Solvability of a Thermoviscoelastic Model of the Motion of Solutions of Polymers Satisfying the Objectivity Principle

A. V. Zvyagin

Matematičeskie zametki, Tome 105 (2019) no. 6, pp. 839-856.

Résumé

The existence of weak solutions of the initial boundary-value problem for a mathematical model describing the motion of weakly concentrated aqueous solutions of polymers is proved. In the model under study, the rheological relation defining the type of the liquid satisfies the objectivity principle. To this end, a smoothed objective Jaumann derivative is considered in the rheological relation. Also, in the mathematical model, the viscosity of the medium depends on temperature, which leads to the appearance of an additional energy balance equation. The proof of the solvability of the problem under consideration is based on the approximation-topological approach to the study of hydrodynamic problems and on the theory of fractional powers of positive operators.

Détail

Keywords: existence theorem, weak solution, non-Newtonian medium, thermoviscoelasticity.

Bibliographie

[1] Ya. I. Voitkunskii, V. B. Amfilokhiev, V. A. Pavlovskii, “Uravneniya dvizheniya zhidkosti s uchetom ee relaksatsionnykh svoistv”, Trudy Leningradskogo ordena Lenina korablestroitelnogo instituta, 69 (1970), 19–26

[2] V. A. Pavlovskii, “K voprosu o teoreticheskom opisanii slabykh vodnykh rastvorov polimerov”, Dokl. AN SSSR, 200:4 (1971), 809–812

[3] V. B. Amfilokhiev, Ya. I. Voitkunskii, N. P. Mazaeva, Ya. S. Khodorkovskii, “Techeniya polimernykh rastvorov pri nalichii konvektivnykh uskorenii”, Trudy Leningradskogo ordena Lenina korablestroitelnogo instituta, 96 (1975), 3–9

[4] V. B. Amfilokhiev, V. A. Pavlovskii, “Eksperimentalnye dannye o laminarno-turbulentnom perekhode pri techenii polimernykh rastvorov v trubakh”, Trudy Leningradskogo ordena Lenina korablestroitelnogo instituta, 104 (1976), 3–5

[5] A. P. Oskolkov, “O nekotorykh kvazilineinykh sistemakh, vstrechayuschikhsya pri izuchenii dvizheniya vyazkikh zhidkostei”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 8, Zap. nauchn. sem. LOMI, 52, Izd-vo «Nauka», Leningrad. otd., L., 1975, 128–157 | MR | Zbl

[6] V. G. Zvyagin, M. V. Turbin, Matematicheskie voprosy gidrodinamiki vyazkouprugikh sred, KRASAND (URSS), M., 2012

[7] A. V. Zvyagin, “Solvability for equations of motion of weak aqueous polymer solutions with objective derivative”, Nonlinear Anal., 90 (2013), 70–85 | DOI | MR | Zbl

[8] A. V. Zvyagin, “Attraktory dlya modeli dvizheniya polimerov s ob'ektivnoi proizvodnoi v reologicheskom sootnoshenii”, Dokl. AN, 453:6 (2013), 599–602 | DOI | Zbl

[9] S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov, Kraevye zadachi mekhaniki neodnorodnykh zhidkostei, Nauka, Novosibirsk, 1983 | MR

[10] A. V. Zvyagin, V. P. Orlov, “Issledovanie razreshimosti zadachi termovyazkouprugosti dlya lineino uprugo-zapazdyvayuschei zhidkosti Foigta”, Matem. zametki, 97:5 (2015), 681–698 | DOI | MR | Zbl

[11] A. V. Zvyagin, V. P. Orlov, “Razreshimost zadachi termovyazkouprugosti dlya odnoi modeli Oskolkova”, Izv. vuzov. Matem., 2014, no. 9, 69–74 | Zbl

[12] A. V. Zvyagin, “Optimalnoe upravlenie s obratnoi svyazyu dlya termovyazkouprugoi modeli dvizheniya zhidkosti Foigta”, Dokl. AN, 468:3 (2016), 251–253 | DOI | Zbl

[13] O. A. Ladyzhenskaya, Matematicheskaya teoriya vyazkoi neszhimaemoi zhidkosti, Nauka, M., 1964

[14] R. Temam, Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, Mir, M., 1981

[15] I. I. Vorovich, V. I. Yudovich, “Statsionarnoe techenie vyazkoi neszhimaemoi zhidkosti”, Matem. sb., 53 (95):4 (1961), 393–428 | MR

[16] A. V. Fursikov, Optimalnoe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya, Universitetskaya seriya, 5, Nauchnaya kniga, Novosibirsk, 1999 | Zbl

[17] V. G. Zvyagin, V. T. Dmitrienko, Approksimatsionno-topologicheskii podkhod k issledovaniyu zadach gidrodinamiki, Editorial URSS, M., 2004

[18] V. G. Zvyagin, “Approksimatsionno-topologicheskii podkhod k issledovaniyu matematicheskikh zadach gidrodinamiki”, SMFN, 46, RUDN, M., 2012, 92–119

[19] V. G. Zvyagin, V. P. Orlov, “Ob odnoi parabolicheskoi zadache dvizheniya termovyazkouprugikh sred”, Matem. zametki, 99:3 (2016), 465–469 | DOI | MR | Zbl

[20] V. G. Zvyagin, V. P. Orlov, “Solvability of a parabolic problem with non-smooth data”, J. Math. Anal. Appl., 453:1 (2017), 589–606 | DOI | MR | Zbl

[21] D. Blanchard, N. Bruyére, O. Guibé, “Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation”, Commun. Pure Appl. Anal., 12:5 (2013), 2213–2227 | DOI | MR | Zbl

[22] D. Blanchard, “A few results on coupled systems of thermomechanics”, On the Notions of Solution to Nonlinear Elliptic Problems. Results and Developments, Quad. Mat., 23, Dept. Math., Seconda Univ. Napoli, Caserta, 2009, 145–182 | MR

[23] J. Simon, “Compact sets in the space $L^p(0,T; B)$”, Ann. Mat. Pura Appl. (4), 146 (1987), 65–96 | DOI | MR | Zbl