Geodesic
Solvability of the initial-boundary value problem for the high-order Oldroyd model
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2022), pp. 79-85.

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The paper considers the solvability in the weak sense of the initial-boundary value problem for the high-order Oldroyd model. For the considered model on the base of the Laplace transform the stress tensor is expressed from the rheological relation. After substituting it into the motion equations, an initial-boundary value problem is obtained for an integro-differential equation with memory along the trajectories of the velocity field. After that, based on the approximation-topological approach to the study of hydrodynamic problems, the existence of a weak solution is proved. In the proof of the assertions, properties of regular Lagrangian flows are essentially used.
Keywords: viscoelastic medium, initial-boundary value problem, weak solution, regular Lagrangian flow.
Mots-clés : motion equation 
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     title = {Solvability of the initial-boundary value problem for the high-order {Oldroyd} model},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
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V. G. Zvyagin; V. P. Orlov; M. V. Turbin. Solvability of the initial-boundary value problem for the high-order Oldroyd model. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2022), pp. 79-85. http://geodesic.mathdoc.fr/item/IVM_2022_7_a7/

[1] Oskolkov A. P., “Nachalno-kraevye zadachi dlya uravnenii dvizheniya zhidkostei Kelvina-Foigta i zhidkostei Oldroita”, Tr. MIAN SSSR, 179, 1988, 126–164

[2] Podlubny I., The Laplace Transform Method for Linear Differential Equations of the Fractional Order, 1997, arXiv: funct-an/9710005v1 | MR

[3] Dyarmati I., Non-equilibrium hydrodynamics. Field theory and variational principles, Springer, Berlin-N.Y., 1970

[4] Zvyagin V. G., Orlov V. P., “Solvability of one non-Newtonian fluid dynamics model with memory”, Nonlinear Anal., 172 (2018), 73–98 | DOI | MR | Zbl

[5] Zvyagin V. G., Orlov V. P., “O slaboi razreshimosti zadachi vyazkouprugosti s pamyatyu”, Differents. uravneniya, 53:2 (2017), 215–220 | Zbl

[6] Zvyagin V., Orlov V., “Weak solvability of fractional Voigt model of visoelasticity”, Discrete and Continuous Dynamical Syst., 38:12 (2018), 6327–6350 | DOI | MR

[7] Temam R., Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, Mir, M., 1987 | MR

[8] Crippa G., de Lellis C., “Estimates and regularity results for the DiPerna-Lions flow”, J. für die Reine und Angewandte Math., 616 (2008), 15–46 | MR | Zbl

[9] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Nauka, glavn. redaktsiya fiz.-matem. lit-ry, M., 1970 | MR

[10] Zvyagin V. G., Dmitrienko V. T., “O slabykh resheniyakh regulyarizovannoi modeli vyazkouprugoi zhidkosti”, Differents. uravneniya, 38:12 (2002), 1633–1645 | MR | Zbl

[11] Zvyagin V. G., “Ob orientirovannoi stepeni odnogo klassa vozmuschenii fredgolmovykh otobrazhenii i bifurkatsii reshenii nelineinoi kraevoi zadachi s nekompaktnymi vozmuscheniyami”, Matem. sb., 182:12 (1991), 1740–1768 | Zbl