November  2018, 23(9): 3855-3877. doi: 10.3934/dcdsb.2018114

On one problem of viscoelastic fluid dynamics with memory on an infinite time interval

Research Institute of Mathematics, Voronezh State University, Universitetskaya pl., 1, Voronezh 394 018, Russia

* Corresponding author: Victor Zvyagin

Received  March 2017 Revised  June 2017 Published  April 2018

Fund Project: The research of the first author was supported by the Russian Science Foundation (project no. 16-11-10125, Lemma 3.4). The research of the second author was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037).

In the present paper we establish the existence of weak solutions of one boundary value problem for one model of a viscoelastic fluid with memory along the trajectories of the velocity field on an infinite time interval. We use solvability of related approximating initial-boundary value problems on finite time intervals and responding pass to the limit.

Citation: Victor Zvyagin, Vladimir Orlov. On one problem of viscoelastic fluid dynamics with memory on an infinite time interval. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3855-3877. doi: 10.3934/dcdsb.2018114
References:
[1]

Yu. Ya. Agranovich and P. E. Sobolewskii, Motion of nonlinear visco-elastic fluid, Nonlinear Analysis: Theory, Methods & Application, 32 (1998), 755-760.  doi: 10.1016/S0362-546X(97)00519-1.  Google Scholar

[2]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in Lecture Notes in Mathematics Calculus of Variations and Nonlinear Differential Equations (CIME series, Cetraro 2005) (eds. B. Dacorogna and P. Marcellini), 1927 (2008), 1-41.  Google Scholar

[3]

J. Astarita and G. Marucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill Inc., New York, 1974. Google Scholar

[4]

C. M. Belonosov and K. A. Cherous, Boundary Value Problems for the Navier-Stokes Equations, Nauka, Moscow, 1985.  Google Scholar

[5]

R. B. Bird and J. M. Wiest, Constitutive equations for polymeric liquids, Annual Review of Fluid Mechanics, 27 (1995), 169-193.   Google Scholar

[6]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and Applied Geophysics, 91 (1971), 134-147.   Google Scholar

[7]

G. Crippa and C. de Lellis, Estimates and regularity results for the diPerna-Lions flow, Journal für Reine und Angewandte Mathematik., 616 (2008), 15-46.   Google Scholar

[8]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[9]

V. T. Dmitrienko and V. G. Zvyagin, The topological degree method for equations of the Navier-Stokes type, Abstract and Applied Analysis, 2 (1997), 1-45.  doi: 10.1155/S1085337597000250.  Google Scholar

[10]

A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylock Press, Albany, NY, 1961.  Google Scholar

[11]

J. A. Nitsche, On Korn's second inequality, RAIRO: Analyse numérique, 15 (1981), 237-248.  doi: 10.1051/m2an/1981150302371.  Google Scholar

[12]

J. G. Oldroyd, Non-Newtonian flow of liquids and solids, in Rheology: Theory and applications (Ed. F. R. Eirich), AP, New York, (1956), 653-682. Google Scholar

[13]

V. P. Orlov and P. E. Sobolevskii, On mathematical models of a viscoelasticity with a memory, Differential Integral Equations, 4 (1991), 103-115.   Google Scholar

[14]

P. A. Rebinder, Physical-Chemical Mechanics, Knowledge, Moscow, 1958. Google Scholar

[15]

P. E. Sobolewskii, Equations of parabolic type in a Banach space, American Mathematical Society Translations, 49 (1966), 1-61.   Google Scholar

[16]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1977.  Google Scholar

[17]

K. Yosida, Functional Analysis, Berlin-Heidelberg-NewYork, Springer-Verlag, 1995. Google Scholar

[18]

V. G. Zvyagin, On the solvability of some initial-boundary value problems for mathematical models of the motion of nonlinearly viscous and viscoelastic fluids, Journal of Mathematical Siences, 124 (2004), 5321-5334.   Google Scholar

[19]

V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, De Gruiter series in nonlinear Analysis and applications, vol. 12, Berlin-New York, 2008.  Google Scholar

[20]

V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of a regularized model of viscoelastic fluid, Differential Equations, 38 (2002), 1731-1744.   Google Scholar

