Existence and large time behavior for generalized Kelvin-Voigt equations governing nonhomogeneous and incompressible fluids

Antont︠s︡ev, S. N.; Oliveira, H. B. de; Khompysh, Kh.

doi:10.1088/1742-6596/1268/1/012008

Cited by 13 publications

(12 citation statements)

References 9 publications

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“…The systems of nonlinear equations of the Sobolev type (Kelvin–Voigt equations) describe flows of viscoelastic fluids. Investigations of the mathematical correctness of such equations are devoted to the works 14–16,32–37 …”

Section: Introductionmentioning

confidence: 99%

“…Investigations of the mathematical correctness of such equations are devoted to the works. [14][15][16][32][33][34][35][36][37] In the work of M. O. Korpusov, A. G. Sveshnikov, 21 a model equation is considered that describes the relaxation of an initial perturbation in a crystalline semiconductor in the case when its electrical conductivity depends nonlocally on the field.…”

Section: Introductionmentioning

confidence: 99%

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An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

Antont︠s︡ev

Aitzhanov

Zhanuzakova

2022

Math Methods in App Sciences

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An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann-Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann-Dirichlet boundary condition are obtained. KEYWORDS asymptotic behavior of the solution, blow-up of the solution, Galerkin method, nonlinear boundary conditions, pseudoparabolic equations, the existence of a solution, uniqueness of the solution MSC CLASSIFICATION

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

Antont︠s︡ev

Aitzhanov

Zhanuzakova

2022

Math Methods in App Sciences

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“…[8][9][10] When F(x, t) = f (t)g(x, t) is given and p = 2, 𝛾 = 0, the various direct problems for classical linear and nonlinear Kelvin-Voigt equations have been studied by Oskolkov. 8,[11][12][13] The nonlinear direct problems for generalized Kelvin-Voigt equations (1.1) and (1.2) perturbed by isotropic or anisotropic diffusion, relaxation, and damping for homogeneous and also nonhomogeneous fluids, have been investigated in previous studies, [14][15][16][17][18][19][20] where the global and local existence and uniqueness, and the quality properties of weak solutions are established. Some inverse problems for linear and nonlinear classical Kelvin-Voigt equations with integral overdetermination condition, which consist of determining a right hand side depending on time or space variable, have been studied by Abylkairov and Khompysh in Abylkairov and Khompysh 21 and Khompysh, 4 where the unique solvability is established.…”

Section: Introductionmentioning

confidence: 99%

“…When F ( x , t ) = f ( t ) g ( x , t ) is given and

p = 2, γ = 0

, the various direct problems for classical linear and nonlinear Kelvin–Voigt equations have been studied by Oskolkov 8,11–13 . The nonlinear direct problems for generalized Kelvin–Voigt equations () and () perturbed by isotropic or anisotropic diffusion, relaxation, and damping for homogeneous and also nonhomogeneous fluids, have been investigated in previous studies, 14–20 where the global and local existence and uniqueness, and the quality properties of weak solutions are established.…”

Section: Introductionmentioning

confidence: 99%

An inverse problem for Kelvin–Voigt equations perturbed by isotropic diffusion and damping

Khompysh

Kenzhebai

2021

Math Methods in App Sciences

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In this paper, we consider an inverse problem of finding a coefficient of right hand side of the following system of Kelvin–Voigt equations perturbed by an isotropic diffusion and damping terms vt+∇π−ϰΔvt−νdiv|D(v)|p−2D(v)=γ|v|m−2v+f(t)g(x,t), divv(x,t)=0. The damping term γ|v|m − 2v in the momentum equation realizes an absorbtion (sink) if γ ≤ 0, and a source if γ > 0. We show how the exponents p, m, the coefficients ν, ϰ, γ, the dimension of the space d, and data of the problem should interact each other for the existence of weak solutions to the problem. We also establish the conditions for uniqueness of the solutions to this problem.

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“…îáçîð â åãî ìîíîãðàôèè [10]) è AE. Ñèìîíà [11], à òàêaeå àêòèâíî ðàçâèâàåòñÿ â íàñòîÿùåå âðåìÿ è äëÿ íåíüþòîíîâñêèõ aeèäêîñòåé (ñì., íàïðèìåð, [12]). Èññëåäîâàíèþ çàäà÷è îïòèìàëüíîãî óïðàâëåíèÿ ñ îáðàòíîé ñâÿçüþ äëÿ îäíîé èç òàêèõ ìîäåëåé ìîäåëè Ôîéãòà ñ ïåðåìåííîé ïëîòíîñòüþ è ïîñâÿùåíà íàñòîÿùàÿ ñòàòüÿ.…”

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The optimal feedback control problem for Voigt model with variable density

Zvyagin¹,

Turbin²

2020

Iz. VUZ. Matematika

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The work is devoted to the study of weak solvability of the optimal feedback control problem for the Voigt model with variable density. The proof is based on the approximationtopological approach. First, the solvability of the approximation problem is proved, and then, based on a priori estimates that are independent of the approximation parameter, it is shown that from the sequence of solutions of the approximation problem, a subsequence converging to the solution of the original problem can be chosen. Then it is shown that among weak solutions to the problem there is at least one that gives a minimum to a given cost functional.

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Existence and large time behavior for generalized Kelvin-Voigt equations governing nonhomogeneous and incompressible fluids

Cited by 13 publications

References 9 publications

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An inverse problem for Kelvin–Voigt equations perturbed by isotropic diffusion and damping

The optimal feedback control problem for Voigt model with variable density

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