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  November 2018 Early access  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 and 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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[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. 

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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. 

[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.

[20]

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

[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.

[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.

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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[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. 

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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. 

[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.

[20]

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

[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.

[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.

[1]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225

[2]

Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873

[3]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[4]

Hua Qiu, Shaomei Fang. A BKM's criterion of smooth solution to the incompressible viscoelastic flow. Communications on Pure and Applied Analysis, 2014, 13 (2) : 823-833. doi: 10.3934/cpaa.2014.13.823

[5]

Baoquan Yuan. Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2211-2219. doi: 10.3934/dcds.2013.33.2211

[6]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116

[7]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357

[8]

Shaoqiang Shang, Yunan Cui. Weak approximative compactness of hyperplane and Asplund property in Musielak-Orlicz-Bochner function spaces. Electronic Research Archive, 2020, 28 (1) : 327-346. doi: 10.3934/era.2020019

[9]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064

[10]

Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375

[11]

A. Jiménez-Casas, Mario Castro, Justine Yassapan. Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid. Conference Publications, 2013, 2013 (special) : 375-384. doi: 10.3934/proc.2013.2013.375

[12]

Kun Wang, Yangping Lin, Yinnian He. Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 657-677. doi: 10.3934/dcds.2012.32.657

[13]

Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001

[14]

Patrick Winkert, Rico Zacher. A priori bounds for weak solutions to elliptic equations with nonstandard growth. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 865-878. doi: 10.3934/dcdss.2012.5.865

[15]

Jeremy LeCrone, Yuanzhen Shao, Gieri Simonett. The surface diffusion and the Willmore flow for uniformly regular hypersurfaces. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3503-3524. doi: 10.3934/dcdss.2020242

[16]

Yinnian He, Yi Li. Asymptotic behavior of linearized viscoelastic flow problem. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 843-856. doi: 10.3934/dcdsb.2008.10.843

[17]

Tong Tang, Yongfu Wang. Strong solutions to compressible barotropic viscoelastic flow with vacuum. Kinetic and Related Models, 2015, 8 (4) : 765-775. doi: 10.3934/krm.2015.8.765

[18]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[19]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[20]

Yinxia Wang, Hengjun Zhao. Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating. Communications on Pure and Applied Analysis, 2018, 17 (2) : 347-374. doi: 10.3934/cpaa.2018020

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (208)
  • HTML views (472)
  • Cited by (1)

Other articles
by authors

[Back to Top]