On a class of Sobolev-type equations

T. G. Sukacheva   ; A. O. Kondyukov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 7 (2014) no. 4, pp. 5-21

Résumé

The article surveys the works of T. G. Sukacheva and her students studying the models of incompressible viscoelastic Kelvin–Voigt fluids in the framework of the theory of semilinear Sobolev-type equations. We focus on the unstable case because of greater generality. The idea is illustrated by an example: the non-stationary thermoconvection problem for the order 0 Oskolkov model. Firstly, we study the abstract Cauchy problem for a semilinear nonautonomous Sobolev-type equation. Then, we treat the corresponding initial-boundary value problem as its concrete realization. We prove the existence and uniqueness of a solution to the stated problem. The solution itself is a quasi-stationary semi-trajectory. We describe the extended phase space of the problem. Other problems of hydrodynamics can also be investigated in this way: for instance, the linearized Oskolkov model, Taylor's problem, as well as some models describing the motion of an incompressible viscoelastic Kelvin–Voigt fluid in the magnetic field of the Earth.

Détail

Keywords: Sobolev type equations; incompressible viscoelastic fluids; relatively $p$-sectorial operators; extended phase spaces.

Bibliographie

[1] Oskolkov A. P., “Initial-Boundary Value Problems for the Equations of the Motion of the Kelvin–Voight and Oldroyd Fluids”, Proceedings of the Steklov Institute of Mathematics, 179, 1988, 137–182 | MR

[2] Oskolkov A. P., “Nonlocal Problems for a Class of Nonlinear Operator Equations that Arise in the Theory of Sobolev Type Equations”, Journal of Soviet Mathematics, 64:1 (1993), 724–735 ; A. P. Oskolkov, “Nelokalnye problemy dlya odnogo klassa nelineinykh operatornykh uravnenii, voznikayuschikh v teorii uravnenii tipa S. L. Soboleva”, Zapiski nauchnykh seminarov LOMI, 198, 1991, 31–48 | DOI | MR | Zbl

[3] Oskolkov A. P., “Some Nonstationary Linear and Quasilinear Systems Occurring in the Investigation of the Motion of Viscous Fluids”, Journal of Soviet Mathematics, 10:2 (1978), 299–335 ; A. P. Oskolkov, “O nekotorykh nestatsionarnykh lineinykh i kvazilineinykh sistemakh, vstrechayuschikhsya pri izuchenii dvizheniya vyazkikh zhidkostei”, Zapiski nauchnykh seminarov LOMI AN SSSR, 59, 1976, 133–177 | DOI | MR | Zbl

[4] Oskolkov A. P., “Theory of Voigt's Fluids”, Journal of Soviet Mathematics, 21:4 (1983), 818–821 ; A. P. Oskolkov, “K teorii zhidkostei Foigta”, Zapiski nauchnykh seminarov LOMI, 96, 1980, 233–236 | DOI | Zbl | MR | Zbl

[5] Sviridyuk G. A., “On the General Theory of Operator Semigroups”, Russian Mathematical Surveys, 49:4 (1994), 45–74 | DOI | MR | Zbl

[6] Sviridyuk G. A., “Solvability of the Thermoconvection Problem of the Viscoelastic Incompressible Fluid”, Soviet Mathematics (Izvestiya VUZ. Matematika), 34:12 (1990), 80–86 | MR | Zbl

[7] Sviridyuk G. A., “Phase Spaces of Semilinear Equations of Sobolev Type with Relatively Strongly Sectorial Operators”, St. Petersburg Mathematical Journal, 6:5 (1994), 1109–1126 | MR | Zbl

[8] Sukacheva T. G., “Solvability of a Nonstationary Thermal Convection Problem of a Viscoelastic Incompressible Fluid”, Differential Equations, 36:8 (2000), 1225–1232 | DOI | MR | Zbl

[9] Sukacheva T. G., Research of Mathematical Models of Incompressible Viscoelastic Fluids, The Dissertation for Scientific Degree of the Doctor of Physical and Mathematical Sciences, Velikiy Novgorod, 2004, 249 pp.

