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 (α = 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 τ (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 → 0), the system is not steady for large time. This implies an interesting physical phenomenon that the term aDu is a bridge between the transformation of kinetic energy u and elastic potential energy τ , but this process is transient for large time, which leads the instability.