Continuous dependence and convergence for a Kelvin–Voigt fluid of order one

Straughan, Brian

doi:10.1007/s11565-021-00381-7

Cited by 17 publications

(4 citation statements)

References 50 publications

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“…An interesting class of them can be found in [12]. Linear and nonlinear versions have been studied [8, 9, 16]. For some of them, the study we have developed can be used.…”

Section: Applications To Some Special Problemsmentioning

confidence: 99%

Uniqueness for a high order ill posed problem

Fernández¹,

Quintanilla²

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this work, we study a high order derivative in time problem. First, we show that there exists a sequence of elements of the spectrum which tends to infinity and therefore, it is ill posed. Then, we prove the uniqueness of solutions for this problem by adapting the logarithmic arguments to this situation. Finally, the results are applied to the backward in time problem for the generalized linear Burgers’ fluid, a couple of heat conduction problems and a viscoelastic model.

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“…An interesting class of them can be found in [12]. Linear and nonlinear versions have been studied [8, 9, 16]. For some of them, the study we have developed can be used.…”

Section: Applications To Some Special Problemsmentioning

confidence: 99%

Uniqueness for a high order ill posed problem

Fernández¹,

Quintanilla²

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…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…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Mariano

2023

Proc. R. Soc. A.

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We prove a claim by Brian Straughan who conjectured that Payne–Song’s and modified Guyer–Krumhansl’s equations can be justified in (and derived from) the general model-building framework for the mechanics of complex bodies, so they emerge from the modelling of microstructural effects. The proof is based on taking into account micro-to-macro spatial scaling and a decomposition of the heat flux into a standard Fourier-type component and one measuring microstructural event with associated and balanced actions.

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“…β is the cooling coefficient. For more papers of the type, one can refer to [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. However, these results above only considered the case of the bounded region.…”

Section: Introductionmentioning

confidence: 99%

Structural Stability on the Boundary Coefficient of the Thermoelastic Equations of Type III

Chen

2022

Mathematics

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This paper investigates the spatial behavior of the solutions of thermoelastic equations of type III in a semi-infinite cylinder by using the partial differential inequalities. By setting an arbitrary positive constant in the energy expression, the fast decay rate of the solutions is obtained. Based on the results of decay, the continuous dependence and the convergence results on the boundary coefficient are established by using the differential inequality technique and the energy analysis method. The main work of this paper is to extend the study of continuous dependence to a semi-infinite cylinder, which can be used as a reference for the study of other types of partial differential equations.

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Continuous dependence and convergence for a Kelvin–Voigt fluid of order one

Abstract: It is shown that the solution to the boundary - initial value problem for a Kelvin–Voigt fluid of order one depends continuously upon the Kelvin–Voigt parameters, the viscosity, and the viscoelastic coefficients. Convergence of a solution is also shown.

Cited by 17 publications

References 50 publications

Uniqueness for a high order ill posed problem

Uniqueness for a high order ill posed problem

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

Structural Stability on the Boundary Coefficient of the Thermoelastic Equations of Type III

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