---

Journal of Applied Nonlinear Dynamics 4(3) (2015) 223--228 | DOI:10.5890/JAND.2015.09.002

V.E. Fedorov; P.N. Davydov

Chelyabinsk State University, Chelyabinsk, Russia

Download Full Text PDF

 

Abstract

By means of the degenerate semigroups theory methods the local existence of a unique solution is proved for initial-boundary value problems to a class of partial differential equations systems of generalized hydrodynamics type. General results are illustrated by examples of a system with the nonlinear viscosity and a weighted system.

Acknowledgments

The first author is supported by Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020). The second author is supported by the grant of Russian Foundation for Basic Research.

References

1. [1]	Oskolkov, A.P. (1998), Initial-boundary value problems for equations of Kelvin-Voigh fluids and Oldroyd fluids motion. Proceedings of the Steklov Institute of Mathematics, 179, 126-164, 1988 [in Russian].

2. [2]	Zvyagin, V.G. and Turbin, M.V. (2010), The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigh fluids , J. of Math. Sciences, 168 (2), 157-308.

3. [3]	Fedorov, V.E., Davydov, P.N.(2013), Semilinear degenerate evolution equations and nonlinear systems of hydrodynamics type, Proceedings of the Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 19 (4), 267-278, 2013 [in Russian].

4. [4]	Showalter, R.E.(1975), Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal., 6 (1), 25-42, 1975.

5. [5]	Sviridyuk, G.A. and Fedorov, V.E. (2003), Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht-Boston.

6. [6]	Sviridyuk, G.A. and Sukacheva, T.G. (1998), On the solvability of a nonstationary problem describing the dynamics of an incompressible viscoelastic fluid, Math. Notes, 63 (3), 388-395.

7. [7]	Ladyzhenskaya, O.A. (1969), The Mathematical Theory of Viscous Incompressible Flow, Mathematics and Its Applications 2 (Revised Second ed.), Gordon and Breach, New York-London-Paris-Montreux-Tokyo- Melbourne.

8. [8]	Ivanova, N.D., Fedorov, V.E., and Komarova, K.M. (2012), Nonlinear inverse problem for the Oskolkov system, linearized in a stationary solution neighborhood. Bulletin of Chelyabinsk State University. Mathematics. Mechanics. Informatics, 15, 49-70 [in Russian].

9. [9]	Abraham, R. and Robbin, J. (1967), Transversal Mappings and Flows, A.Benjamin Inc., New York.

10. [10]	Hassard, B.D., Kazarinoff, N.D., and Wan. Y.H. (1981), Theory and Applications of Hopf Bifurcation, London Math. Society, Lecture Notes, Ser. 41, Cambridge University Press, Cambridge-London-New York- New Rochelle-Melbourne-Sydney.