Global solutions of the 3D inhomogeneous incompressible viscoelastic system without structure assumptions

In this paper, we prove the global existence of strong solutions for the inhomogeneous incompressible viscoelastic system without any additional structure assumptions on ${\mathbb{R}}^3$. Unlike the time weighted energy method presented by Ai and Wang (Nonlinear Anal. 254 (2025), 113747.), by replacing $H^{-1}$ conditions with certain $L^1$ conditions on initial data, we need to develop some new transformation techniques for the system (1.1) and make use of elegant spectral analysis method to capture an enhanced time-decay rate of the velocity field $u$, which is essential to establish the uniform bounds of the density and deformation tensor.