×

Dissipative solvability of Jeffreys-Oldroyd-\(\alpha\) model. (English) Zbl 1477.35176

Summary: We study the initial-boundary value problem for the so-called alpha model of Jeffreys-Oldroyd fluids motion in 2D and 3D dimensions. In this paper the global in time existence of dissipative solutions to this problem is obtained. For this the topological approximation method to the initial-boundary value problem is applied.

MSC:

35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D.D. Holm, C. Jeffery, S. Kurien, D. Livescu, M.A. Taylor and B.A. Wingate, The LANS-\( \alpha\) model for computing turbulence origins, results, and open problems, Los Alamos Science 29 (2005), 152-172.
[2] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincare models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett. 349 (1998), 4173-4177.
[3] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincare equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), 1-81. · Zbl 0951.37020
[4] A.A. Ilyin, E. Lunasin and E.S. Titi, A modified Leray-\( \alpha\) subgrid scale model of turbulence, Nonlinearity 19 (2006), 879-897. · Zbl 1106.35050
[5] M.A. Krasnosel’skiı and P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 263, Springer-Verlag, Berlin, 1984. · Zbl 0546.47030
[6] J. Leray, Essai sur le mouvement d’un fluide visqueux emplissant l’space, Acta Math. 63 (1934), 193-248.
[7] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Oxford University Press, Oxford, 1996. · Zbl 0866.76002
[8] N.G. Lloyd, Degree Theory, Cambridge University Press, 1978. · Zbl 0367.47001
[9] J.G. Oldroyd, Non-Newtonian flow of liquids and solids, Rheology: Theory and Applications, Vol. 1, Academic Press, New York, 1956.
[10] M. Reiner, Reology, Handbuch der Physik (S. Flugge, ed.), Bd. VI, Springer, 1958. · Zbl 0102.42102
[11] J. Simon, Compact sets in the space \(L^p (0, T ; B)\), Ann. Mat. Pura Appl. 146 (1987), 65-96. · Zbl 0629.46031
[12] R. Temam, Navier-Stokes Equations. Studies in Mathematics and its Application, NorthHolland, Amsterdam, 1979. · Zbl 0426.35003
[13] D.A. Vorotnikov, Dissipative solutions for equations of viscoelastic diffusion in polymers, J. Math. Anal. Appl. 339 (2008), 876-888. · Zbl 1128.35355
[14] D.A. Vorotnikov, The second boundary value problem for equations of viscoelastic diffusion in polymers, Adv. Math. Res. 10 (2010), 249-271.
[15] D.A. Vorotnikov, Global generalized solutions for Maxwell-alpha and Euler-alpha equations, Nonlinearity 25 (2012), 309-327. · Zbl 1252.35232
[16] A.V. Zvyagin, Solvability for equations of motion of weak aqueous polymer solutions with objective derivative, Nonlinear Anal. 90 (2013), 70-85. · Zbl 1404.76016
[17] V.G. Zvyagin, Topological approximation approach to study of mathematical problems of hydrodynamics, J. Math. Sci. 201 (2014), 830-858. · Zbl 1310.35124
[18] V.G. Zvyagin and S.K. Kondratyev, Attractors of equations of non-Newtonian fluid dynamics, Uspekhi Mat. Nauk. 69 (2014), 81-156; English transl.: Russ. Math. Surv. 69 (2014), 845-913. · Zbl 1347.35004
[19] V.G. Zvyagin and S.K. Kondratyev, Pullback attractors of the Jeffreys-Oldroyd equations, J. Differentail Equations 260 (2016), 5026-5042. · Zbl 1331.35055
[20] V.G. Zvyagin, V.V. Obukhovskii and A V. Zvyagin, On inclusions with multivalued operators and their applications to some optimization problems, J. Fixed Point Theory Appl. 16 (2014), 27-82. · Zbl 1316.49025
[21] V.G. Zvyagin and V.P. Orlov, On certain mathematical models in continuum thermomechanics, J. Fixed Point Theory Appl. 15 (2014), 3-47. · Zbl 1315.80004
[22] A.V. Zvyagin and D.M. Polyakov, On the solvability of the Jeffreys-Oldroyd-alpha model, Differ. Uravn. 52 (2016), 782-787; English transl.: Differential Equations 52 (2016), 761-766. · Zbl 1352.35134
[23] V.G. Zvyagin and D.A. Vorotnikov, Approximating-topological methods in some problems of hydrodynamics, J. Fixed Point Theory Appl. 3 (2008), 23-49. · Zbl 1158.35407
[24] V.G. Zvyagin and D.A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, de Gruyter Series in Nonlinear Analysis and Applications, vol. 12, Walter de Gruyter, Berlin, 2008. · Zbl 1155.76004
[25] A.V. Zvyagin, V.G. Zvyagin and D.M. Polyakov, On solvability of a fluid flow alphamodel with memory, Izv. Vyssh. Uchebn. Zaved. Mat. 6 (2018), 78-84; English transl.: Russian Math. 62 (2018), 69-74. · Zbl 1433.76014
[26] A.V. Zvyagin, V.G. Zvyagin and D.M. Polyakov, Dissipative solvability of an alpha model of fluid flow with memory, Zh. Vychisl. Mat. Mat. Fiz. 59 (2019), 1243-1257; English transl.: Comput. Math. Math. Phys. 59 (2019), 1185-1198. · Zbl 1435.35318
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.