Asymptotic Behavior for an Oldroyd-B Model in Two Dimensions

Abstract

The paper is devoted to asymptotic behavior for a compressible Oldroyd-B model in $\mathbb{T}^{2}$ . We prove that the weak solution will converge to the strong solution as the rough initial data of the former tend to the smooth initial data of the latter. The proof relies on a relative entropy method. This work can be viewed as a generalization of the weak-strong uniqueness where the initial data for the weak solution and the strong solution are smooth and the same. The main challenges focus on a $L^{2}$ -type estimate for the extra stress tensor difference $\tau^{\theta}-\tau^{*}$ due to the absence of $L^{2}$ -norm of $\tau^{\theta}$ in the entropy estimate and that the initial data of the weak solution are rough here. To handle these, some commutator estimates are adopted.