The Generalized Linearized Model of Incompressible Viscoelastic Fluid of Nonzero Order

T. G. Sukacheva

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 11 (2012), pp. 75-87

Résumé

In the frames of the non-autonomous sobolev type equations The Cauchy–Dirichlet problem for the generalized linearized Oskolkov's system modeling thermoconvection of the incompressible viscoelastic fluid of the nonzero order is considered. The theorem of the existence of the unique solution of this problem is proved and the description of its extended phase space is received.

Détail

Mots-clés : Sobolev type equation
Keywords: an incompressible viscoelastic fluid, Oskolkov models, extended phase space.

Bibliographie

[1] Oskolkov A. P., “Initial-boundary Value Problems for Equations of Motion Kelvin–Voight and Oldroyd Fluids”, Trudy Mat. In-ta AN SSSR, 179, 1988, 126–164 | MR

[2] Oskolkov A. P., “Nonlocal Problems for One Class of Nonlinear Operator Equations That Arise in the Theory of Sobolev Type Equations”, J. of Mathematical Sciences, 64:1 (1993), 724–735 | Zbl

[3] Karazeeva N. A., Kotsiolis A. A., Oskolkov A. P., On Attractors and Dynamical Systems Generated by the Initial-boundary Value Problems for Equations of Motion Linear Viscoelastic Fluids, Sankt-Peterburg, 1988, 58 pp.

[4] Sviridyuk G. A., “On the General Theory of Operator Semigroups”, Russ. Math. Surv., 49:4 (1994), 45–74 | MR | Zbl

[5] Oskolkov A. P., “Some Quasilinear Systems Occurring in the Study of the Motion of Viscous Fluids”, J. of Mathematical Sciences, 9:5 (1978), 765–790 | Zbl

[6] Oskolkov A. P., “Theory of Voight Fluids”, J. of Mathematical Sciences, 21:5 (1983), 818–821 | MR | Zbl

[7] Sviridyuk G. A., “Solubility of the Thermal Convection of Viscoelastic Incompressible Fluid”, Russian Mathematics, 1990, no. 12, 65–70 | MR

[8] Sviridyuk G. A., “Phase Spaces of Semilinear Sobolev Type Equations with Relatively Strong Sectorial Operator”, St. Petersburg Mathematical J., 6:5 (1994), 216–237

[9] Sukacheva T. G., The Study of Mathematical Models of Incompressible Viscoelastic Fluids, dis. ... Dr. Science, Velikiy Novgorod, 2004, 249 pp.

[10] Sukacheva T. G., “Unsteady Linearized Model of the Motion of an Incompressible Viscoelastic Fluid”, Vestn. Chelyab. gos. un-ta. Ser. Matematika. Mekhanika. Informatika, 2009, no. 20 (158), issue 11, 77–83

[11] Sukacheva T. G., “Unsteady Linearized Model of the Motion of an Incompressible Viscoelastic Fluid of the High Order”, Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie», 2009, no. 17 (150), issue 3, 86–93

[12] G. A. Sviridyuk, V. E. Fedorov, Sobolev type equations and degenerate semigroups of operators, VSP, Utrecht–Boston–Köln–Tokyo, 2003, 179 pp. | MR | Zbl

[13] Sukacheva T. G., “The Problem of Thermal Convection for a Linearized Model of the Motion of an Incompressible Viscoelastic Fluid”, Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie», 2010, no. 16 (192), issue 5, 83–93

[14] Sukacheva T. G., “The Problem of Thermal Convection for the Linearized Model of the Motion of an Incompressible Viscoelastic Fluid”, Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie», 2011, no. 37 (254), issue 10, 40–53

[15] Sviridyuk G. A., “Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type”, Izvestiya: Mathematics, 42:3 (1994), 601–614 | DOI | MR | Zbl

[16] H. A. Levine, “Some nonexistance and instability theorems for solutions of formally parabolic equations of the form $Du_t=-Au+F(u)$”, Arch. Rat. Mech. Anal., 51:5 (1973), 371–386 | DOI | MR | Zbl

[17] Sviridyuk G. A., Sukacheva T. G., “Cauchy Problem for a Class of Semilinear Equations of Sobolev Type”, Siberian Mathematical J., 31:5 (1990), 794–802 | DOI | MR | Zbl

[18] Sviridyuk G. A., “The Phase Space of a Class of Operator Equations”, Differential Equations, 26:2 (1990), 250–258 | MR | Zbl

[19] Borisovich Yu. G., Zvyagin V. G., Sapronov Yu. I., “Non-linear Fredholm Maps and Leray-Schauder Theory”, Russian Mathematical Surveys, 32:4 (1977), 1–55 | DOI | MR | MR

[20] Marsden Dzh., Mak-Kraken M., Hopf Bifurcation and Its Applications, Mir, M., 1980, 368 pp. | MR

[21] Bokareva T. A., Investigation of Phase Space of Sobolev Type Equations with Relatively Sectorial Operators, Dis. ... cand. Science, Sankt-Peterburg, 1993, 107 pp.

[22] Sviridyuk G. A., “A Model of Weakly Viscoelastic Fluid”, Russian Mathematics, 1994, no. 1, 62–70 | MR

[23] Sukacheva T. G., “On the Solvability of a Nonstationary Problem of the Dynamics of Incompressible Viscoelastic Kelvin–Voight Fluid of Nonzero Order”, Russian Mathematics, 1998, no. 3 (430), 47–54

[24] Sviridyuk G. A., “Semilinear Sobolev Type Equation with Relatively Bounded Operator”, DAN SSSR, 18:4 (1991), 828–831

[25] Sviridyuk G. A., “Semilinear Sobolev Type Equations with Relatively Sectorial Operators”, Dokl. RAN, 329:3 (1993), 274–277

[26] Sviridyuk G. A., Fedorov V. E., “On the Identities of Analytic Semigroups of Operators with Kernels”, Siberian Mathematical J., 39:3 (1998), 522–533 | MR | Zbl

[27] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Springer Verlag, Berlin–Heidelberg–N.Y., 1981, 840 pp. | DOI | MR | MR | Zbl