Global Regularity and instability for the incompressible non-viscous Oldroyd-B model

TL;DR

This paper proves global existence and explores instability in the 2D non-viscous Oldroyd-B model, revealing how stress decay and parameter ratios influence system stability and energy transformation.

Contribution

It demonstrates global existence for large initial data when the stress tensor decays exponentially and analyzes the instability near the critical ratio, providing new insights into energy dynamics.

Findings

Global existence with large initial data when stress decays exponentially

Instability occurs when the ratio approaches 1, indicating transient energy transformation

The term involving the ratio acts as a bridge between kinetic and elastic energies

Abstract

In this paper, we consider the 2-dimensional non-viscous Oldroyd-B model. In the case of the ratio equal 1~($\alpha=0$), it is a difficult case since the velocity field $u(t,x)$ is no longer decay. Fortunately, by {observing the exponential decay} of the stress tensor $\tau(t,x)$, we succeeded in proving the global existence for this system with some large initial data. Moreover, we give an unsteady result: when the ratio is close to 1~($a\rightarrow 0$), the system is not steady for large time. This implies an interesting physical phenomenon that the term $a\mathbb{D}u$ is a bridge between the transformation of kinetic energy $u$ and elastic potential energy $\tau$, but this process is transient for large time, which leads the instability.