Rotation-Strain Decomposition for the Incompressible Viscoelasticity in Two Dimensions

TL;DR

This paper revisits the global well-posedness of 2D incompressible viscoelasticity models by using a new strain matrix identity, relaxing previous smallness conditions on the deformation tensor.

Contribution

It introduces a novel approach that avoids using the rotation angle equation, broadening the class of initial conditions for global solutions.

Findings

Established global classical solutions under weaker initial conditions.

Developed a new identity for the strain matrix.

Demonstrated the deformation tensor can be far from equilibrium in maximum norm.

Abstract

In \cite{Lei}, the author derived an exact rotation-strain model in two dimensions for the motion of incompressible viscoelastic materials via the polar decomposition of the deformation tensor. Based on the rotation-strain model, the author constructed a family of large global classical solutions for the 2D incompressible viscoelasticity. To get such a global well-posedness result, the equation for the rotation angle was essential to explore the underlying weak dissipative structure of the whole viscoelastic system even though the momentum equation for the velocity field and the transport equation for the strain tensor have already formed a closed subsystem. In this paper, we revisit such a result without making use of the equation of the rotation angle. The proof relies on a new identity satisfied by the strain matrix. The smallness assumptions are only imposed on the $H^2$ norm of initial velocity field and the initial strain matrix, which implies that the deformation tensor is allowed being away from the equilibrium of 2 in the maximum norm.