Abstract
In this work, we consider initial-boundary-value problems for motion equations of a viscoelastic medium with the Jeffreys constitutive law and for motion equations of the regularized Jeffreys model. We obtain a theorem on the convergence of weak solutions of initial-boundary-value problems for the regularized model to weak solutions of the original problem as the regularization parameter tends to zero.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. T. Dmitrienko and V. G. Zvyagin, “Investigation of a regularized model of motion of a viscoelastic medium,” in: O. Rozanova, ed., Analytical Approaches to Multidimensional Balance Laws, Nova Science, New York (2004).
R. V. Goldstein and V. A. Gorodtsov, Mechanics of Continuous Media, Part I [in Russian], Nauka, Fizmatlit, Moscow (2000).
C. Guillopé and J. C. Saut, “Mathematical problems arising in differential models for viscoelastic fluids,” in: Mathematical Topics in Fluid Mechanics, Longman, Harlow (1993), pp. 64–92.
W. G. Litvinov, A Model and General Problem on Plastic Flow under Deformations, Bericht, Universität Stuttgart (1999).
J. G. Oldroyd, “Non-Newtonian flow of fluids and solids,” in: F. R. Eirich, ed., Rheology: Theory and Applications, Academic Press (1956).
M. Reiner, “Rheology,” in: S. Flugge, ed., Handbuch der Physik, Bd. VI, Springer (1958).
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland (1979).
D. A. Vorotnikov and V. G. Zvyagin, “On the existence of weak solutions for the initial-boundary value problem in the Jeffreys model of motion of a viscoelastic medium,” Abstr. Appl. Anal., 10, 815–829 (2004).
V. G. Zvyagin and V. T. Dmitrienko, “On the weak solutions of the regularized model of a viscoelastic fluid,” Differ. Uravn., 38, No. 12, 1633–1645 (2002).
V. G. Zvyagin and D. A. Vorotnikov, Mathematical Models of Non-Newtonian Fluids [in Russian], VSU, Voronezh (2004).
Author information
Authors and Affiliations
Additional information
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 49–63, 2005.
Rights and permissions
About this article
Cite this article
Vorotnikov, D.A., Zvyagin, V.G. On the convergence of solutions of the regularized problem for motion equations of Jeffreys viscoelastic medium to solutions of the original problem. J Math Sci 144, 4398–4408 (2007). https://doi.org/10.1007/s10958-007-0277-0
Issue Date:
DOI: https://doi.org/10.1007/s10958-007-0277-0