Journal of Applied Mathematics
Volume 2005 (2005), Issue 1, Pages 59-80
Solvability of initial boundary value problems for equations describing motions of linear viscoelastic fluids
Petersburg Department, V. A. Steklov Institute of Mathematics, 27 Fontanka, St., Petersburg 191011, Russia
Received 16 April 2003; Revised 28 July 2004
Copyright © 2005 N. A. Karazeeva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The nonlinear parabolic equations describing motion of incompressible media are investigated. The rheological equations of most general type are considered. The deviator of the stress tensor is expressed as a nonlinear continuous positive definite operator applied to the rate of strain tensor. The global-in-time estimate of solution of initial boundary value problem is obtained. This estimate is valid for systems of equations of any non-Newtonian fluid. Solvability of initial boundary value problems for such equations is proved under some additional hypothesis. The application of this theory makes it possible to prove the existence of global-in-time solutions of two-dimensional initial boundary value problems for generalized linear viscoelastic liquids, that is, for liquids with linear integral rheological equation, and for third-grade liquids.