Abstract
In this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.
Highlights
1 Introduction The Oldroyd model for an incompressible viscoelastic fluid is governed by the following system of equations in R3:
U denotes the fluid velocity, F := (Fij)3×3 stands for the deformation tensor, p represents the fluids pressure, and μ > 0 is a viscosity constant
There have been several interesting works on the initial value problem of (1.1), for instance, the short time existence of a smooth solution and the global existence of a smooth solution that is initially small have been established in various settings [10,11,12]
Summary
For large (rough) initial data, the global existence of weak solutions to (1.1) has been achieved by [13, 14] in dimension two. It is not hard to establish the existence of a global in time weak solution of (1.2) by following the scheme of [16] on the incompressible Navier–Stokes equations. Motivated by [15, 21], we investigate the global existence of the classical solutions to the following Cauchy problem in R3: For the last term M5, by integration by parts, we can obtain
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