Abstract
We establish the existence and uniqueness (the latter only in the plane case) of a weak solution to an initial-boundary value problem for the system of the equations of motion of a viscoelastic fluid, namely, for the anti-Zener model whose constitutive law contains fractional derivatives. We use the approximation of this problem by a sequence of regularized Navier–Stokes systems and passage to the limit.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Gyarmati I., Non-Equilibrium Thermodynamics: Field Theory and Variational Principles, Springer-Verlag, Berlin and Heidelberg (1970).
Mainardi F. and Spada G., “Creep, relaxation and viscosity properties for basic fractional models in rheology,” Eur. Phys. J. Special Topics, vol. 193, No. 1, 133–160 (2011).
Ogorodnikov E. N., Radchenko V. P., and Yashagin N. S., “Rheological model of viscoelastic body with memory and differential equations of fractional oscillators,” Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, vol. 22, No. 1, 255–268 (2011).
Zvyagin V. G., “On solvability of some initial-boundary problems for mathematical models of the motion of nonlinearly viscous and viscoelastic fluids,” J. Math. Sci., vol. 124, No. 5, 5321–5334 (2004).
Zvyagin V. G. and Turbin M. V., Mathematical Problems for Hydrodynamic Viscoelastic Media [Russian], Krassand (URSS), Moscow (2012).
Orlov V. P., Rode D. A., and Pliev M. A., “Weak solvability of the generalized Voigt viscoelasticity model,” Sib. Math. J., vol. 58, No. 5, 859–874 (2017).
Zvyagin V. G. and Orlov V. P., “Weak solvability of a fractional Voigt viscoelasticity model,” Dokl. Math., vol. 96, No. 2, 491–493 (2017).
Temam R., Navier–Stokes Equations. Theory and Numerical Analysis, Amer. Math. Soc., Providence (2001).
Samko S. G., Kilbas A. A., and Marichev O. M., Integrals and Fractional-Order Derivatives and Some of Their Applications, Gordon and Breach, Amsterdam (1993).
Krasnoselskii M. A., Zabreiko P. P., Pustylnik E. I., and Sobolevskii P. E., Integral Operators in Spaces of Summable Functions, Noordhoff Int. Publ., Leyden (1976).
Ashyralyev A., “A note on fractional derivatives and fractional powers of operators,” J. Math. Anal. Appl., vol. 357, 232–236 (2009).
Zvyagin V. G. and Dmitrienko V. T., “On weak solutions of a regularized model of a viscoelastic fluid,” Differ. Equ., vol. 38, No. 12, 1731–1744 (2002).
Zvyagin V. G. and Dmitrienko V. T., Topological Approximation Approach to the Study of the Problems of Hydrodynamics. A Navier–Stokes System [Russian], Èditorial URSS, Moscow (2004).
Orlov V. P. and Sobolevskii P. E., “On mathematical models of a viscoelasticity with a memory,” Differ. Integral Equ., vol. 4, No. 1, 103–115 (1991).
Yosida K., Functional Analysis, Springer-Verlag, Berlin (1994).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © 2018 Zvyagin V.G. and Orlov V.P.
__________
Voronezh. Translated from Sibirskii Matematicheskii Zhurnal, vol. 59, no. 6, pp. 1351–1369, November–December, 2018; DOI: 10.17377/smzh.2018.59.610.
Rights and permissions
About this article
Cite this article
Zvyagin, V.G., Orlov, V.P. On Solvability of an Initial-Boundary Value Problem for a Viscoelasticity Model with Fractional Derivatives. Sib Math J 59, 1073–1089 (2018). https://doi.org/10.1134/S0037446618060101
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446618060101