Analysis of the Avalos-Triggiani Problem for the Linear Oskolkov System of the Highest Order and a System of Wave Equations

The Avalos-Triggiani problem for a system of wave equations and a linear Oskolkov system of the highest order is investigated. The mathematical model contains a linear Oskolkov system describing the flow of an incompressible viscoelastic Kelvin-Voigt fluid of of the highest order, and a wave vector equation corresponding to some structure immersed in the specified fluid. Based on the method proposed by the authors of this problem, the theorem of the existence of the unique solution to the Avalos-Triggiani problem for the indicated systems is proved.

Keywords

Avalos-Triggiani problem; incompressible viscoelastic fluid; linear Oskolkov systems.

References

1. Avalos G., Lasiecka I., Triggiani R. Higher Regularity of a Coupled Parabolic-Hyperbolic Fluid-Structure Interactive System. Georgian Mathematical Journal, 2008, vol. 15, no. 3, pp. 403-437. DOI:10.1515/GMJ.2008.403
2. Avalos G., Triggiani R. Backward Uniqueness of the S.C. Semigroup Arising in Parabolic-Hyperbolic Fluid-Structure Interaction. Differential Equations, 2008, vol. 245, no. 3, pp. 737-761. DOI:10.1016/j.jde.2007.10.036
3. Oskolkov A.P. Initial-Boundary Value Problems for Equations of Motion of Kelvin-Voight Fluids and Oldroyd fluids. Proceedings of the Steklov Institute of Mathematics, 1989, vol. 179, pp. 137-182.
4. Sviridyuk G.A., Sukacheva T.G. The Avalos-Triggiani Problem for the Linear Oskolkov System and a System of Wave Equations. Computational Mathematics and Mathematical Physics, 2022, vol. 62, no. 3, pp. 427-431.
5. Sukacheva T.G., Sviridyuk G.A. The Avalos-Triggiani Problem for the Linear Oskolkov System and a System of Wave Equaions. II. Journal of Computational and Engineering Mathematics, 2022, vol. 9, no. 2, pp. 67-72. DOI:10.14529/jcem220206
6. Kondyukov A.O., Sukacheva T.G. The Linear Oskolkov System of Nonzero Order in the Avalos-Triggiani Problem. Journal of Computational and Engineering Mathematics, 2023, vol. 10, no. 4, pp. 17-23. DOI: 10.14529/jcem230302
7. Sukacheva T.G., Kondyukov A.O. Analysis of the Avalos-Triggiani Problem for the Linear Oskolkov System of Nonzero Order and a System of Wave Equations. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2023, vol. 16, no. 4, pp. 93-98. DOI: 10.14529/mmp230407
8. Sukacheva T.G. Unsteady Linearized Model of the Motion of an Incompressible High-Order Viscoelastic Fluid. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2009, no. 17(150), issue 3, pp. 86-93.
9. Oskolkov A.P. Some Nonstationary Linear and Quasilinear Systems Occurring in the Investigation of the Motion of Viscous Fluids. Journal of Soviet Mathematics, 1978, vol. 10, pp. 299-335. DOI: 10.1007/BF01566608
10. Sviridyuk G.A., Sukacheva T.G. Phase Spaces of a Class of Operator Semilinear Equations of Sobolev Type. Differential Equations, 1990, vol. 26, no. 2, pp. 188-195.
11. Sviridyuk, G.A., Sukacheva, T.G. On the Solvability of a Nonstationary Problem Describing the Dynamics of an Incompressible Viscoelastic Fluid. Mathematical Notes, 1998, vol. 63, no. 3, pp. 388-395. DOI:10.1007/BF02317787
12. Kondyukov A.O., Sukacheva T.G. Phase Space of the Initial-Boundary Value Problem for the Oskolkov System of Nonzero Order. Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 5, pp. 823-828. DOI: 10.1134/S0965542515050127
13. Vasyuchkova K.V., Manakova N.A., Sviridyuk G.A. Some Mathematical Models with a Relatively Bounded Operator and Additive "White Noise". Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 4, pp. 5-14. DOI: 10.14529/mmp170401
14. Sviridyuk G.A., Zamyshlyaeva A.A., Zagrebina S.A. Multipoint Initial-Final Value for one Class of Sobolev Type Models of Higher Order with Additive "White Noise". Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 3, pp. 103-117. DOI: 10.14529/mmp180308
15. Favini A., Zagrebina S.A., Sviridyuk G.A. Multipoint Initial-Final Value Problems for Dinamical Sobolev-Type Equations in the Space of Noises. Electronic Journal of Differential Equations, 2018, vol. 2018, no. 128, pp. 1-10.