Global well-posedness, stability and instability for the non-viscous Oldroyd-B model

TL;DR

This paper investigates the global behavior of solutions to the 3D incompressible Oldroyd-B model, revealing how the coupling coefficient influences stability, steady states, and energy dissipation, with implications for physical phenomena.

Contribution

It establishes global existence results for different coupling coefficients and analyzes the impact of the coupling coefficient on solution stability and energy dissipation.

Findings

Solutions are globally steady when $k^m o k > 0$.

Energy exhibits a jump as $k o 0$, indicating non-steady behavior.

Smaller $k$ enhances energy dissipation but cannot be zero, or dissipation vanishes.

Abstract

In this paper we consider the 3-dimensional incompressible Oldroyd-B model. First, we establish two results of the global existence for different kinds of the coupling coefficient $k$. Then, we prove that the solutions $(u,\tau)$ are globally steady when $k^m\rightarrow k>0$, though $(u,\tau)$ corresponds to different decays for different kinds of $k>0~$. Finally, we show that the energy of $u(t,x)$ will have a jump when $k\rightarrow 0$ in large time, which implies a non-steady phenomenon. In a word, we find an interesting physical phenomenon of \eqref{1} such that smaller coupling coefficient $k$ will have a better impact for the energy dissipation of $(u,\tau)$, but $k$ can't be too small to zero, or the dissipation will vanish instantly. While the damping term $\tau$ and $\mathbb{D}u$ always bring the well impact for the energy dissipation.