Stability and large-time behavior for the N-Dimensional Euler-FENE dumbbell model near an equilibrium

This paper studies the N-dimensional FENE dumbbell model without velocity dissipation, focusing on the stability and decay of perturbations near the steady solution $(0,\text{pin})$. Due to the lack of velocity dissipation, the above problems are highly challenging. In fact, without coupling, the corresponding N-dimensional Euler equation near u=0 is well known to be unstable. To overcome this difficulty, we analyze the wave structure arising in the system governing perturbations around the steady state, which originates from the equilibrium configuration and the coupling effects. This wave structure enables us to establish the global stability in the $H^s$-type Sobolev norms. Also, we highlight the critical role of wave structure in the decay estimates of the Euler-FENE dumbbell model. By combining this property with the Fourier splitting method, we derive the decay rate, which is identical to that of the general FENE dumbbell with velocity dissipation.