Stability for the Kelvin–Voigt variable order equations backward in time

Abstract

Continuous dependence upon the initial data is established for a solution to the equations for a Kelvin-Voigt fluid of variable order, for the backward in time problem. Relatively weak restrictions are required on the base velocity field for a potentially improperly posed problem. KEYWORDS backward in time, continuous dependence, Kelvin-Voigt fluid, stability MSC CLASSIFICATION 35B30; 76D03 INTRODUCTIONFlow of a Newtonian, or linearly viscous, fluid is the subject of much research. However, it is becoming increasingly important to study flow of non-Newtonian fluids. Among such non-Newtonian fluids are viscoelastic fluids which typically display history-dependent behaviour. Analyses of fading memory fluids or general viscoelastic fluids may be found in, for example, previous works. [1][2][3][4][5][6][7][8][9][10][11] The subject of this article is a particular class of fading memory fluids attached to the names of Kelvin and of Voigt, see, for example, previous studies. [12][13][14][15][16][17] We are especially interested in the class of Kelvin -Voigt fluids succinctly described and analysed particularly in the Russian literature; see Karazeeva et al, 18 Oskolkov, 19,20 Oskolkov and Shadiev, 21 and Sviridyuk and Sukachevaand. 22 The topic of continuous dependence of a solution to a boundary-initial value problem is very important. This subject, including continuous dependence upon the models themselves is analysed by previous works. [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] An increasingly important class of continuous dependence analyses are those pertaining to improperly posed problems, or non-well-posed problems; see previous literature. 12,13,24,[40][41][42][43][44][45][46][47][48][49][50][51][52][53] Improperly posed problems have been amenable to analyse especially with the aid of the famous paper of John. 54 John 54 suggested imposing an a priori bound on a quantity, or a class of quantities, and this has proved invaluable in the subsequent analysis.We now concentrate on the nonlinear hierarchy of Kelvin-Voigt equations of variable order, 1, … , L. We analyse the backward in time problem for these equations, which in general will lead to a series of improperly posed problems. Such problems are of practical value in extrapolating from the past and computational methods may be based on analytical results, as explained in detail by Carasso. 45 KELVIN-VOIGT VARIABLE ORDER EQUATIONSLet Ω ⊂ R 3 be a bounded domain with boundary Γ sufficiently smooth to allow the use of the divergence theorem. The inner product and norm on L 2 (Ω) will be denoted by (• , •) and || • ||, respectively. Throughout this article, we employ standard indicial notation together with the Einstein summation convention, so that a repeated Roman index denotes

Second gradient Green–Naghdi type thermo-elasticity and viscoelasticity

“…In order to describe non local anisotropic G-N thermoelastic and viscoelastic effects, we let ψ be continuously differentiable with respect to all the independent variables at the current time t, representing the state σ = (k, ∇u, ∇∇u, ∇ k, ∇∇ k). Upon evaluation of ψ and substitution in (20), also generalizing the standard thermodynamics arguments, we immediately find…”

“…As a first step we avoid the last term in (20) due to the non local behavior and rewrite the fourth addendum as − 1 θ 2 ∇θ • q. Finally, we neglect the terms due to the mechanical structure and we focus on a modified free energy potential, now depending on θ and ∇α.…”

“…There is also awareness that, due to their employment in many areas of continuum mechanics with interesting applications to micro-and nanothermal devices, a reexamination of their local/non local constitutive features is worthwhile, see [16,17,18] The introduction of suitable retardation parameters, responsible for memory effects, leads to a non local thermo-viscoelastic theory of Kelvin-Voigt (shortly K-V) type, which might be easily generalized by incorporating the dependence on past histories. Nowadays, K-V type viscoelasticity has been increasingly occupying attention in literature, also within generalized theories (see e.g [20,10] and references therein). Our constitutive theories are going to allow for memory effects at micro-and nano-scales interacting phenomena, where different size effects on heat conduction and elastic deformations become essential and standard local theories are proven to fail [22].…”

Proof of Straughan’s claim on Payne–Song’s and modified Guyer–Krumhansl’s equations

“…In an elegant analysis of stationary and oscillatory convection in a Brinkman-Darcy-Kelvin-Voigt fluid, Straughan [1] (see also [2][3][4][5]) considered a system of evolution laws including-beyond a regularized balance of momentum-Payne-Song's equation [6] in Eulerian representation, that is an energy balance given by…”

“…In an elegant analysis of stationary and oscillatory convection in a Brinkman–Darcy–Kelvin–Voigt fluid, Straughan [1] (see also [2–5]) considered a system of evolution laws including—beyond a regularized balance of momentum—Payne–Song’s equation [6] in Eulerian representation, that is an energy balance given by T˙=κnormalΔT−divq,where the superposed dot indicates from now on total derivative with respect to time; T is the absolute temperature, κ the conductivity, taken to be constant, and q the heat flux, which satisfies, per se , a version of Guyer–Krumhansl’s equation [7,8], given by ℓDqDt=−q−κnormal∇T+ς^1normalΔq+ς^2normal∇divq,with ℓ a time delay and scriptD/scriptDt a generic objective derivative, which he considers in the analysis to be the Lie derivative with respect to the macroscopic fluid velocity.…”