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 and 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), 349-374. doi: 10.1090/S0033-569X-10-01185-3.

[2]

G. Amendola and M. Fabrizio, Thermal convection in a simple fluid with fading memory, J. Math. Anal. Appl., 366 (2010), 444-459. doi: 10.1016/j.jmaa.2009.11.043.

[3]

G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.

[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), 144-158.

[5]

C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuous mechanics, in Applications of Methods of Functional Analysis to Problems in Mechanics, Lectures Notes in Mathematics, 503, Springer-Verlag, Berlin-Heidelberg, 1976, 295-306. doi: 10.1007/BFb0088765.

[6]

R. Datko, Extending a theorem of A. M. Lyapunov to Hilbert space, J. Math. Anal. Appl., 32 (1970), 610-616. doi: 10.1016/0022-247X(70)90283-0.

[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), 43-96. doi: 10.1007/s00205-005-0406-1.

[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), 92-96. doi: 10.1016/j.jnnfm.2006.01.005.

[9]

M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal., 125 (1994), 341-373. doi: 10.1007/BF00375062.

[10]

M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids, Diff. Integral Equat., 6 (1993), 491-505.

[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), 161-176. doi: 10.1016/S0377-0257(03)00096-X.

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Lectures Notes in Mathematics, 10, University of Maryland, 1974.

[13]

L. Preziosi and S. Rionero, Energy stability of steady shear flows of a viscoelastic fluid, Int. J. Eng. Sci., 27 (1989), 1167-1181. doi: 10.1016/0020-7225(89)90096-7.

[14]

M. Slemrod, An energy stability method for simple fluids, Arch. Rational Mech. Anal., 68 (1978), 1-18. doi: 10.1007/BF00276175.

[15]

B. Straughan, The Energy Method, Stability, and Non Linear Convection, $2^{nd}$ edition, Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21740-6.

show all references

References:
[1]

G. Amendola, Free energies for incompressible viscoelastic fluids, Quart. Appl. Math., 68 (2010), 349-374. doi: 10.1090/S0033-569X-10-01185-3.

[2]

G. Amendola and M. Fabrizio, Thermal convection in a simple fluid with fading memory, J. Math. Anal. Appl., 366 (2010), 444-459. doi: 10.1016/j.jmaa.2009.11.043.

[3]

G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.

[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), 144-158.

[5]

C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuous mechanics, in Applications of Methods of Functional Analysis to Problems in Mechanics, Lectures Notes in Mathematics, 503, Springer-Verlag, Berlin-Heidelberg, 1976, 295-306. doi: 10.1007/BFb0088765.

[6]

R. Datko, Extending a theorem of A. M. Lyapunov to Hilbert space, J. Math. Anal. Appl., 32 (1970), 610-616. doi: 10.1016/0022-247X(70)90283-0.

[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), 43-96. doi: 10.1007/s00205-005-0406-1.

[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), 92-96. doi: 10.1016/j.jnnfm.2006.01.005.

[9]

M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal., 125 (1994), 341-373. doi: 10.1007/BF00375062.

[10]

M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids, Diff. Integral Equat., 6 (1993), 491-505.

[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), 161-176. doi: 10.1016/S0377-0257(03)00096-X.

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Lectures Notes in Mathematics, 10, University of Maryland, 1974.

[13]

L. Preziosi and S. Rionero, Energy stability of steady shear flows of a viscoelastic fluid, Int. J. Eng. Sci., 27 (1989), 1167-1181. doi: 10.1016/0020-7225(89)90096-7.

[14]

M. Slemrod, An energy stability method for simple fluids, Arch. Rational Mech. Anal., 68 (1978), 1-18. doi: 10.1007/BF00276175.

[15]

B. Straughan, The Energy Method, Stability, and Non Linear Convection, $2^{nd}$ edition, Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21740-6.

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