[10] Sviridyuk G. A., Fedorov V. E., Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston–Köln, 2003, 179 pp. | MR | Zbl

[11] Sviridyuk G. A., “Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type”, Russian Academy of Sciences. Izvestiya Mathematics, 42:3 (1994), 601–614 ; G. A. Sviridyuk, “Kvazistatsionarnye traektorii polulineinykh dinamicheskikh uravnenii tipa Soboleva”, Izvestiya RAN. Seriya matematicheskaya, 57:3 (1993), 192–207 | DOI | MR | Zbl

[12] Levine H. A., “Some Nonexistance and Instability Theorems for Solutions of Formally Parabolic Equations of the Form $Du_t=-Au+F(u)$”, Archive for Rational Mechanics and Analysis, 51:5 (1973), 371–386 | DOI | MR | Zbl

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

[14] Sviridyuk G. A., Sukacheva T. G., “Phase Space of One Class of Operator Equations”, Differential Equations, 26:2 (1990), 250–258 (in Russian) | MR | Zbl

[15] Sviridyuk G. A., Sukacheva T. G., “Some Mathematical Problems of the Dynamics of Viscoelastic Incompressible media”, Vestnik MaGU. Matematika, 2005, no. 8, 5–33 (in Russian)

[16] Borisovich Yu. G., Zvyagin V. G., Sapronov Y. I., “Non-Linear Fredholm Maps and Leray–Schauder Theory”, Russian Mathematical Surveys, 32:4 (1977), 1–54 (in Russian) | DOI | MR | Zbl

[17] Marsden J. E., McCracken M., The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976 ; Dzh. Marsden, M. Mak-Kraken, Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, Mir, M., 1980, 368 pp. | MR | Zbl | MR

[18] Bokareva T. A., Research of Phase Space of Sobolev Type Equations with Relatively Sectorial Operators, The Dissertation for Scientific Degree of the Kandidat of Physical and Mathematical Sciences, St. Petersburg, 1993, 107 pp.

[19] Ladyzhenskaya O. A., Mathematical Problems of Dynamics of Viscous Incompressible Fluid, Nauka, M., 1970, 288 pp.

[20] Sviridyuk G. A., “On a Model of Weakly Viscoelastic Fluid”, Russian Mathematics (Izvestiya VUZ. Matematika), 38:1 (1994), 59–68 | MR | Zbl

[21] Sviridyuk G. A., “Semilinear Equations of Sobolev Type with Relatively Bounded Operator”, Doklady Akademii Nauk, 318:4 (1991), 828–831 (in Russian) | MR | Zbl

[22] Sviridyuk G. A., “Semilinear Equations of Sobolev Type with Relatively Sectorial Operators”, Doklady RAN, 329:3 (1993), 274–277 (in Russian) | MR | Zbl

[23] Sviridyuk G. A., Fedorov V. E., “Analytic Semigroups with Kernel and Linear Equations of Sobolev Type”, Siberian Mathematical Journal, 36:5 (1995), 973–987 | DOI | MR | Zbl

[24] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer, Berlin, 1981 ; D. Khenri, Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Per. s angl., Mir, M., 1985, 376 pp. | DOI | MR | Zbl | MR

[25] Sviridyuk G. A., Sukacheva T. G., “On the Solvability of a Nonstationary Problem Describing the Dynamics of an Incompressible Viscoelastic Fluid”, Mathematical Notes, 63:3–4 (1998), 388–395 | DOI | MR | Zbl

[26] Sukacheva T. G., “On a Certain Model of Motion of an Incompressible Visco-Elastic Kelvin–Voight Fluid of Nonzero Order”, Differential Equations, 33:4 (1997), 557–562 | MR | Zbl

[27] Sukacheva T. G., “On the Solvability of the Non-Stationary Problem of Dynamics of Incompressible Viscoelastic Kelvin–Voight Fluid of Nonzero Order”, Russian Mathematics (Izvestiya VUZ. Matematika), 42:3 (1998), 44–51 | MR | Zbl

