Global solutions of the 2D inhomogeneous incompressible viscoelastic system

In this paper, we investigate the global existence of strong solutions for the inhomogeneous incompressible viscoelastic system with only velocity dissipation on $\mathbb{R}^{2}$. Due to the criticality of the time-weight, the methods for the corresponding problem on $\mathbb{R}^{3}$ cannot be directly applied to the two-dimensional case. To overcome the main difficulties, we first transform the original system into a suitable dissipative system by introducing an effective tensor. Then we develop a new fractional time-weighted energy framework, combined with elegant commutator and bilinear estimates, to prove the global existence of strong solutions without the help of the common ``div-curl" structure on the viscoelastic system.