Abstract and Applied Analysis

On dynamics of viscoelastic multidimensional medium with variable boundary

V. P. Orlov

Abstract

We prove the existence and uniqueness theorems for solutions of an initial-boundary value problem to the system of equations, which describes dynamics of viscoelastic continuous medium with a variable boundary and a memory along the trajectories of particles in classes of summable functions.

Article information

Source
Abstr. Appl. Anal., Volume 7, Number 9 (2002), 475-495.

Dates
First available in Project Euclid: 14 April 2003

https://projecteuclid.org/euclid.aaa/1050348353

Digital Object Identifier
doi:10.1155/S1085337502203097

Mathematical Reviews number (MathSciNet)
MR1934202

Zentralblatt MATH identifier
1034.35141

Subjects
Primary: 35Q99: None of the above, but in this section

Citation

Orlov, V. P. On dynamics of viscoelastic multidimensional medium with variable boundary. Abstr. Appl. Anal. 7 (2002), no. 9, 475--495. doi:10.1155/S1085337502203097. https://projecteuclid.org/euclid.aaa/1050348353

References

• O. V. Besov, V. P. Il'in, and S. M. Nikol'skiï, Integral Representations of Functions and Embedding Theorems, Nauka, Moscow, 1975 (Russian).
• V. T. Dmitrienko and V. G. Zvyagin, On the solvability of a boundary value problem for a mathematical model of steady flows of a nonlinearly viscous fluid, Mat. Zametki 69 (2001), no. 6, 843–853. \CMP1+861+5661 861 566
• S. K. Godunov, Introduction in Continuous Mechanics, Nauka, Moscow, 1982.
• O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equation of Parabolic Type, Nauka, Moscow, 1973.
• V. G. L\=\itv\=\inov, Movement of a Nonlinear Viscoelastic Fluid, Nauka, Moscow, 1982.
• ––––, Operator equations that describe steady flows of a nonlinear viscoelastic fluid, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint (1988), no. 46, 58.
• ––––, A model and a general problem on plastic flow under great deformations, bericht 99/07, sonderforschungsbereich 404, Universität Stuttgart, 1999.
• V. P. Orlov, Stability of a zero solution of mathematical model of multidimensional viscoelastic medium, Dokl. Akad. Nauk Ukrain. SSR Ser. A (1995), no. 12, 15–17.
• ––––, Stability of zero solution of mathematical model of multidimensional barotropic viscoelastic medium, Nonlinear Anal. 26 (1996), no. 12, 1937–1950.
• V. P. Orlov and P. E. Sobolevskiĭ, Investigation of mathematical models of multidimensional viscoelastic media, Dokl. Akad. Nauk Ukrain. SSR Ser. A (1989), no. 10, 31–35.
• ––––, On mathematical models of a viscoelasticity with a memory, Differential Integral Equations 4 (1991), no. 1, 103–115.
• P. E. Sobolevskiĭ, Coerciveness inequalities for abstract parabolic equations, Dokl. Akad. Nauk SSSR 157 (1964), 52–55.
• V. A. Solonnikov, Estimates in ${L}\sb{p}$ of solutions of elliptic and parabolic systems, Trudy Mat. Inst. Steklov. 102 (1967), 137–160.
• ––––, Solvability of the problem of evolution of an isolated amount of a viscous incompressible capillary fluid, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 140 (1984), 179–186.
• V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of an initial-boundary value problem for the equation of motion of a viscoelastic fluid, Dokl. Akad. Nauk 380 (2001), no. 3, 308–311. \CMP1+868+1221 868 122