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

TL;DR

This paper proves the global existence of strong solutions for the 3D inhomogeneous incompressible viscoelastic system without structural assumptions, using spectral analysis and new transformation techniques to improve decay rate estimates.

Contribution

It introduces novel transformation methods and spectral analysis to establish global solutions without additional structural assumptions on the system.

Findings

Established global existence of strong solutions in 3D.

Developed new techniques for decay rate analysis.

Achieved uniform bounds for density and deformation tensor.

Abstract

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.