Amendola, Giovambattista; Fabrizio, Mauro; Golden, John; Manes, Adele Energy stability for thermo-viscous fluids with a fading memory heat flux. (English) Zbl 1381.35131 Evol. Equ. Control Theory 4, No. 3, 265-279 (2015). 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. Cited in 2 Documents 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 Keywords:Bénard problem; viscous fluid with memory; exponential stability PDFBibTeX XMLCite \textit{G. Amendola} et al., Evol. Equ. Control Theory 4, No. 3, 265--279 (2015; Zbl 1381.35131) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.