Global well-posedness and uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model

Abstract

In this paper, we consider global strong solutions and the uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model in $\R^d$, where $d=2$ and $3$. The well-recognized problem of the global existence of smooth solutions for the 2D inviscid Oldroyd-B model without smallness assumptions is open due to the complex structure of $Q(\na u,τ)\neq 0$. Therefore, improving the smallness assumptions, especially in the lower regularity class, is the core question in the area of fluid models. On the other hand, long-time behaviors of solutions including temporal decay and uniform-in-time damping stability are also of deep significance. To tackle these challenges, we first establish local well-posedness in the sense of Hadamard with critical regularity. Then, by virtue of the sharp commutator estimate for Calderon--Zygmund operators, we establish global well-posedness for $d=2$ with damping in the low regularity class $(L^2\cap B^1_{\infty,1})\times (L^2\cap B^0_{\infty,1})$. Furthermore, in both the 2D and 3D cases, we prove global existence of solutions to the inviscid Oldroyd-B model uniformly in the damping parameter $a\in[0,1]$. In addition, we obtain optimal temporal decay rates and time integrability by improving the existing Fourier splitting method and developing a novel temporal decomposition strategy. These results lead to the major contribution of the present paper: the uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model. More specifically, we discover the correlation between the sharp vanishing damping rate and the temporal decay rate. On periodic domains, we also prove exponential relaxation toward spatially homogeneous equilibria: the nonzero Fourier modes decay at a rate uniform in $a\in[0,1]$, while the decay rate of the stress zero mode necessarily degenerates as $a\to0$. We support our findings by providing numerical evidence.