Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows

TL;DR

This paper studies the long-term behavior of incompressible viscoelastic flows, proving global existence, decay rates, and weak-strong uniqueness for solutions in the whole space, using a new decomposition method.

Contribution

It introduces a novel Helmholtz projection-based decomposition to prove global smooth solutions and establishes optimal decay rates and weak-strong uniqueness for the system.

Findings

Global smooth solutions exist near equilibrium.

Optimal decay rates for solutions and derivatives.

Weak-strong uniqueness in finite energy class.

Abstract

We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of global smooth solutions near equilibrium. Then under additional assumptions that the initial data belong to $L^1$ and their Fourier modes do not degenerate at low frequencies, we obtain the optimal $L^2$ decay rates for the global smooth solutions and their spatial derivatives. At last, we establish the weak-strong uniqueness property in the class of finite energy weak solutions for the incompressible viscoelastic system.