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

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~($a\to 0$), 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.

Oldroyd-B model; exponential decay; global solutions; instability. arXiv admin note: substantial text overlap with arXiv:2110.08475

References in corpus (3)

• The estimate of lifespan and local well-posedness for the non-resistive MHD equations in homogeneous Besov spaces
• Global well-posedness and inviscid limits of the generalized Oldroyd type models
• Global well-posedness for the non-viscous MHD equations with magnetic diffusion in critical Besov spaces