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Energy stability for thermo-viscous fluids with a fading memory heat flux. (English) Zbl 1381.35131

Summary: 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35B35 Stability in context of PDEs
45K05 Integro-partial differential equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
80A17 Thermodynamics of continua
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[1] G. Amendola, Free energies for incompressible viscoelastic fluids,, Quart. Appl. Math., 68, 349 (2010) · Zbl 1425.76024
[2] G. Amendola, Thermal convection in a simple fluid with fading memory,, J. Math. Anal. Appl., 366, 444 (2010) · Zbl 1379.80004
[3] G. Amendola, <em>Thermodynamics of Materials with Memory: Theory and Applications</em>,, Springer (2012) · Zbl 1237.80001
[4] G. Amendola, On energy stability for a thermal convection in viscous fluids with memory,, Palestine Journal of Mathematics, 2, 144 (2013) · Zbl 1343.45012
[5] C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuous mechanics,, in Applications of Methods of Functional Analysis to Problems in Mechanics, 295 (1976) · Zbl 0345.47032
[6] R. Datko, Extending a theorem of A. M. Lyapunov to Hilbert space,, J. Math. Anal. Appl., 32, 610 (1970) · Zbl 0211.16802
[7] L. Deseri, The concept of a minimal state in viscoelasticity: New free energies and applications to \(PDE_S\),, Arch. Rational Mech. Anal., 181, 43 (2006) · Zbl 1152.74323
[8] C. R. Doering, Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers,, J. Non-Newtonian Fluid Mech., 135, 92 (2006) · Zbl 1195.76174
[9] M. Fabrizio, Free energies and dissipation properties for systems with memory,, Arch. Rational Mech. Anal., 125, 341 (1994) · Zbl 0806.73006
[10] M. Fabrizio, On asymptotic stability for linear viscoelastic fluids,, Diff. Integral Equat., 6, 491 (1993) · Zbl 0776.76010
[11] A. Lozinski, An energy estimate for the Oldroyd-B model: Theory and applications,, J. Non-Newtonian Fluid Mech., 112, 161 (2003) · Zbl 1065.76018
[12] A. Pazy, <em>Semigroups of Linear Operators and Applications to Partial Differential Equations</em>,, Lectures Notes in Mathematics (1974) · Zbl 0516.47023
[13] L. Preziosi, Energy stability of steady shear flows of a viscoelastic fluid,, Int. J. Eng. Sci., 27, 1167 (1989) · Zbl 0693.76017
[14] M. Slemrod, An energy stability method for simple fluids,, Arch. Rational Mech. Anal., 68, 1 (1978) · Zbl 0417.76005
[15] B. Straughan, <em>The Energy Method, Stability, and Non Linear Convection</em>,, \(2^{nd}\) edition (2004) · Zbl 1066.35084
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