benard problem: Existence and physical bounds on their fractal dimension, Nonlinear Anal. Theory, Methods & Applications, 11, 939-967, 1987.
  • Foias, C., Prodi, G., Sur le comportement global des solutions non stationnaires des equations de Navier-Stokes en dimension two, Rend. Sem. Mat. Univ., Padova, 39, 1-34, 1967.
  • Hale, J.K., Raugel, G., A damped hyperbolic equation on thin domains, Trans. Amer. Math. Soc., 329, 185-219, 1992.
  • Hale, J.K., Raugel, G., Partial differential equations on thin domains, Differ. Eq. Math. Phys., Birmingham, AL, 1990, Academic Press, Boston, 63-97, 1992.
  • Hale, J. K., Raugel, G., Reaction - Diffusion equation on thin domains, J. Math. Pures Appl., 71, 33-95, 1992.
  • Hoang, L.T., Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (I), J. Math. Fluid Mech., 12, No. 3, 435-472, 2010.
  • Hoang, L.T., Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (II), J. Math. Fluid Mech., 15, 361-395, 2013.
  • Hu, C., Navier-Stokes equations in 3D thin domains with Navier friction boundary condition, J. Differ. Equ., 236, No. 1, 133-163, 2007.
  • Hu, C., Global strong solutions of Navier-Stokes equations with interface boundary in three-dimensional thin domains, Nonlinear Anal. 74, No. 12, 3964-3997, 2011.
  • Iftimie, D., The 3D Navier - Stokes equations seen as a perturbation of the 2D Navier - Stokes equations, Bull. Soc. Math., France, 127, 473-517, 1999.
  • Iftimie, D. and Raugel, G., Some results on the Navier-Stokes equations in thin 3D domains, J. Differ. Equ., 169, 281-331, 2001.
  • Iftimie, D., Raugel, G., Sell, G.R., Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J., 56, No. 3, 1083-1156, 2007.
  • Jones, D.A., and Titi, E.S., Determination of the solutions of the Navier - Stokes equations by finite volume elements, Phys. D, 60, 165-174, 1992.
  • Jones, D., Titi, E.S., Upper bounds on the number of determining modes, nodes, and volume elements for the Navier - Stokes equations, Indiana Univ. Math. J., 42, No. 3, 875-887, 1993.
  • Kagei, Y., On weak solutions of nonstationary Boussinesq equations, Diff. Integral Equ., 6, 587-611, 1993.
  • Kapustyan, O.V., Melnik, V.S., Valero, J., A weak attractor and properties of solutions for the three-dimensional Benard problem, Discr. Contin. Dyn. Syst. Ser. A, 18, 449-481, 2007.
  • Kapustyan, O.V., Pankov, A.V., Global $\varphi$-attractor for a modified 3D Benard system on channel-like domains, Nonauton. Dyn. Syst., 1, Issue 1, 1-9, 2014.
  • Kapustyan, O.V., Pankov, A.V., Valero, J., On global attractors of multivalued semiflows generated by the 3D Benard system, Set-Valued and Variat. Anal., 20, 445-465, 2012.
  • Kaya, M. and Çelebi, A.O., Existence of weak solutions of the g-Kelvin-Voight equation, Math. Comput. Modelling, 49, 497-504, 2009.
  • Kaya, M. and Çelebi, A.O., Global attractor for the regularized Benard problem, Appl. Anal., 93, Issue 9, 1989-2001, 2014.
  • Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, 2nd Edition, Gordon and Breach, New York, 1969.
  • Moise, I., Temam, R., Ziane, M., Asymptotic analysis of the Navier - Stokes equations in thin domains, Topol. Methods Nonlinear Anal., 10, 249-282, 1997.
  • Montgomery, S., Global regularity of the Navier-Stokes equations on thin three-dimensional domains with periodic boundary conditions, Electron. J. Differ. Equ., 11, 1-19, 1999.
  • Morimoto, H., Non-stationary Boussinesq equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 9, 61-75, 1992.
  • Olson, E., Titi, E.S., Determining modes for continuous data assimilation in 2D turbulence, J. Stat. Physics, 113, No.516, 799-840, 2003.
  • Olson, E., Titi, E.S., Determining modes and Grashof number in 2D turbulence: a numerical case study, Theor. Comput. Fluid Dyn., 22, Issue 5, 327-339, 2008.
  • Raugel, G., Sell, G.R., Navier - Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6, 503-568, 1993.
  • Raugel, G., Sell, G.R., Navier - Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions, Nonlinear Partial Differential Equations and Their Applications, College de France Seminar, vol. XI, Longman, Harlow, , 205-247, 1994.
  • Roh, J., g-Navier-Stokes equations, Thesis, University of Minnesota, 2001.
  • Roh, J., Dynamics of the g-Navier-Stokes equations, J. Differ. Equ., 211, No. 2, 452-484, 2005.
  • Roh, J., Geometry of $L^2(\Omega,g)$, J. Chungcheong Math. Soc., 19, No.3, 283-289, 2006.
  • Temam, R., Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 41, SIAM, Philadelphia, 1983.
  • Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis, vol. 2 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 3rd edition, 1984.
  • Temam, R. and Ziane, M., Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differ. Equ., 1, 499-546, 1996.
  • Temam, R. and Ziane, M., Navier-Stokes equations in thin spherical domains, Contemp. Math., 209, 281-314, 1997.
  • Wu, D., On the Dimension of the Pullback Attractors for g-Navier-Stokes Equations, Discrete Dyn. Nature Soc., 2010, Article ID 893240, 16 pages.
There are 42 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Muharrem Özlük This is me

Meryem Kaya

Publication Date December 12, 2018
Published in Issue Year 2018 Volume: 47 Issue: 6

Cite

APA Özlük, M., & Kaya, M. (2018). On the weak solutions and determining modes of the g-Benard problem. Hacettepe Journal of Mathematics and Statistics, 47(6), 1453-1466.
AMA Özlük M, Kaya M. On the weak solutions and determining modes of the g-Benard problem. Hacettepe Journal of Mathematics and Statistics. December 2018;47(6):1453-1466.
Chicago Özlük, Muharrem, and Meryem Kaya. “On the Weak Solutions and Determining Modes of the G-Benard Problem”. Hacettepe Journal of Mathematics and Statistics 47, no. 6 (December 2018): 1453-66.
EndNote Özlük M, Kaya M (December 1, 2018) On the weak solutions and determining modes of the g-Benard problem. Hacettepe Journal of Mathematics and Statistics 47 6 1453–1466.
IEEE M. Özlük and M. Kaya, “On the weak solutions and determining modes of the g-Benard problem”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, pp. 1453–1466, 2018.
ISNAD Özlük, Muharrem - Kaya, Meryem. “On the Weak Solutions and Determining Modes of the G-Benard Problem”. Hacettepe Journal of Mathematics and Statistics 47/6 (December 2018), 1453-1466.
JAMA Özlük M, Kaya M. On the weak solutions and determining modes of the g-Benard problem. Hacettepe Journal of Mathematics and Statistics. 2018;47:1453–1466.
MLA Özlük, Muharrem and Meryem Kaya. “On the Weak Solutions and Determining Modes of the G-Benard Problem”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, 2018, pp. 1453-66.
Vancouver Özlük M, Kaya M. On the weak solutions and determining modes of the g-Benard problem. Hacettepe Journal of Mathematics and Statistics. 2018;47(6):1453-66.

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