[28] Sukacheva T. G., “Solvability of a Nonstationary Thermoconvection Convection Problem for a Viscoelastic Incompressible Fluid”, Differential Equations, 36:8 (2000), 1225–1232 | DOI | MR | Zbl

[29] Sukacheva T. G., Matveeva O. P., “The Thermoconvection Problem of the Incompressible Viscoelastic Kelvin–Voight Fluid of the Nonzero Order”, Russian Mathematics (Izvestiya VUZ. Matematika), 45:11 (2001), 44–51 | MR

[30] Sukacheva T. G., Matveeva O. P., “Quasi-Stationary Semi-Trajectories in the Non-Stationary Model of the Thermoconvection of the Viscoelastic Incompressible Fluid of the High Order”, INPRIM-98, Izd. Inst. Math., Novosibirsk, 1998, 98–105 (in Russian) | MR | Zbl

[31] Sukacheva T. G., “Non-Stationary Linearized Model of the Motion of an Incompressible Viscoelastic Fluid”, Vestnik Chelyabinskogo gosudarstvennogo universiteta. Seriya Matematika. Mekhanika. Informatika, 20(158):11 (2009), 77–83 (in Russian) | MR

[32] Sukacheva T. G., Daugavet M. N., “Linearized Model of the Motion of an Incompressible Viscoelastic Kelvin–Voigt Fluid of Nonzero Order”, Journal of Applied and Industrial Mathematics, 6:4 (2003), 111–118 (in Russian) | MR | Zbl

[33] Sukacheva T. G., “Non-Stationary Linearized Model of the Motion of an Incompressible Viscoelastic Fluid of the High Order”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming Computer Software, 17(150) (2009), 86–93 (in Russian) | MR | Zbl

[34] Sukacheva T. G., “The Thermoconvection Problem for the Linearizied Model of the Incompressible Viscoelastic Fluid”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming Computer Software, 16(192):5 (2010), 83–93 (in Russian) | Zbl

[35] Sukacheva T. G., “The Thermoconvection Problem for the Linearizied Model of the Incompressible Viscoelastic Fluid of the Nonzero Order”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming Computer Software, 37(254):10 (2011), 40–53 (in Russian)

[36] Sukacheva T. G., “The Generalizied Linearizied Thermoconvection Problem for the Model of the Incompressible Viscoelastic Fluid of the Nonzero Order”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming Computer Software, 5(264):11 (2012), 75–87 (in Russian) | Zbl

[37] Sukacheva T. G., Extended Phase Spaces of Oskolkov Models, LAP, 2011

[38] Matveeva O. P., Sukacheva T. G., “Quasi-Stationary Trajectories of the Teylor Problem for the Generalizied Model of the Incompressible Viscoelastic Fluid”, Bulletin of the Novgorod State University. Series “Physical and Mathematical Sciences”, 2 (2013), 34–37 (in Russian)

[39] Matveeva O. P., Sukacheva T. G., The Mathematical Models of a Viscoelastic Incompressible Fluid of Nonzero Order, Publishing Center of South Ural State University, Chelyabinsk, 2014 (in Russian)

[40] Sukacheva T. G., Kondyukov A. O., “The Phase Space of a Model of the Magnetohydrodynamics”, IV International School-Seminar “Nonlinear Analysis and Extremal Problems” (Irkutsk, 2014), 30 (in Russian) [Т. Г. Сукачева, А. О. Кондюков, “Фазовое пространство одной модели магнитогидродинамики”, IV Международная школа-семинар «Нелинейный анализ и экстремальные задачи» (Иркутск, 22–28 июня 2014 г.), Иркутск, 2014, 30] | MR | Zbl

[41] Kondyukov A. O., Sukacheva T. G., “Quasi-Stationary Semitrajectories in a Model of the Magnetohydrodynamics”, International Conference on Differential Equations and Dynamical Systems (Suzdal, 2014), 91–92 (in Russian)