Global Existence with Small Initial Data for Three-Dimensional Incompressible Isotropic Viscoelastic Materials

TL;DR

This paper proves the global existence of solutions for a 3D incompressible isotropic viscoelastic material model with small initial data, using advanced PDE techniques independent of viscosity size.

Contribution

It establishes global existence results for the nonlinear PDE system modeling viscoelastic materials with small initial data, independent of viscosity magnitude.

Findings

Global existence for small initial data is proven.

The proof employs vector fields, energy decay, and Sobolev inequalities.

Results are applicable to 3D incompressible isotropic viscoelastic materials.

Abstract

Global existence for a system of nonlinear partial differential equations (PDE) modeling an isotropic incompressible viscoelastic material is proved. The structure of the PDE is derived through constitutive assumptions on the material. Restriction on the size of the initial displacement and velocity for the model is specified independent of the size of the viscosity of the material. The proof of global existence combines use of vector fields, local energy decay estimates, generalized Sobolev inequalities, and hyperbolic energy estimates.