Geodesic
On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids
Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 2, Tome 2 (2003), pp. 57-69 
V. G. Zvyagin. On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids. Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 2, Tome 2 (2003), pp. 57-69. http://geodesic.mathdoc.fr/item/CMFD_2003_2_a2/
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     author = {V. G. Zvyagin},
     title = {On {Solvability} of {Some} {Initial-Boundary} {Problems} for {Mathematical} {Models} of the {Motion} of {Nonlinearly} {Viscous} and {Viscoelastic} {Fluids}},
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     volume = {2},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In the paper, the settings of initial-boundary and initial value problems arising in a number of models of movement of nonlinearly viscous or viscoelastic incompressible fluid are considered, and existence theorems for these problems are presented. In particular, the settings of initial-boundary value problems appearing in the regularized model of the movement of viscoelastic fluid with Jeffris constitutive relation are described. The theorems for the existence of weak and strong solutions for these problems in bounded domains are given. The initial value problem for a nonlinearly viscous fluid on the whole space is considered. The estimates on the right-hand side and initial conditions under which there exist local and global solutions of this problem are presented. The modification of Litvinov's model for laminar and turbulent flows with a memory is described. The existence theorem for weak solutions of initial-boundary value problem appearing in this model is given.

[1] Agranovich Yu. A., Sobolevskii P. E., “Dvizhenie nelineino-vyazkoi zhidkosti”, Dokl. AN SSSR, 314:3 (1990), 521–525 | Zbl

[2] Vorotnikov D. A., “O dvizhenii nelineino-vyazkoi zhidkosti v $\mathbb R^n$”, Vestnik VGU, cer. Fiz., Mat., 2002, no. 1, 102–120 | Zbl

[3] Zvyagin V. G., Dmitrienko V. T., “Gomotopicheskaya klassifikatsiya odnogo klassa nepreryvnykh otobrazhenii”, Mat. zametki, 31:5 (1982), 801–812 | MR | Zbl

[4] Zvyagin V. G., Dmitrienko V. T., “O slabykh resheniyakh nachalno-kraevoi zadachi dlya uravneniya dvizheniya vyazkouprugoi zhidkosti”, Dokl. RAN, 380:3 (2001), 308–311 | MR | Zbl

[5] Litvinov V. G., Dvizhenie nelineino-vyazkoi zhidkosti, Nauka, M., 1982, 376 pp.

[6] Oldroit Dzh. G., “Nenyutonovskie techeniya zhidkostei i tverdykh tel”, Reologiya: Teoriya i prilozheniya, IL, M., 1962, 757–793

[7] Agranovich Yu. Ya., Sobolevskii P. E., “Motion of nonlinear visco-elastic fluid”, Nonlinear Anal., Theory Methods Appl., 32:6 (1998), 755–760 | DOI | MR | Zbl

[8] Litvinov W. G., “Some models and problems for laminar and turbulent flows of viscous and nonlinear viscous fluids”, J. Math. Phys. Sci., 30:3 (1996), 101–157 | MR | Zbl

[9] Litvinov W. G., “General nonlocal model describing the laminar and turbulent flows of viscous and nonlinear viscous fluids and its investigation”, Math. Nachr., 220 (2000), 79–110 | 3.0.CO;2-3 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl