On Strong Solutions of Regularized Model of a Viscoelastic Medium with Variable Boundary

Abstract

We consider the initial-value problem for systems of equations describing the evolution of a viscoelastic medium with variable boundary with memory along the trajectories of a velocity field, which generalizes the Navier-Stokes system of equations. Nonlocal existence and uniqueness theorem of strong solutions containing senior square-integrable derivatives in the planar case are established.

On strong solutions of fractional nonlinear viscoelastic model of Voigt type

“…For a number of non‐Newtonian systems, similar results were obtained in Orlov and Sobolevskii 7 and Orlov 8 . Let us note that fractional analogues of Newton, Maxwell, Kelvin–Voigt, Jeffrey, Oldroyd, Zener, anti‐Zener, and Burgers models are widely known and used (see, e.g., Gorenflo & Mainardi, 1 Bagleys & Torvik, 2 Caputo & Mainardi, 3,4 Mainardi & Spada, 9 Scott‐Blair, 10 Zvyagin, 11 and references therein).…”

“…In this paper, a similar result is obtained for one of the fractional nonlinear problem described above. In a particular case, one variant of this result in a simpler situation was obtained in Orlov 8 …”

“…In the case of N =2, the existence and uniqueness of strong solutions ( , ) to regularized Problems () and () for μ 1 ( s )=const, μ 2 =0 was established in Orlov 8 . The examples of nonlinear non‐Newtonian fluids and nonlinearities μ 1 ( s ) considered in the paper can be found in Litvinov 14 ,.…”

“…For the operator equation, a family of approximation equations is defined. Existence of solutions of the approximation equation is established by iterative process on the basis of the results of Orlov and Sobolevskii 7 and Orlov 8 . The solution of the original operator equation is obtained as the limit of solutions of approximation equations.…”

On a problem in the dynamics of a thermoviscoelastic medium with memory

“…In [10], for a regularized viscoelasticity model (problem (1.1), (1.2) with θ = 0 and μ 1 (s) = const), non local (n = 2) and local (n = 3) theorems on the existence and uniqueness of strong solutions were proved by applying an approximative topological method. Another method was used in [11] to establish the same result for a more general model with nonlinear viscosity μ 1 .…”