American Institute of Mathematical Sciences

September  2015, 4(3): 265-279. doi: 10.3934/eect.2015.4.265

Energy stability for thermo-viscous fluids with a fading memory heat flux

 1 Dipartimento di Matematica, Largo Bruno Pontecorvo 5, Pisa, 56127, Italy, Italy 2 Dipartimento di Matematica, Piazza di Porta S. Donato 5, Bologna, 40127, Italy 3 School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received  April 2015 Revised  July 2015 Published  September 2015

In this work we consider the thermal convection problem in arbitrary bounded domains of a three-dimensional space for incompressible viscous fluids, with a fading memory constitutive equation for the heat flux. With the help of a recently proposed free energy, expressed in terms of a minimal state functional for such a system, we prove an existence and uniqueness theorem for the linearized problem. Then, assuming some restrictions on the Rayleigh number, we also prove exponential decay of solutions.
Citation: Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden, Adele Manes. Energy stability for thermo-viscous fluids with a fading memory heat flux. Evolution Equations & Control Theory, 2015, 4 (3) : 265-279. doi: 10.3934/eect.2015.4.265
References:
 [1] G. Amendola, Free energies for incompressible viscoelastic fluids,, Quart. Appl. Math., 68 (2010), 349. doi: 10.1090/S0033-569X-10-01185-3. Google Scholar [2] G. Amendola and M. Fabrizio, Thermal convection in a simple fluid with fading memory,, J. Math. Anal. Appl., 366 (2010), 444. doi: 10.1016/j.jmaa.2009.11.043. Google Scholar [3] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications,, Springer, (2012). doi: 10.1007/978-1-4614-1692-0. Google Scholar [4] G. Amendola, M. Fabrizio and A. Manes, On energy stability for a thermal convection in viscous fluids with memory,, Palestine Journal of Mathematics, 2 (2013), 144. Google Scholar [5] C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuous mechanics,, in Applications of Methods of Functional Analysis to Problems in Mechanics, (1976), 295. doi: 10.1007/BFb0088765. Google Scholar [6] R. Datko, Extending a theorem of A. M. Lyapunov to Hilbert space,, J. Math. Anal. Appl., 32 (1970), 610. doi: 10.1016/0022-247X(70)90283-0. Google Scholar [7] L. Deseri, M. Fabrizio and J. M. Golden, The concept of a minimal state in viscoelasticity: New free energies and applications to $PDE_S$,, Arch. Rational Mech. Anal., 181 (2006), 43. doi: 10.1007/s00205-005-0406-1. Google Scholar [8] C. R. Doering, B. Eckhardt and J. Schumacher, Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers,, J. Non-Newtonian Fluid Mech., 135 (2006), 92. doi: 10.1016/j.jnnfm.2006.01.005. Google Scholar [9] M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory,, Arch. Rational Mech. Anal., 125 (1994), 341. doi: 10.1007/BF00375062. Google Scholar [10] M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids,, Diff. Integral Equat., 6 (1993), 491. Google Scholar [11] A. Lozinski and R. G. Owens, An energy estimate for the Oldroyd-B model: Theory and applications,, J. Non-Newtonian Fluid Mech., 112 (2003), 161. doi: 10.1016/S0377-0257(03)00096-X. Google Scholar [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Lectures Notes in Mathematics, (1974). Google Scholar [13] L. Preziosi and S. Rionero, Energy stability of steady shear flows of a viscoelastic fluid,, Int. J. Eng. Sci., 27 (1989), 1167. doi: 10.1016/0020-7225(89)90096-7. Google Scholar [14] M. Slemrod, An energy stability method for simple fluids,, Arch. Rational Mech. Anal., 68 (1978), 1. doi: 10.1007/BF00276175. Google Scholar [15] B. Straughan, The Energy Method, Stability, and Non Linear Convection,, $2^{nd}$ edition, (2004). doi: 10.1007/978-0-387-21740-6. Google Scholar

