Abstract
The existence and uniqueness of a strong solution to the initial-boundary value problem for a system of fluid dynamics equations that is a fractional analogue of the Voigt viscoelasticity model in the plane case are established. The rheological equation of this model involves fractional derivatives.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
I. Gyarmati, Nonequilibrium Thermodynamics: Field Theory and Variational Principles (Springer-Verlag, Berlin, 1970).
E. N. Ogorodnikov, V. P. Radchenko, and N. S. Yashagin, “Rheological model of viscoelastic body with memory and differential equations of fractional oscillator,” Vestn. Samar. Gos. Tekh. Univ. Ser. Fiz.-Mat. Nauki 22 (1), 255–268 (2011).
F. Mairandi and G. Spada, “Creep, relaxation and viscosity properties for basic fractional models in rheology,” Eur. Phys. J. Special Topics 193, 133–160 (2011).
V. G. Zvyagin, “On the solvability of some initial–boundary value problems for mathematical models of motion of nonlinear viscous and viscoelastic fluids,” Sovrem. Mat. Fundam. Napravlen. 2, 57–69 (2003).
V. G. Zvyagin and M. V. Turbin, Mathematical Issues of Viscoelastic Fluid Dynamics (Krasand URSS, Moscow, 2012) [in Russian].
V. G. Zvyagin and V. P. Orlov, “Some mathematical models in thermomechanics of continua,” J. Fixed Point Theory Appl. 15 (1), 3–47 (2014).
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order: Theory and Applications (Nauka i Tekhnika, Minsk, 1987; Gordon and Breach, London, 1993).
V. P. Orlov, D. A. Rode, and M. A. Pliev, “Weak solvability of the generalized Voigt viscoelasticity model,” Sib. Math. J. 58 (5), 859–874 (2017).
V. G. Zvyagin and V. P. Orlov, “On solvability of an initial-boundary value problem for a viscoelasticity model with fractional derivatives,” Sib. Math. J. 59 (6), 1073–1089 (2018).
V. Zvyagin and V. Orlov, “Weak solvability of fractional Voigt model of viscoelasticity,” Discrete Continuous Dyn. Syst. Ser. A 38 (12), 6327–6350 (2018).
V. Zvyagin and V. Orlov, “On one problem of viscoelastic fluid dynamics with memory on an infinite time interval,” Discrete Continuous Dyn. Syst. Ser. 23 (9), 3855–3877 (2003).
V. G. Zvyagin and V. P. Orlov, “On the weak solvability of a fractional viscoelasticity model,” Dokl. Math. 98 (3), 568–570 (2018).
V. G. Zvyagin and V. P. Orlov, “Weak solvability of a fractional Voigt viscoelasticity model,” Dokl. Math. 96 (2), 491–493 (2017).
G. Seregin, Lecture Notes on Regularity Theory for the Navier–Stokes Equations (World Scientific, Singapore, 2015).
J. C. Robinson, J. L. Rodrigo, and W. Sadowski, The Three-Dimensional Navier–Stokes Equations (Cambridge Univ. Press, Cambridge, 2016).
V. P. Orlov and P. E. Sobolevskii, “On smoothness of weak solutions of the equation of motion of nearly Newtonian fluids,” Chislen. Metody Mekh. Sploshnoi sredy 16 (1), 107–119 (1985).
V. P. Orlov and P. E. Sobolevskii, “On mathematical modes of a viscoelasticity with a memory,” Differ. Integral Equations 4 (1), 103–115 (1991).
V. P. Orlov, “On the strong solutions of a regularized model of a nonlinear visco-elastic medium,” Math. Notes 84 (2), 224–238 (2008).
V. P. Orlov and M. I. Parshin, “On a problem in the dynamics of a thermoviscoelastic medium with memory,” Comput. Math. Math. Phys. 55 (4), 650–665 (2015).
V. G. Zvyagin and V. P. Orlov, “On a model of thermoviscoelasticity of Jeffreys–Oldroyd type,” Comput. Math. Math. Phys. 56 (10), 1803–1812 (2016).
A. V. Zvyagin, V. G. Zvyagin, and D. M. Polyakov, “Dissipative solvability of an alpha model of fluid flow with memory,” Comput. Math. Math. Phys. 59 (7), 1185–1198 (2019).
V. T. Dmitrienko and V. G. Zvyagin, “On strong solutions of an initial-boundary value problem for a regularized model of an incompressible viscoelastic medium,” Russ. Math. 48 (9), 21–37 (2004).
V. G. Zvyagin and D. A. Vorotnikov, “Survey of results and open problems concerning mathematical models of motion of Jeffreys-type viscoelastic media,” Vestn. Voronezh. Gos. Univ. Ser. Fiz. Mat., No. 2, 30–50 (2009).
R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1979).
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow (Gordon and Breach, New York, 1969; Nauka, Moscow, 1970).
P. P. Zabreiko, A. I. Koshelev, M. A. Krasnosel’skii, S. G. Mikhlin, A. S. Rakovshchik, and V. Ya. Stetsenko, Integral Equations: A Reference Text (Nauka, Moscow, 1968; Noordhoff, Leyden, 1975).
V. G. Zvyagin and V. T. Dmitrienko, “On weak solutions of a regularized model of a viscoelastic fluid,” Differ. Equations 38 (12), 1731–1744 (2002).
V. G. Zvyagin and V. T. Dmitrienko, Approximative-Topological Approach to the Study of Hydrodynamic Problems (URSS, Moscow, 2004) [in Russian].
K. Yosida, Functional Analysis (Springer-Verlag, Berlin, 1965).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, R.I., 1968).
Funding
This work was supported by the Russian Science Foundation, project no. 19-11-00146.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Zvyagin, V.G., Orlov, V.P. On Regularity of Weak Solutions to a Generalized Voigt Model of Viscoelasticity. Comput. Math. and Math. Phys. 60, 1872–1888 (2020). https://doi.org/10.1134/S0965542520110159
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542520110159