On the initial‐boundary value problem of the incompressible viscoelastic fluid system

Abstract

In this paper, we shall establish the local well-posedness of the initial-boundary value problem of the viscoelastic fluid system of the Oldroyd model. We shall also prove that the local solutions can be extended globally and that the global solutions decay exponentially fast as time goes to infinity provided that the initial data are sufficiently close to the equilibrium state.

Global Solutions to the Oldroyd-B Model with a Class of Large Initial Data

“…Besides, we would like to mention that Constantin and Kliegl [9] proved the global regularity of solutions in two dimensional case for the Oldroyd-B fluids with diffusive stress. An approach based on Lagrangian particle dynamics can be found in [19,20,21,23,24,25,30,32].…”

The global existence of small solutions to the Oldroyd-B model

“…[4,11,13,15] dealt with the inertial Oldroyd-B model for the deformation tensor and established global-in-time existence for small initial data.…”

Existence Results for a Polymer Melt With an Evolving Natural Configuration

“…Since all the obtained are uniform with respect to the viscosity, the incompressible limit of the isotropic elastodynamics is also examined. Finally Fanghua Lin and Ping Zhang 16 have obtained global-in-time existence of solutions for an Oldroyd model by adapting the hyperbolic type approach, provided that the initial data are close enough to the equilibrium state. It should be mentioned that all these models involve nonlocal (in time) constitutive equations.…”