Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism

TL;DR

This paper proves the global existence of small smooth solutions for the 3D incompressible Oldroyd-B model without damping, overcoming the challenge of partial dissipation through time-weighted energies and structural analysis.

Contribution

It introduces a novel energy method based on the coupled structure to establish global solutions without damping or div-curl assumptions.

Findings

Global small solutions are proven to exist for the 3D Oldroyd-B model without damping.

The method applies to viscoelastic systems with Hookean elasticity, removing previous physical assumptions.

Partial dissipation of stress and full dissipation of velocity are demonstrated.

Abstract

In this paper, we prove the global existence of small smooth solutions to the three-dimensional incompressible Oldroyd-B model without damping on the stress tensor. The main difficulty is the lack of full dissipation in stress tensor. To overcome it, we construct some time-weighted energies based on the special coupled structure of system. Such type energies show the partial dissipation of stress tensor and the strongly full dissipation of velocity. In the view of treating "nonlinear term" as a "linear term", we also apply this result to 3D incompressible viscoelastic system with Hookean elasticity and then prove the global existence of small solutions without the physical assumption (div-curl structure) as previous works.