[21]

V. G. Zvyagin and V. P. Orlov, On the weak solvability of the problem of viscoelasticity with memory, Differential Equations, 53 (2017), 212-217.  doi: 10.1134/S0012266117020070.  Google Scholar

[22]

V. G. Zvyagin and V. P. Orlov, On certain mathematical models in continuum thermomechanics, Journal of Fixed Point Theory and Applications, 15 (2014), 3-47.  doi: 10.1007/s11784-014-0179-y.  Google Scholar

show all references

References:
[1]

Yu. Ya. Agranovich and P. E. Sobolewskii, Motion of nonlinear visco-elastic fluid, Nonlinear Analysis: Theory, Methods & Application, 32 (1998), 755-760.  doi: 10.1016/S0362-546X(97)00519-1.  Google Scholar

[2]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in Lecture Notes in Mathematics Calculus of Variations and Nonlinear Differential Equations (CIME series, Cetraro 2005) (eds. B. Dacorogna and P. Marcellini), 1927 (2008), 1-41.  Google Scholar

[3]

J. Astarita and G. Marucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill Inc., New York, 1974. Google Scholar

[4]

C. M. Belonosov and K. A. Cherous, Boundary Value Problems for the Navier-Stokes Equations, Nauka, Moscow, 1985.  Google Scholar

[5]

R. B. Bird and J. M. Wiest, Constitutive equations for polymeric liquids, Annual Review of Fluid Mechanics, 27 (1995), 169-193.   Google Scholar

[6]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and Applied Geophysics, 91 (1971), 134-147.   Google Scholar

[7]

G. Crippa and C. de Lellis, Estimates and regularity results for the diPerna-Lions flow, Journal für Reine und Angewandte Mathematik., 616 (2008), 15-46.   Google Scholar

[8]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[9]

V. T. Dmitrienko and V. G. Zvyagin, The topological degree method for equations of the Navier-Stokes type, Abstract and Applied Analysis, 2 (1997), 1-45.  doi: 10.1155/S1085337597000250.  Google Scholar

[10]

A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylock Press, Albany, NY, 1961.  Google Scholar

[11]

J. A. Nitsche, On Korn's second inequality, RAIRO: Analyse numérique, 15 (1981), 237-248.  doi: 10.1051/m2an/1981150302371.  Google Scholar

[12]

J. G. Oldroyd, Non-Newtonian flow of liquids and solids, in Rheology: Theory and applications (Ed. F. R. Eirich), AP, New York, (1956), 653-682. Google Scholar

[13]

V. P. Orlov and P. E. Sobolevskii, On mathematical models of a viscoelasticity with a memory, Differential Integral Equations, 4 (1991), 103-115.   Google Scholar

[14]

P. A. Rebinder, Physical-Chemical Mechanics, Knowledge, Moscow, 1958. Google Scholar

[15]

P. E. Sobolewskii, Equations of parabolic type in a Banach space, American Mathematical Society Translations, 49 (1966), 1-61.   Google Scholar

[16]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1977.  Google Scholar

[17]

K. Yosida, Functional Analysis, Berlin-Heidelberg-NewYork, Springer-Verlag, 1995. Google Scholar

[18]

V. G. Zvyagin, On the solvability of some initial-boundary value problems for mathematical models of the motion of nonlinearly viscous and viscoelastic fluids, Journal of Mathematical Siences, 124 (2004), 5321-5334.   Google Scholar

[19]

V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, De Gruiter series in nonlinear Analysis and applications, vol. 12, Berlin-New York, 2008.  Google Scholar

[20]

V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of a regularized model of viscoelastic fluid, Differential Equations, 38 (2002), 1731-1744.   Google Scholar

[21]

V. G. Zvyagin and V. P. Orlov, On the weak solvability of the problem of viscoelasticity with memory, Differential Equations, 53 (2017), 212-217.  doi: 10.1134/S0012266117020070.  Google Scholar

[22]

V. G. Zvyagin and V. P. Orlov, On certain mathematical models in continuum thermomechanics, Journal of Fixed Point Theory and Applications, 15 (2014), 3-47.  doi: 10.1007/s11784-014-0179-y.  Google Scholar

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