# Global existence and decay rate of strong solution to incompressible Oldroyd type model equations

## TL;DR

This paper proves the global existence and decay rates over time for solutions to an incompressible Oldroyd model with damping, including decay in higher Sobolev norms, for small initial data.

## Contribution

It establishes the global existence of solutions and derives sharp decay rates in both $L^{2}$ and higher Sobolev norms for the Oldroyd model with damping.

## Key findings

- Global existence for small initial data
- Sharp decay rates in $L^{2}$ norm
- Decay rates for higher order Sobolev norms

## Abstract

This paper investigates the global existence and the decay rate in time of a solution to the Cauchy problem for an incompressible Oldroyd model with a deformation tensor damping term. There are three major results. The first is the global existence of the solution for small initial data. Second, we derive the sharp time decay of the solution in $L^{2}-$norm. Finally, the sharp time decay of the solution of higher order Sobolev norms is obtained.