show all references

References:
 [1] G. Amendola, Free energies for incompressible viscoelastic fluids,, Quart. Appl. Math., 68 (2010), 349. doi: 10.1090/S0033-569X-10-01185-3. Google Scholar [2] G. Amendola and M. Fabrizio, Thermal convection in a simple fluid with fading memory,, J. Math. Anal. Appl., 366 (2010), 444. doi: 10.1016/j.jmaa.2009.11.043. Google Scholar [3] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications,, Springer, (2012). doi: 10.1007/978-1-4614-1692-0. Google Scholar [4] G. Amendola, M. Fabrizio and A. Manes, On energy stability for a thermal convection in viscous fluids with memory,, Palestine Journal of Mathematics, 2 (2013), 144. Google Scholar [5] C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuous mechanics,, in Applications of Methods of Functional Analysis to Problems in Mechanics, (1976), 295. doi: 10.1007/BFb0088765. Google Scholar [6] R. Datko, Extending a theorem of A. M. Lyapunov to Hilbert space,, J. Math. Anal. Appl., 32 (1970), 610. doi: 10.1016/0022-247X(70)90283-0. Google Scholar [7] L. Deseri, M. Fabrizio and J. M. Golden, The concept of a minimal state in viscoelasticity: New free energies and applications to $PDE_S$,, Arch. Rational Mech. Anal., 181 (2006), 43. doi: 10.1007/s00205-005-0406-1. Google Scholar [8] C. R. Doering, B. Eckhardt and J. Schumacher, Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers,, J. Non-Newtonian Fluid Mech., 135 (2006), 92. doi: 10.1016/j.jnnfm.2006.01.005. Google Scholar [9] M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory,, Arch. Rational Mech. Anal., 125 (1994), 341. doi: 10.1007/BF00375062. Google Scholar [10] M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids,, Diff. Integral Equat., 6 (1993), 491. Google Scholar [11] A. Lozinski and R. G. Owens, An energy estimate for the Oldroyd-B model: Theory and applications,, J. Non-Newtonian Fluid Mech., 112 (2003), 161. doi: 10.1016/S0377-0257(03)00096-X. Google Scholar [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Lectures Notes in Mathematics, (1974). Google Scholar [13] L. Preziosi and S. Rionero, Energy stability of steady shear flows of a viscoelastic fluid,, Int. J. Eng. Sci., 27 (1989), 1167. doi: 10.1016/0020-7225(89)90096-7. Google Scholar [14] M. Slemrod, An energy stability method for simple fluids,, Arch. Rational Mech. Anal., 68 (1978), 1. doi: 10.1007/BF00276175. Google Scholar [15] B. Straughan, The Energy Method, Stability, and Non Linear Convection,, $2^{nd}$ edition, (2004). doi: 10.1007/978-0-387-21740-6. Google Scholar
 [1] Marco Cabral, Ricardo Rosa, Roger Temam. Existence and dimension of the attractor for the Bénard problem on channel-like domains. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 89-116. doi: 10.3934/dcds.2004.10.89 [2] O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449 [3] Björn Birnir, Nils Svanstedt. Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 53-74. doi: 10.3934/dcds.2004.10.53 [4] Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801 [5] Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control & Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321 [6] Monica Conti, Elsa M. Marchini, Vittorino Pata. Exponential stability for a class of linear hyperbolic equations with hereditary memory. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1555-1565. doi: 10.3934/dcdsb.2013.18.1555 [7] Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure & Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721 [8] Victor Zvyagin, Vladimir Orlov. On one problem of viscoelastic fluid dynamics with memory on an infinite time interval. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3855-3877. doi: 10.3934/dcdsb.2018114 [9] Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138 [10] Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic & Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673 [11] B. A. Wagner, Andrea L. Bertozzi, L. E. Howle. Positive feedback control of Rayleigh-Bénard convection. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 619-642. doi: 10.3934/dcdsb.2003.3.619 [12] Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure & Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591 [13] Quan Wang. Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 543-563. doi: 10.3934/dcdsb.2014.19.543 [14] Takayuki Kubo, Yoshihiro Shibata, Kohei Soga. On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3741-3774. doi: 10.3934/dcds.2016.36.3741 [15] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424 [16] Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062 [17] Yuming Qin, T. F. Ma, M. M. Cavalcanti, D. Andrade. Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid. Communications on Pure & Applied Analysis, 2005, 4 (3) : 635-664. doi: 10.3934/cpaa.2005.4.635 [18] Tingyuan Deng. Three-dimensional sphere $S^3$-attractors in Rayleigh-Bénard convection. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 577-591. doi: 10.3934/dcdsb.2010.13.577 [19] Toshiyuki Ogawa. Bifurcation analysis to Rayleigh-Bénard convection in finite box with up-down symmetry. Communications on Pure & Applied Analysis, 2006, 5 (2) : 383-393. doi: 10.3934/cpaa.2006.5.383 [20] Jungho Park. Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 591-604. doi: 10.3934/dcdsb.2006.6.591

2018 Impact Factor: 1.048

Metrics

• HTML views (0)
• Cited by (2)

• on AIMS